3.8.65 \(\int \frac {1}{x^3 (a+b x^2)^2 (c+d x^2)^{5/2}} \, dx\)
Optimal. Leaf size=304 \[ -\frac {b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}+\frac {(5 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{7/2}}-\frac {d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{6 a^2 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {d (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{2 a^2 c^3 \sqrt {c+d x^2} (b c-a d)^3}-\frac {b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \]
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Rubi [A] time = 0.48, antiderivative size = 304, normalized size of antiderivative = 1.00,
number of steps used = 10, 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 (2 b c-a d) \left (5 a^2 d^2-a b c d+b^2 c^2\right )}{2 a^2 c^3 \sqrt {c+d x^2} (b c-a d)^3}-\frac {d \left (5 a^2 d^2-6 a b c d+6 b^2 c^2\right )}{6 a^2 c^2 \left (c+d x^2\right )^{3/2} (b c-a d)^2}-\frac {b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}+\frac {(5 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{7/2}}-\frac {b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
-(d*(6*b^2*c^2 - 6*a*b*c*d + 5*a^2*d^2))/(6*a^2*c^2*(b*c - a*d)^2*(c + d*x^2)^(3/2)) - (b*(2*b*c - a*d))/(2*a^
2*c*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - 1/(2*a*c*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) - (d*(2*b*c - a*d
)*(b^2*c^2 - a*b*c*d + 5*a^2*d^2))/(2*a^2*c^3*(b*c - a*d)^3*Sqrt[c + d*x^2]) + ((4*b*c + 5*a*d)*ArcTanh[Sqrt[c
+ d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2)) - (b^(7/2)*(4*b*c - 9*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*
d]])/(2*a^3*(b*c - a*d)^(7/2))
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^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (4 b c+5 a d)+\frac {7 b d x}{2}}{x (a+b x)^2 (c+d x)^{5/2}} \, dx,x,x^2\right )}{2 a c}\\ &=-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c-a d) (4 b c+5 a d)+\frac {5}{2} b d (2 b c-a d) x}{x (a+b x) (c+d x)^{5/2}} \, dx,x,x^2\right )}{2 a^2 c (b c-a d)}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {3}{4} (b c-a d)^2 (4 b c+5 a d)-\frac {3}{4} b d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{x (a+b x) (c+d x)^{3/2}} \, dx,x,x^2\right )}{3 a^2 c^2 (b c-a d)^2}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {3}{8} (b c-a d)^3 (4 b c+5 a d)+\frac {3}{8} b d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right ) x}{x (a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{3 a^2 c^3 (b c-a d)^3}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\left (b^4 (4 b c-9 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3 (b c-a d)^3}-\frac {(4 b c+5 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a^3 c^3}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {\left (b^4 (4 b c-9 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^2}\right )}{2 a^3 d (b c-a d)^3}-\frac {(4 b c+5 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 a^3 c^3 d}\\ &=-\frac {d \left (6 b^2 c^2-6 a b c d+5 a^2 d^2\right )}{6 a^2 c^2 (b c-a d)^2 \left (c+d x^2\right )^{3/2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}-\frac {d (2 b c-a d) \left (b^2 c^2-a b c d+5 a^2 d^2\right )}{2 a^2 c^3 (b c-a d)^3 \sqrt {c+d x^2}}+\frac {(4 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{7/2}}-\frac {b^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 190, normalized size = 0.62 \begin {gather*} \frac {b^2 c^2 x^2 \left (a+b x^2\right ) (4 b c-9 a d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b \left (d x^2+c\right )}{b c-a d}\right )-(b c-a d) \left (x^2 \left (a+b x^2\right ) \left (-5 a^2 d^2+a b c d+4 b^2 c^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {d x^2}{c}+1\right )-3 a c \left (a^2 d+a b \left (d x^2-c\right )-2 b^2 c x^2\right )\right )}{6 a^3 c^2 x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
(b^2*c^2*(4*b*c - 9*a*d)*x^2*(a + b*x^2)*Hypergeometric2F1[-3/2, 1, -1/2, (b*(c + d*x^2))/(b*c - a*d)] - (b*c
- a*d)*(-3*a*c*(a^2*d - 2*b^2*c*x^2 + a*b*(-c + d*x^2)) + (4*b^2*c^2 + a*b*c*d - 5*a^2*d^2)*x^2*(a + b*x^2)*Hy
pergeometric2F1[-3/2, 1, -1/2, 1 + (d*x^2)/c]))/(6*a^3*c^2*(b*c - