3.4.20 \(\int \frac {1}{x^4 (a+b x^3)^2 (c+d x^3)^{3/2}} \, dx\)
Optimal. Leaf size=241 \[ -\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 (b c-a d)^{5/2}}+\frac {(3 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{5/2}}-\frac {d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{3 a^2 c^2 \sqrt {c+d x^3} (b c-a d)^2}-\frac {b (2 b c-a d)}{3 a^2 c \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 241, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used =
{446, 103, 151, 152, 156, 63, 208} \begin {gather*} -\frac {d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{3 a^2 c^2 \sqrt {c+d x^3} (b c-a d)^2}-\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 (b c-a d)^{5/2}}+\frac {(3 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{5/2}}-\frac {b (2 b c-a d)}{3 a^2 c \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^4*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
-(d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*Sqrt[c + d*x^3]) - (b*(2*b*c - a*d))/(3*a^2*
c*(b*c - a*d)*(a + b*x^3)*Sqrt[c + d*x^3]) - 1/(3*a*c*x^3*(a + b*x^3)*Sqrt[c + d*x^3]) + ((4*b*c + 3*a*d)*ArcT
anh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^3*c^(5/2)) - (b^(5/2)*(4*b*c - 7*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqr
t[b*c - a*d]])/(3*a^3*(b*c - a*d)^(5/2))
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]
Rule 152
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
Rule 156
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 446
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (4 b c+3 a d)+\frac {5 b d x}{2}}{x (a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )}{3 a c}\\ &=-\frac {b (2 b c-a d)}{3 a^2 c (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c-a d) (4 b c+3 a d)+\frac {3}{2} b d (2 b c-a d) x}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{3 a^2 c (b c-a d)}\\ &=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^3}}-\frac {b (2 b c-a d)}{3 a^2 c (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {1}{4} (b c-a d)^2 (4 b c+3 a d)-\frac {1}{4} b d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^3}}-\frac {b (2 b c-a d)}{3 a^2 c (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {\left (b^3 (4 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3 (b c-a d)^2}-\frac {(4 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^3\right )}{6 a^3 c^2}\\ &=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^3}}-\frac {b (2 b c-a d)}{3 a^2 c (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {\left (b^3 (4 b c-7 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^3 d (b c-a d)^2}-\frac {(4 b c+3 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 a^3 c^2 d}\\ &=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^3}}-\frac {b (2 b c-a d)}{3 a^2 c (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {1}{3 a c x^3 \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {(4 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{5/2}}-\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 a^3 (b c-a d)^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.16, size = 189, normalized size = 0.78 \begin {gather*} \frac {b^2 c^2 x^3 \left (a+b x^3\right ) (4 b c-7 a d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \left (d x^3+c\right )}{b c-a d}\right )-(a d-b c) \left (x^3 \left (a+b x^3\right ) \left (3 a^2 d^2+a b c d-4 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^3}{c}+1\right )+a c \left (a^2 d+a b \left (d x^3-c\right )-2 b^2 c x^3\right )\right )}{3 a^3 c^2 x^3 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^4*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
(b^2*c^2*(4*b*c - 7*a*d)*x^3*(a + b*x^3)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^3))/(b*c - a*d)] - (-(b*c
) + a*d)*(a*c*(a^2*d - 2*b^2*c*x^3 + a*b*(-c + d*x^3)) + (-4*b^2*c^2 + a*b*c*d + 3*a^2*d^2)*x^3*(a + b*x^3)*Hy
pergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^3)/c]))/(3*a^3*c^2*(b*c - a*d)^2*x^3*(a + b*x^3)*Sqrt[c + d*x^3])
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.95, size = 272, normalized size = 1.13 \begin {gather*} \frac {\left (7 a b^{5/2} d-4 b^{7/2} c\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3} \sqrt {a d-b c}}{b c-a d}\right )}{3 a^3 (a d-b c)^{5/2}}+\frac {(3 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^3}}{\sqrt {c}}\right )}{3 a^3 c^{5/2}}+\frac {-a^3 c d^2-3 a^3 d^3 x^3+2 a^2 b c^2 d+a^2 b c d^2 x^3-3 a^2 b d^3 x^6-a b^2 c^3+a b^2 c^2 d x^3+2 a b^2 c d^2 x^6-2 b^3 c^3 x^3-2 b^3 c^2 d x^6}{3 a^2 c^2 x^3 \left (a+b x^3\right ) \sqrt {c+d x^3} (a d-b c)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x^4*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x]
[Out]
(-(a*b^2*c^3) + 2*a^2*b*c^2*d - a^3*c*d^2 - 2*b^3*c^3*x^3 + a*b^2*c^2*d*x^3 + a^2*b*c*d^2*x^3 - 3*a^3*d^3*x^3
- 2*b^3*c^2*d*x^6 + 2*a*b^2*c*d^2*x^6 - 3*a^2*b*d^3*x^6)/(3*a^2*c^2*(-(b*c) + a*d)^2*x^3*(a + b*x^3)*Sqrt[c +
d*x^3]) + ((-4*b^(7/2)*c + 7*a*b^(5/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^3])/(b*c - a*d)])/(3
*a^3*(-(b*c) + a*d)^(5/2)) + ((4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^3]/Sqrt[c]])/(3*a^3*c^(5/2))
________________________________________________________________________________________
fricas [B] time = 2.00, size = 2384, normalized size = 9.89
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/(b*x^3+a)^2/(d*x^3+c)^(3/2),x, algorithm="fricas")
[Out]
[-1/6*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^9 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^6 + (4*a*b^3*c
^5 - 7*a^2*b^2*c^4*d)*x^3)*sqrt(b/(b*c - a*d))*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*(b*c - a*d)*sqrt
(b/(b*c - a*d)))/(b*x^3 + a)) - ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^9 + (4*b^4*
c^4 - a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^6 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*
b*c^2*d^2 + 3*a^4*c*d^3)*x^3)*sqrt(c)*log((d*x^3 + 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*(a^2*b^2*c^4 - 2*
a^3*b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^6 + (2*a*b^3*c^4 - a^2*b^2*c
^3*d - a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^3)*sqrt(d*x^3 + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)
*x^9 + (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^6 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^
4*d^2)*x^3), -1/6*(2*((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^9 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^
6 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*d)*x^3)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^3 + c)*(b*c - a*d)*sqrt(-b/(b*c
- a*d))/(b*d*x^3 + b*c)) - ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^9 + (4*b^4*c^4
- a*b^3*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^6 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^
2*d^2 + 3*a^4*c*d^3)*x^3)*sqrt(c)*log((d*x^3 + 2*sqrt(d*x^3 + c)*sqrt(c) + 2*c)/x^3) + 2*(a^2*b^2*c^4 - 2*a^3*
b*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^6 + (2*a*b^3*c^4 - a^2*b^2*c^3*d
- a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^3)*sqrt(d*x^3 + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^9
+ (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^6 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^
2)*x^3), -1/6*(2*((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^9 + (4*b^4*c^4 - a*b^3*c^3
*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^6 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 + 3*a
^4*c*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c) + ((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^9 + (4*b^4*c^5
- 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^6 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*d)*x^3)*sqrt(b/(b*c - a*d))*log((b*d*
x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*(b*c - a*d)*sqrt(b/(b*c - a*d)))/(b*x^3 + a)) + 2*(a^2*b^2*c^4 - 2*a^3*b
*c^3*d + a^4*c^2*d^2 + (2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^6 + (2*a*b^3*c^4 - a^2*b^2*c^3*d
- a^3*b*c^2*d^2 + 3*a^4*c*d^3)*x^3)*sqrt(d*x^3 + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^9
+ (a^3*b^3*c^6 - a^4*b^2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^6 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2
)*x^3), -1/3*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^9 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^6 + (4*
a*b^3*c^5 - 7*a^2*b^2*c^4*d)*x^3)*sqrt(-b/(b*c - a*d))*arctan(-sqrt(d*x^3 + c)*(b*c - a*d)*sqrt(-b/(b*c - a*d)
)/(b*d*x^3 + b*c)) + ((4*b^4*c^3*d - 5*a*b^3*c^2*d^2 - 2*a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x^9 + (4*b^4*c^4 - a*b^3
*c^3*d - 7*a^2*b^2*c^2*d^2 + a^3*b*c*d^3 + 3*a^4*d^4)*x^6 + (4*a*b^3*c^4 - 5*a^2*b^2*c^3*d - 2*a^3*b*c^2*d^2 +
3*a^4*c*d^3)*x^3)*sqrt(-c)*arctan(sqrt(d*x^3 + c)*sqrt(-c)/c) + (a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2 +
(2*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b*c*d^3)*x^6 + (2*a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + 3*a^4
*c*d^3)*x^3)*sqrt(d*x^3 + c))/((a^3*b^3*c^5*d - 2*a^4*b^2*c^4*d^2 + a^5*b*c^3*d^3)*x^9 + (a^3*b^3*c^6 - a^4*b^
2*c^5*d - a^5*b*c^4*d^2 + a^6*c^3*d^3)*x^6 + (a^4*b^2*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^3)]
________________________________________________________________________________________
giac [A] time = 0.22, size = 367, normalized size = 1.52 \begin {gather*} \frac {{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{3} + c\right )}^{2} b^{3} c^{2} d - 2 \, {\left (d x^{3} + c\right )} b^{3} c^{3} d - 2 \, {\left (d x^{3} + c\right )}^{2} a b^{2} c d^{2} + 3 \, {\left (d x^{3} + c\right )} a b^{2} c^{2} d^{2} + 3 \, {\left (d x^{3} + c\right )}^{2} a^{2} b d^{3} - 7 \, {\left (d x^{3} + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \, {\left (d x^{3} + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{3 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left ({\left (d x^{3} + c\right )}^{\frac {5}{2}} b - 2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} b c + \sqrt {d x^{3} + c} b c^{2} + {\left (d x^{3} + c\right )}^{\frac {3}{2}} a d - \sqrt {d x^{3} + c} a c d\right )}} - \frac {{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c}}{\sqrt {-c}}\right )}{3 \, a^{3} \sqrt {-c} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/(b*x^3+a)^2/(d*x^3+c)^(3/2),x, algorithm="giac")
[Out]
1/3*(4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^2*c^2 - 2*a^4*b*c*d + a^5*d^2
)*sqrt(-b^2*c + a*b*d)) - 1/3*(2*(d*x^3 + c)^2*b^3*c^2*d - 2*(d*x^3 + c)*b^3*c^3*d - 2*(d*x^3 + c)^2*a*b^2*c*d
^2 + 3*(d*x^3 + c)*a*b^2*c^2*d^2 + 3*(d*x^3 + c)^2*a^2*b*d^3 - 7*(d*x^3 + c)*a^2*b*c*d^3 + 2*a^2*b*c^2*d^3 + 3
*(d*x^3 + c)*a^3*d^4 - 2*a^3*c*d^4)/((a^2*b^2*c^4 - 2*a^3*b*c^3*d + a^4*c^2*d^2)*((d*x^3 + c)^(5/2)*b - 2*(d*x
^3 + c)^(3/2)*b*c + sqrt(d*x^3 + c)*b*c^2 + (d*x^3 + c)^(3/2)*a*d - sqrt(d*x^3 + c)*a*c*d)) - 1/3*(4*b*c + 3*a
*d)*arctan(sqrt(d*x^3 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^2)
________________________________________________________________________________________
maple [C] time = 0.25, size = 1067, normalized size = 4.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^4/(b*x^3+a)^2/(d*x^3+c)^(3/2),x)
[Out]
1/a^2*(-1/3*(d*x^3+c)^(1/2)/c^2/x^3-2/3/((x^3+c/d)*d)^(1/2)/c^2*d+d*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(5/2))+
2/a^3*b^2*(-2/3/(a*d-b*c)/((x^3+c/d)*d)^(1/2)-1/3*I*b/d^2*2^(1/2)*sum(1/(-a*d+b*c)/(a*d-b*c)*(-c*d^2)^(1/3)*(1
/2*I*(2*x+(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3))/d)/(-c*d^2)^(1/3)*d)^(1/2)*((x-(-c*d^2)^(1/3)/d)/(-3*(-c*
d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3))*d)^(1/2)*(-1/2*I*(2*x+(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3))/d)/(-c*d^
2)^(1/3)*d)^(1/2)/(d*x^3+c)^(1/2)*(2*_alpha^2*d^2+I*(-c*d^2)^(1/3)*3^(1/2)*_alpha*d-(-c*d^2)^(1/3)*_alpha*d-I*
3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2*(-c*d^2)^(1/3)/d-1/2*I*3^(1/2)*(-c*d^2
)^(1/3)/d)*3^(1/2)/(-c*d^2)^(1/3)*d)^(1/2),1/2*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d+I*3^(1/2)*c*d-3*c*d-I*(-
c*d^2)^(2/3)*3^(1/2)*_alpha-3*(-c*d^2)^(2/3)*_alpha)/(a*d-b*c)*b/d,(I*3^(1/2)*(-c*d^2)^(1/3)/(-3/2*(-c*d^2)^(1
/3)/d+1/2*I*3^(1/2)*(-c*d^2)^(1/3)/d)/d)^(1/2)),_alpha=RootOf(_Z^3*b+a)))+1/a^2*b^2*(-1/3/(a*d-b*c)^2*(d*x^3+c
)^(1/2)/(b*x^3+a)*b-2/3/(a*d-b*c)^2/((x^3+c/d)*d)^(1/2)*d+1/2*I*b/d*2^(1/2)*sum(1/(a*d-b*c)^3*(-c*d^2)^(1/3)*(
1/2*I*(2*x+(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3))/d)/(-c*d^2)^(1/3)*d)^(1/2)*((x-(-c*d^2)^(1/3)/d)/(-3*(-c
*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3))*d)^(1/2)*(-1/2*I*(2*x+(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3))/d)/(-c*d
^2)^(1/3)*d)^(1/2)/(d*x^3+c)^(1/2)*(2*_alpha^2*d^2+I*(-c*d^2)^(1/3)*3^(1/2)*_alpha*d-(-c*d^2)^(1/3)*_alpha*d-I
*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2*(-c*d^2)^(1/3)/d-1/2*I*3^(1/2)*(-c*d^
2)^(1/3)/d)*3^(1/2)/(-c*d^2)^(1/3)*d)^(1/2),1/2*(2*I*(-c*d^2)^(1/3)*3^(1/2)*_alpha^2*d+I*3^(1/2)*c*d-3*c*d-I*(
-c*d^2)^(2/3)*3^(1/2)*_alpha-3*(-c*d^2)^(2/3)*_alpha)/(a*d-b*c)*b/d,(I*3^(1/2)*(-c*d^2)^(1/3)/(-3/2*(-c*d^2)^(
1/3)/d+1/2*I*3^(1/2)*(-c*d^2)^(1/3)/d)/d)^(1/2)),_alpha=RootOf(_Z^3*b+a)))-2*b/a^3*(2/3/((x^3+c/d)*d)^(1/2)/c-
2/3*arctanh((d*x^3+c)^(1/2)/c^(1/2))/c^(3/2))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{3} + a\right )}^{2} {\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^4/(b*x^3+a)^2/(d*x^3+c)^(3/2),x, algorithm="maxima")
[Out]
integrate(1/((b*x^3 + a)^2*(d*x^3 + c)^(3/2)*x^4), x)
________________________________________________________________________________________
mupad [B] time = 19.63, size = 18847, normalized size = 78.20
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^4*(a + b*x^3)^2*(c + d*x^3)^(3/2)),x)
[Out]
(2*b*log(1/x^6))/(3*a^3*c^(3/2)) - (c + d*x^3)^(1/2)/(3*a^2*c^2*x^3) + (d*log(1/x^6))/(2*a^2*c^(5/2)) + (2*b*l
og(c^(3/2)*(c + d*x^3)^(1/2) - c^(1/2)*(c + d*x^3)^(3/2) + d^2*x^6 + 2*c*d*x^3 + 3*c^(1/2)*d*x^3*(c + d*x^3)^(
1/2)))/(3*a^3*c^(3/2)) + (d*log(c^(3/2)*(c + d*x^3)^(1/2) - c^(1/2)*(c + d*x^3)^(3/2) + d^2*x^6 + 2*c*d*x^3 +
3*c^(1/2)*d*x^3*(c + d*x^3)^(1/2)))/(2*a^2*c^(5/2)) - (b^7*c^9*x^4*(c + d*x^3)^(1/2))/(2*(2*a^9*c^6*d^5*x + 2*
a^9*c^5*d^6*x^4 + a^5*b^4*c^9*d^2*x^4 + a^6*b^3*c^8*d^3*x^4 - 3*a^7*b^2*c^7*d^4*x^4 + a^5*b^4*c^8*d^3*x^7 - 3*
a^7*b^2*c^6*d^5*x^7 - 3*a^8*b*c^7*d^4*x + a^6*b^3*c^9*d^2*x - a^8*b*c^6*d^5*x^4 + 2*a^8*b*c^5*d^6*x^7)) - (5*a
^9*d^7*x^4*(c + d*x^3)^(1/2))/(4*(a^6*b^5*c^9*x + a^5*b^6*c^9*x^4 - 3*a^7*b^4*c^7*d^2*x^4 - a^8*b^3*c^6*d^3*x^
4 + 2*a^9*b^2*c^5*d^4*x^4 - 3*a^7*b^4*c^6*d^3*x^7 + 2*a^8*b^3*c^5*d^4*x^7 - 3*a^8*b^3*c^7*d^2*x + 2*a^9*b^2*c^
6*d^3*x + a^6*b^5*c^8*d*x^4 + a^5*b^6*c^8*d*x^7)) + (3*a^2*d^2*x*(c + d*x^3)^(1/2))/(a^2*b^2*c^4*x^4 + 2*a^4*c
^2*d^2*x^4 + a^3*b*c^4*x + 2*a^4*c^3*d*x + 3*a^3*b*c^3*d*x^4 + a^2*b^2*c^3*d*x^7 + 2*a^3*b*c^2*d^2*x^7) + (2*b
^2*c^2*x*(c + d*x^3)^(1/2))/(a^2*b^2*c^4*x^4 + 2*a^4*c^2*d^2*x^4 + a^3*b*c^4*x + 2*a^4*c^3*d*x + 3*a^3*b*c^3*d
*x^4 + a^2*b^2*c^3*d*x^7 + 2*a^3*b*c^2*d^2*x^7) - (b^(7/2)*c*log((a^6*b^(15/2)*c^10*36i)/(a*(a*d - b*c)^(1/2)
+ b*x^3*(a*d - b*c)^(1/2)) - (a^7*b^(13/2)*c^9*d*198i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^12
*b^(3/2)*c^4*d^6*18i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (a^11*b^(5/2)*c^5*d^5*126i)/(a*(a*d -
b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^10*b^(7/2)*c^6*d^4*360i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(
1/2)) - (a^9*b^(9/2)*c^7*d^3*540i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^8*b^(11/2)*c^8*d^2*450
i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^6*b^(15/2)*c^9*d*x^3*18i)/(a*(a*d - b*c)^(1/2) + b*x^3
*(a*d - b*c)^(1/2)) - (a^11*b^(5/2)*c^4*d^6*x^3*18i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^10*b
^(7/2)*c^5*d^5*x^3*90i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (a^9*b^(9/2)*c^6*d^4*x^3*180i)/(a*(a
*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^8*b^(11/2)*c^7*d^3*x^3*180i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d
- b*c)^(1/2)) - (a^7*b^(13/2)*c^8*d^2*x^3*90i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (36*a^6*b^7*
c^9*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (360*a^8*b^5*c^7*d^
2*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (360*a^9*b^4*c^6*d^3*
(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (180*a^10*b^3*c^5*d^4*(
c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (36*a^11*b^2*c^4*d^5*(c
+ d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (180*a^7*b^6*c^8*d*(c + d*
x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)))*2i)/(3*a^3*(a*d - b*c)^(5/2)) +
(b^(5/2)*d*log((a^6*b^(15/2)*c^10*36i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (a^7*b^(13/2)*c^9*d*
198i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^12*b^(3/2)*c^4*d^6*18i)/(a*(a*d - b*c)^(1/2) + b*x^
3*(a*d - b*c)^(1/2)) - (a^11*b^(5/2)*c^5*d^5*126i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^10*b^(
7/2)*c^6*d^4*360i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (a^9*b^(9/2)*c^7*d^3*540i)/(a*(a*d - b*c)
^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^8*b^(11/2)*c^8*d^2*450i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)
) + (a^6*b^(15/2)*c^9*d*x^3*18i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (a^11*b^(5/2)*c^4*d^6*x^3*1
8i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^10*b^(7/2)*c^5*d^5*x^3*90i)/(a*(a*d - b*c)^(1/2) + b*
x^3*(a*d - b*c)^(1/2)) - (a^9*b^(9/2)*c^6*d^4*x^3*180i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (a^8
*b^(11/2)*c^7*d^3*x^3*180i)/(a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (a^7*b^(13/2)*c^8*d^2*x^3*90i)/(
a*(a*d - b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (36*a^6*b^7*c^9*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d
- b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (360*a^8*b^5*c^7*d^2*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d -
b*c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (360*a^9*b^4*c^6*d^3*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*
c)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) + (180*a^10*b^3*c^5*d^4*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c
)^(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (36*a^11*b^2*c^4*d^5*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^
(1/2) + b*x^3*(a*d - b*c)^(1/2)) - (180*a^7*b^6*c^8*d*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2))/(a*(a*d - b*c)^(1/2
) + b*x^3*(a*d - b*c)^(1/2)))*7i)/(6*a^2*(a*d - b*c)^(5/2)) + (5*a^4*d^4*x^4*(c + d*x^3)^(1/2))/(2*(a^4*b^2*c^
6*x + a^3*b^3*c^6*x^4 + 2*a^5*b*c^5*d*x + 2*a^4*b^2*c^4*d^2*x^7 + 3*a^4*b^2*c^5*d*x^4 + 2*a^5*b*c^4*d^2*x^4 +
a^3*b^3*c^5*d*x^7)) - (65*a^3*d^3*x^4*(c + d*x^3)^(1/2))/(24*(a^3*b^2*c^5*x^4 + 2*a^5*c^3*d^2*x^4 + a^4*b*c^5*
x + 2*a^5*c^4*d*x + 3*a^4*b*c^4*d*x^4 + a^3*b^2*c^4*d*x^7 + 2*a^4*b*c^3*d^2*x^7)) - (8*b^3*c^3*x^4*(c + d*x^3)
^(1/2))/(3*(a^3*b^2*c^5*x^4 + 2*a^5*c^3*d^2*x^4 + a^4*b*c^5*x + 2*a^5*c^4*d*x + 3*a^4*b*c^4*d*x^4 + a^3*b^2*c^
4*d*x^7 + 2*a^4*b*c^3*d^2*x^7)) + (14*b^3*c^4*x^4*(c + d*x^3)^(1/2))/(a^3*b^2*c^6*x^4 + 2*a^5*c^4*d^2*x^4 + a^
4*b*c^6*x + 2*a^5*c^5*d*x + 3*a^4*b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 + 2*a^4*b*c^4*d^2*x^7) - (5*a^7*c^2*d^5*x*(c
+ d*x^3)^(1/2))/(2*(a^5*b^4*c^9*x + a^4*b^5*c^9*x^4 - 3*a^6*b^3*c^7*d^2*x^4 - a^7*b^2*c^6*d^3*x^4 - 3*a^6*b^3
*c^6*d^3*x^7 + 2*a^7*b^2*c^5*d^4*x^7 + 2*a^8*b*c^6*d^3*x - 3*a^7*b^2*c^7*d^2*x + a^5*b^4*c^8*d*x^4 + 2*a^8*b*c
^5*d^4*x^4 + a^4*b^5*c^8*d*x^7)) - (5*a^7*c*d^6*x^4*(c + d*x^3)^(1/2))/(2*(a^5*b^4*c^9*x + a^4*b^5*c^9*x^4 - 3
*a^6*b^3*c^7*d^2*x^4 - a^7*b^2*c^6*d^3*x^4 - 3*a^6*b^3*c^6*d^3*x^7 + 2*a^7*b^2*c^5*d^4*x^7 + 2*a^8*b*c^6*d^3*x
- 3*a^7*b^2*c^7*d^2*x + a^5*b^4*c^8*d*x^4 + 2*a^8*b*c^5*d^4*x^4 + a^4*b^5*c^8*d*x^7)) - (3*a^8*c^2*d^5*x*(c +
d*x^3)^(1/2))/(8*(a^6*b^4*c^9*x + a^5*b^5*c^9*x^4 - 3*a^7*b^3*c^7*d^2*x^4 - a^8*b^2*c^6*d^3*x^4 - 3*a^7*b^3*c
^6*d^3*x^7 + 2*a^8*b^2*c^5*d^4*x^7 + 2*a^9*b*c^6*d^3*x - 3*a^8*b^2*c^7*d^2*x + a^6*b^4*c^8*d*x^4 + 2*a^9*b*c^5
*d^4*x^4 + a^5*b^5*c^8*d*x^7)) - (3*a^8*c*d^6*x^4*(c + d*x^3)^(1/2))/(8*(a^6*b^4*c^9*x + a^5*b^5*c^9*x^4 - 3*a
^7*b^3*c^7*d^2*x^4 - a^8*b^2*c^6*d^3*x^4 - 3*a^7*b^3*c^6*d^3*x^7 + 2*a^8*b^2*c^5*d^4*x^7 + 2*a^9*b*c^6*d^3*x -
3*a^8*b^2*c^7*d^2*x + a^6*b^4*c^8*d*x^4 + 2*a^9*b*c^5*d^4*x^4 + a^5*b^5*c^8*d*x^7)) + (23*a^9*c^3*d^5*x*(c +
d*x^3)^(1/2))/(8*(a^7*b^4*c^10*x + a^6*b^5*c^10*x^4 - 3*a^8*b^3*c^8*d^2*x^4 - a^9*b^2*c^7*d^3*x^4 - 3*a^8*b^3*
c^7*d^3*x^7 + 2*a^9*b^2*c^6*d^4*x^7 + 2*a^10*b*c^7*d^3*x - 3*a^9*b^2*c^8*d^2*x + a^7*b^4*c^9*d*x^4 + 2*a^10*b*
c^6*d^4*x^4 + a^6*b^5*c^9*d*x^7)) - (a*b^6*c^9*x*(c + d*x^3)^(1/2))/(2*(2*a^9*c^6*d^5*x + 2*a^9*c^5*d^6*x^4 +
a^5*b^4*c^9*d^2*x^4 + a^6*b^3*c^8*d^3*x^4 - 3*a^7*b^2*c^7*d^4*x^4 + a^5*b^4*c^8*d^3*x^7 - 3*a^7*b^2*c^6*d^5*x^
7 - 3*a^8*b*c^7*d^4*x + a^6*b^3*c^9*d^2*x - a^8*b*c^6*d^5*x^4 + 2*a^8*b*c^5*d^6*x^7)) + (3*a^2*b^6*c^9*x^4*(c
+ d*x^3)^(1/2))/(4*(2*a^10*c^7*d^4*x + 2*a^10*c^6*d^5*x^4 + a^7*b^3*c^9*d^2*x^4 - 3*a^8*b^2*c^8*d^3*x^4 + a^6*
b^4*c^9*d^2*x^7 - 3*a^8*b^2*c^7*d^4*x^7 + a^7*b^3*c^10*d*x - 3*a^9*b*c^8*d^3*x + a^6*b^4*c^10*d*x^4 - a^9*b*c^
7*d^4*x^4 + 2*a^9*b*c^6*d^5*x^7)) - (5*a^6*c^2*d^3*x*(c + d*x^3)^(1/2))/(2*(a^6*b^2*c^7*x + a^5*b^3*c^7*x^4 +
2*a^7*b*c^6*d*x + 2*a^6*b^2*c^5*d^2*x^7 + 3*a^6*b^2*c^6*d*x^4 + 2*a^7*b*c^5*d^2*x^4 + a^5*b^3*c^6*d*x^7)) - (5
*a^6*c*d^4*x^4*(c + d*x^3)^(1/2))/(2*(a^6*b^2*c^7*x + a^5*b^3*c^7*x^4 + 2*a^7*b*c^6*d*x + 2*a^6*b^2*c^5*d^2*x^
7 + 3*a^6*b^2*c^6*d*x^4 + 2*a^7*b*c^5*d^2*x^4 + a^5*b^3*c^6*d*x^7)) + (4*a^2*b^4*c^6*x*(c + d*x^3)^(1/2))/(a^5
*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3
*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 + a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7) +
(8*a*b^5*c^6*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4
- 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 +
a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) - (14*a^2*b^4*c^7*x*(c + d*x^3)^(1/2))/(a^5*b^3*c^9*x + 2*a^8*c^6*d^
3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x
+ a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*c^5*d^4*x^7) - (14*a*b^5*c^7*x^4*(c + d*
x^3)^(1/2))/(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3
*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b
*c^5*d^4*x^7) + (a^3*b^4*c^7*x*(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^
9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*
c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) + (269*a^4*b^4*c^8*x*(c + d*x^3)^(1/2))/(24*(a^7*b^3*c
^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x
^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) + (
65*a^6*c^2*d^4*x*(c + d*x^3)^(1/2))/(8*(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4
- 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 +
a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) + (65*a^6*c*d^5*x^4*(c + d*x^3)^(1/2))/(24*(a^5*b^3*c^8*x + 2*a^8*c^
5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^
2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 + a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) - (41*a^6*c^3*d^4*x*(c
+ d*x^3)^(1/2))/(3*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2
*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 +
2*a^7*b*c^5*d^4*x^7)) - (5*a^7*c^3*d^4*x*(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9
*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x
^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) + (47*a^8*c^4*d^4*x*(c + d*x^3)^(1/2))/(6*(
a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*
c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*
x^7)) - (34*a^3*b^2*c^5*x*(c + d*x^3)^(1/2))/(3*(a^5*b^2*c^7*x^4 + 2*a^7*c^5*d^2*x^4 + a^6*b*c^7*x + 2*a^7*c^6
*d*x + 3*a^6*b*c^6*d*x^4 + a^5*b^2*c^6*d*x^7 + 2*a^6*b*c^5*d^2*x^7)) + (47*a^3*c^2*d^2*x*(c + d*x^3)^(1/2))/(3
*(a^3*b^2*c^6*x^4 + 2*a^5*c^4*d^2*x^4 + a^4*b*c^6*x + 2*a^5*c^5*d*x + 3*a^4*b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 +
2*a^4*b*c^4*d^2*x^7)) + (38*a^3*c*d^3*x^4*(c + d*x^3)^(1/2))/(3*(a^3*b^2*c^6*x^4 + 2*a^5*c^4*d^2*x^4 + a^4*b*c
^6*x + 2*a^5*c^5*d*x + 3*a^4*b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 + 2*a^4*b*c^4*d^2*x^7)) - (257*a^5*c^3*d^2*x*(c +
d*x^3)^(1/2))/(24*(a^5*b^2*c^7*x^4 + 2*a^7*c^5*d^2*x^4 + a^6*b*c^7*x + 2*a^7*c^6*d*x + 3*a^6*b*c^6*d*x^4 + a^
5*b^2*c^6*d*x^7 + 2*a^6*b*c^5*d^2*x^7)) - (5*a^9*c*d^6*x*(c + d*x^3)^(1/2))/(4*(a^6*b^5*c^9*x + a^5*b^6*c^9*x^
4 - 3*a^7*b^4*c^7*d^2*x^4 - a^8*b^3*c^6*d^3*x^4 + 2*a^9*b^2*c^5*d^4*x^4 - 3*a^7*b^4*c^6*d^3*x^7 + 2*a^8*b^3*c^
5*d^4*x^7 - 3*a^8*b^3*c^7*d^2*x + 2*a^9*b^2*c^6*d^3*x + a^6*b^5*c^8*d*x^4 + a^5*b^6*c^8*d*x^7)) + (23*a^9*c^2*
d^6*x^4*(c + d*x^3)^(1/2))/(8*(a^7*b^4*c^10*x + a^6*b^5*c^10*x^4 - 3*a^8*b^3*c^8*d^2*x^4 - a^9*b^2*c^7*d^3*x^4
- 3*a^8*b^3*c^7*d^3*x^7 + 2*a^9*b^2*c^6*d^4*x^7 + 2*a^10*b*c^7*d^3*x - 3*a^9*b^2*c^8*d^2*x + a^7*b^4*c^9*d*x^
4 + 2*a^10*b*c^6*d^4*x^4 + a^6*b^5*c^9*d*x^7)) + (a^2*b^6*c^10*x*(c + d*x^3)^(1/2))/(2*(2*a^10*c^7*d^5*x + 2*a
^10*c^6*d^6*x^4 + a^6*b^4*c^10*d^2*x^4 + a^7*b^3*c^9*d^3*x^4 - 3*a^8*b^2*c^8*d^4*x^4 + a^6*b^4*c^9*d^3*x^7 - 3
*a^8*b^2*c^7*d^5*x^7 - 3*a^9*b*c^8*d^4*x + a^7*b^3*c^10*d^2*x - a^9*b*c^7*d^5*x^4 + 2*a^9*b*c^6*d^6*x^7)) + (a
*b^7*c^10*x^4*(c + d*x^3)^(1/2))/(2*(2*a^10*c^7*d^5*x + 2*a^10*c^6*d^6*x^4 + a^6*b^4*c^10*d^2*x^4 + a^7*b^3*c^
9*d^3*x^4 - 3*a^8*b^2*c^8*d^4*x^4 + a^6*b^4*c^9*d^3*x^7 - 3*a^8*b^2*c^7*d^5*x^7 - 3*a^9*b*c^8*d^4*x + a^7*b^3*
c^10*d^2*x - a^9*b*c^7*d^5*x^4 + 2*a^9*b*c^6*d^6*x^7)) + (a^2*b^5*c^7*x^4*(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x
+ 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a
^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) + (269*a^3*
b^5*c^8*x^4*(c + d*x^3)^(1/2))/(24*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4
- 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 +
a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) - (26*a^6*c^2*d^5*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^9*x + 2*a^8*c
^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d
^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) - (5*a^7*c^2*d^5*x^4*
(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d
^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7
+ 2*a^8*b*c^5*d^4*x^7)) + (79*a^8*c^3*d^5*x^4*(c + d*x^3)^(1/2))/(12*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6
*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b
^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) - (34*a^2*b^3*c^5*x^4*(c + d*x^3)
^(1/2))/(3*(a^5*b^2*c^7*x^4 + 2*a^7*c^5*d^2*x^4 + a^6*b*c^7*x + 2*a^7*c^6*d*x + 3*a^6*b*c^6*d*x^4 + a^5*b^2*c^
6*d*x^7 + 2*a^6*b*c^5*d^2*x^7)) - (239*a^5*c^2*d^3*x^4*(c + d*x^3)^(1/2))/(24*(a^5*b^2*c^7*x^4 + 2*a^7*c^5*d^2
*x^4 + a^6*b*c^7*x + 2*a^7*c^6*d*x + 3*a^6*b*c^6*d*x^4 + a^5*b^2*c^6*d*x^7 + 2*a^6*b*c^5*d^2*x^7)) - (3*a^2*b^
5*c^8*x*(c + d*x^3)^(1/2))/(4*(2*a^9*c^6*d^4*x + 2*a^9*c^5*d^5*x^4 + a^6*b^3*c^8*d^2*x^4 - 3*a^7*b^2*c^7*d^3*x
^4 + a^5*b^4*c^8*d^2*x^7 - 3*a^7*b^2*c^6*d^4*x^7 + a^6*b^3*c^9*d*x - 3*a^8*b*c^7*d^3*x + a^5*b^4*c^9*d*x^4 - a
^8*b*c^6*d^4*x^4 + 2*a^8*b*c^5*d^5*x^7)) - (3*a*b^6*c^8*x^4*(c + d*x^3)^(1/2))/(4*(2*a^9*c^6*d^4*x + 2*a^9*c^5
*d^5*x^4 + a^6*b^3*c^8*d^2*x^4 - 3*a^7*b^2*c^7*d^3*x^4 + a^5*b^4*c^8*d^2*x^7 - 3*a^7*b^2*c^6*d^4*x^7 + a^6*b^3
*c^9*d*x - 3*a^8*b*c^7*d^3*x + a^5*b^4*c^9*d*x^4 - a^8*b*c^6*d^4*x^4 + 2*a^8*b*c^5*d^5*x^7)) + (3*a^3*b^5*c^9*
x*(c + d*x^3)^(1/2))/(4*(2*a^10*c^7*d^4*x + 2*a^10*c^6*d^5*x^4 + a^7*b^3*c^9*d^2*x^4 - 3*a^8*b^2*c^8*d^3*x^4 +
a^6*b^4*c^9*d^2*x^7 - 3*a^8*b^2*c^7*d^4*x^7 + a^7*b^3*c^10*d*x - 3*a^9*b*c^8*d^3*x + a^6*b^4*c^10*d*x^4 - a^9
*b*c^7*d^4*x^4 + 2*a^9*b*c^6*d^5*x^7)) + (5*a^4*c*d^3*x*(c + d*x^3)^(1/2))/(2*(a^4*b^2*c^6*x + a^3*b^3*c^6*x^4
+ 2*a^5*b*c^5*d*x + 2*a^4*b^2*c^4*d^2*x^7 + 3*a^4*b^2*c^5*d*x^4 + 2*a^5*b*c^4*d^2*x^4 + a^3*b^3*c^5*d*x^7)) -
(8*a*b^4*c^5*x*(c + d*x^3)^(1/2))/(3*(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*a^7*c^3*d^4*x^4 -
3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*b*c^4*d^3*x^4 + a
^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7)) - (5*a^5*c*d^4*x*(c + d*x^3)^(1/2))/(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x
+ a^3*b^4*c^7*x^4 + 2*a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^
4*b^3*c^6*d*x^4 - a^6*b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7) + (5*a^10*c^2*d^6*x*(c + d*x^3)
^(1/2))/(4*(a^7*b^5*c^10*x + a^6*b^6*c^10*x^4 - 3*a^8*b^4*c^8*d^2*x^4 - a^9*b^3*c^7*d^3*x^4 + 2*a^10*b^2*c^6*d
^4*x^4 - 3*a^8*b^4*c^7*d^3*x^7 + 2*a^9*b^3*c^6*d^4*x^7 - 3*a^9*b^3*c^8*d^2*x + 2*a^10*b^2*c^7*d^3*x + a^7*b^5*
c^9*d*x^4 + a^6*b^6*c^9*d*x^7)) + (5*a^10*c*d^7*x^4*(c + d*x^3)^(1/2))/(4*(a^7*b^5*c^10*x + a^6*b^6*c^10*x^4 -
3*a^8*b^4*c^8*d^2*x^4 - a^9*b^3*c^7*d^3*x^4 + 2*a^10*b^2*c^6*d^4*x^4 - 3*a^8*b^4*c^7*d^3*x^7 + 2*a^9*b^3*c^6*
d^4*x^7 - 3*a^9*b^3*c^8*d^2*x + 2*a^10*b^2*c^7*d^3*x + a^7*b^5*c^9*d*x^4 + a^6*b^6*c^9*d*x^7)) - (11*a*b^2*c^3
*x*(c + d*x^3)^(1/2))/(3*(a^3*b^2*c^5*x^4 + 2*a^5*c^3*d^2*x^4 + a^4*b*c^5*x + 2*a^5*c^4*d*x + 3*a^4*b*c^4*d*x^
4 + a^3*b^2*c^4*d*x^7 + 2*a^4*b*c^3*d^2*x^7)) + (14*a*b^2*c^4*x*(c + d*x^3)^(1/2))/(a^3*b^2*c^6*x^4 + 2*a^5*c^
4*d^2*x^4 + a^4*b*c^6*x + 2*a^5*c^5*d*x + 3*a^4*b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 + 2*a^4*b*c^4*d^2*x^7) + (11*a
*b*d^2*x^4*(c + d*x^3)^(1/2))/(2*(a^2*b^2*c^4*x^4 + 2*a^4*c^2*d^2*x^4 + a^3*b*c^4*x + 2*a^4*c^3*d*x + 3*a^3*b*
c^3*d*x^4 + a^2*b^2*c^3*d*x^7 + 2*a^3*b*c^2*d^2*x^7)) - (143*a^3*c*d^2*x*(c + d*x^3)^(1/2))/(24*(a^3*b^2*c^5*x
^4 + 2*a^5*c^3*d^2*x^4 + a^4*b*c^5*x + 2*a^5*c^4*d*x + 3*a^4*b*c^4*d*x^4 + a^3*b^2*c^4*d*x^7 + 2*a^4*b*c^3*d^2
*x^7)) + (22*b^2*c*d*x^4*(c + d*x^3)^(1/2))/(3*(a^2*b^2*c^4*x^4 + 2*a^4*c^2*d^2*x^4 + a^3*b*c^4*x + 2*a^4*c^3*
d*x + 3*a^3*b*c^3*d*x^4 + a^2*b^2*c^3*d*x^7 + 2*a^3*b*c^2*d^2*x^7)) + (11*a^3*b^2*c^3*d^2*x*(c + d*x^3)^(1/2))
/(3*(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2
*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4
*x^7)) - (31*a^4*b^2*c^4*d^2*x*(c + d*x^3)^(1/2))/(2*(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^
8*c^4*d^4*x^4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*
c^5*d^3*x^4 + a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) + (28*a^2*b^4*c^5*d*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3
*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7
- 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 + a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) + (29
*a^5*b*c^2*d^4*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^
4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4
+ a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) + (23*a^4*b^2*c^5*d^2*x*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^9*x + 2*a
^8*c^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c
^7*d^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) + (7*a^2*b^4*c^6*
d*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2
*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8
*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) - (7*a^5*b*c^3*d^4*x^4*(c + d*x^3)^(1/2))/(4*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x +
a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5
*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) - (9*a^5*b^2*c^5*d^2*x*(c + d*x
^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 -
3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8
*b*c^5*d^4*x^7)) - (9*a^3*b^4*c^6*d*x^4*(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x
^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4
- a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) - (9*a^6*b*c^3*d^4*x^4*(c + d*x^3)^(1/2))/(8*
(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6
*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7
)) + (23*a^6*b^2*c^6*d^2*x*(c + d*x^3)^(1/2))/(24*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^
10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b
*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) - (31*a^4*b^4*c^7*d*x^4*(c + d*x^3)^(1/2))/(8*(a^7*b^
3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^
3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7))
- (21*a^7*b*c^4*d^4*x^4*(c + d*x^3)^(1/2))/(8*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c
^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7
*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) + (209*a^2*b*c^2*d^2*x^4*(c + d*x^3)^(1/2))/(12*(a^3*b^2*
c^6*x^4 + 2*a^5*c^4*d^2*x^4 + a^4*b*c^6*x + 2*a^5*c^5*d*x + 3*a^4*b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 + 2*a^4*b*c^
4*d^2*x^7)) - (89*a^3*b^2*c^4*d*x^4*(c + d*x^3)^(1/2))/(6*(a^5*b^2*c^7*x^4 + 2*a^7*c^5*d^2*x^4 + a^6*b*c^7*x +
2*a^7*c^6*d*x + 3*a^6*b*c^6*d*x^4 + a^5*b^2*c^6*d*x^7 + 2*a^6*b*c^5*d^2*x^7)) - (109*a^4*b*c^3*d^2*x^4*(c + d
*x^3)^(1/2))/(12*(a^5*b^2*c^7*x^4 + 2*a^7*c^5*d^2*x^4 + a^6*b*c^7*x + 2*a^7*c^6*d*x + 3*a^6*b*c^6*d*x^4 + a^5*
b^2*c^6*d*x^7 + 2*a^6*b*c^5*d^2*x^7)) - (33*a^2*b*c^2*d*x*(c + d*x^3)^(1/2))/(2*(a^3*b^2*c^5*x^4 + 2*a^5*c^3*d
^2*x^4 + a^4*b*c^5*x + 2*a^5*c^4*d*x + 3*a^4*b*c^4*d*x^4 + a^3*b^2*c^4*d*x^7 + 2*a^4*b*c^3*d^2*x^7)) + (329*a^
2*b*c^3*d*x*(c + d*x^3)^(1/2))/(12*(a^3*b^2*c^6*x^4 + 2*a^5*c^4*d^2*x^4 + a^4*b*c^6*x + 2*a^5*c^5*d*x + 3*a^4*
b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 + 2*a^4*b*c^4*d^2*x^7)) - (205*a^4*b*c^4*d*x*(c + d*x^3)^(1/2))/(12*(a^5*b^2*c
^7*x^4 + 2*a^7*c^5*d^2*x^4 + a^6*b*c^7*x + 2*a^7*c^6*d*x + 3*a^6*b*c^6*d*x^4 + a^5*b^2*c^6*d*x^7 + 2*a^6*b*c^5
*d^2*x^7)) + (6*a^2*b^3*c^3*d^2*x^4*(c + d*x^3)^(1/2))/(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*
a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*
b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7) - (a^3*b^2*c^2*d^3*x^4*(c + d*x^3)^(1/2))/(a^4*b^3*c^
7*x + 2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 -
3*a^6*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7) - (19*a^3
*b^3*c^4*d^2*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4
- 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 +
a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) + (10*a^4*b^2*c^3*d^3*x^4*(c + d*x^3)^(1/2))/(3*(a^5*b^3*c^8*x + 2*a
^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c
^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 + a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) - (a^3*b^3*c^5*d^
2*x^4*(c + d*x^3)^(1/2))/(2*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2
*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8
*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) - (97*a^4*b^2*c^4*d^3*x^4*(c + d*x^3)^(1/2))/(6*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*
x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x +
a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) - (9*a^4*b^3*c^5*d^2*x^4*(c
+ d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*
x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 +
2*a^8*b*c^5*d^4*x^7)) - (9*a^5*b^2*c^4*d^3*x^4*(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^
4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^
8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) + (31*a^5*b^3*c^6*d^2*x^4*(c + d*x^3)^
(1/2))/(24*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4
- 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^
9*b*c^6*d^4*x^7)) + (415*a^6*b^2*c^5*d^3*x^4*(c + d*x^3)^(1/2))/(24*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b
^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3
*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) + (11*a*b*c*d*x*(c + d*x^3)^(1/2))/
(2*(a^2*b^2*c^4*x^4 + 2*a^4*c^2*d^2*x^4 + a^3*b*c^4*x + 2*a^4*c^3*d*x + 3*a^3*b*c^3*d*x^4 + a^2*b^2*c^3*d*x^7
+ 2*a^3*b*c^2*d^2*x^7)) - (10*a^2*b^3*c^4*d*x*(c + d*x^3)^(1/2))/(3*(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x + a^3*b^4
*c^7*x^4 + 2*a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^4*b^3*c^6
*d*x^4 - a^6*b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7)) - (19*a^4*b*c^2*d^3*x*(c + d*x^3)^(1/2)
)/(6*(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^
2*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^
4*x^7)) - (8*a*b^4*c^4*d*x^4*(c + d*x^3)^(1/2))/(a^4*b^3*c^7*x + 2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*a^7*c^3
*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 - 3*a^6*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*b*c^4*d
^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7) - (15*a^4*b*c*d^4*x^4*(c + d*x^3)^(1/2))/(2*(a^4*b^3*c^7*x +
2*a^7*c^4*d^3*x + a^3*b^4*c^7*x^4 + 2*a^7*c^3*d^4*x^4 - 3*a^5*b^2*c^5*d^2*x^4 - 3*a^5*b^2*c^4*d^3*x^7 - 3*a^6
*b*c^5*d^2*x + a^4*b^3*c^6*d*x^4 - a^6*b*c^4*d^3*x^4 + a^3*b^4*c^6*d*x^7 + 2*a^6*b*c^3*d^4*x^7)) + (12*a^3*b^3
*c^5*d*x*(c + d*x^3)^(1/2))/(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4 - 3*a^6*b^2
*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 + a^4*b^4*c^7
*d*x^7 + 2*a^7*b*c^4*d^4*x^7) + (51*a^5*b*c^3*d^3*x*(c + d*x^3)^(1/2))/(4*(a^5*b^3*c^8*x + 2*a^8*c^5*d^3*x + a
^4*b^4*c^8*x^4 + 2*a^8*c^4*d^4*x^4 - 3*a^6*b^2*c^6*d^2*x^4 - 3*a^6*b^2*c^5*d^3*x^7 - 3*a^7*b*c^6*d^2*x + a^5*b
^3*c^7*d*x^4 - a^7*b*c^5*d^3*x^4 + a^4*b^4*c^7*d*x^7 + 2*a^7*b*c^4*d^4*x^7)) - (14*a^3*b^3*c^6*d*x*(c + d*x^3)
^(1/2))/(3*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x + a^4*b^4*c^9*x^4 + 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*
a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5*b^3*c^8*d*x^4 - a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*
c^5*d^4*x^7)) - (169*a^5*b*c^4*d^3*x*(c + d*x^3)^(1/2))/(12*(a^5*b^3*c^9*x + 2*a^8*c^6*d^3*x + a^4*b^4*c^9*x^4
+ 2*a^8*c^5*d^4*x^4 - 3*a^6*b^2*c^7*d^2*x^4 - 3*a^6*b^2*c^6*d^3*x^7 - 3*a^7*b*c^7*d^2*x + a^5*b^3*c^8*d*x^4 -
a^7*b*c^6*d^3*x^4 + a^4*b^4*c^8*d*x^7 + 2*a^7*b*c^5*d^4*x^7)) - (9*a^4*b^3*c^6*d*x*(c + d*x^3)^(1/2))/(8*(a^6
*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3
*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4 + a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) -
(9*a^6*b*c^4*d^3*x*(c + d*x^3)^(1/2))/(8*(a^6*b^3*c^9*x + 2*a^9*c^6*d^3*x + a^5*b^4*c^9*x^4 + 2*a^9*c^5*d^4*x
^4 - 3*a^7*b^2*c^7*d^2*x^4 - 3*a^7*b^2*c^6*d^3*x^7 - 3*a^8*b*c^7*d^2*x + a^6*b^3*c^8*d*x^4 - a^8*b*c^6*d^3*x^4
+ a^5*b^4*c^8*d*x^7 + 2*a^8*b*c^5*d^4*x^7)) - (7*a^5*b^3*c^7*d*x*(c + d*x^3)^(1/2))/(8*(a^7*b^3*c^10*x + 2*a^
10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b
*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) + (77*a^7*b*c^5
*d^3*x*(c + d*x^3)^(1/2))/(8*(a^7*b^3*c^10*x + 2*a^10*c^7*d^3*x + a^6*b^4*c^10*x^4 + 2*a^10*c^6*d^4*x^4 - 3*a^
8*b^2*c^8*d^2*x^4 - 3*a^8*b^2*c^7*d^3*x^7 - 3*a^9*b*c^8*d^2*x + a^7*b^3*c^9*d*x^4 - a^9*b*c^7*d^3*x^4 + a^6*b^
4*c^9*d*x^7 + 2*a^9*b*c^6*d^4*x^7)) - (41*a*b^2*c^2*d*x^4*(c + d*x^3)^(1/2))/(3*(a^3*b^2*c^5*x^4 + 2*a^5*c^3*d
^2*x^4 + a^4*b*c^5*x + 2*a^5*c^4*d*x + 3*a^4*b*c^4*d*x^4 + a^3*b^2*c^4*d*x^7 + 2*a^4*b*c^3*d^2*x^7)) - (71*a^2
*b*c*d^2*x^4*(c + d*x^3)^(1/2))/(6*(a^3*b^2*c^5*x^4 + 2*a^5*c^3*d^2*x^4 + a^4*b*c^5*x + 2*a^5*c^4*d*x + 3*a^4*
b*c^4*d*x^4 + a^3*b^2*c^4*d*x^7 + 2*a^4*b*c^3*d^2*x^7)) + (133*a*b^2*c^3*d*x^4*(c + d*x^3)^(1/2))/(6*(a^3*b^2*
c^6*x^4 + 2*a^5*c^4*d^2*x^4 + a^4*b*c^6*x + 2*a^5*c^5*d*x + 3*a^4*b*c^5*d*x^4 + a^3*b^2*c^5*d*x^7 + 2*a^4*b*c^
4*d^2*x^7))
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a + b x^{3}\right )^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**4/(b*x**3+a)**2/(d*x**3+c)**(3/2),x)
[Out]
Integral(1/(x**4*(a + b*x**3)**2*(c + d*x**3)**(3/2)), x)
________________________________________________________________________________________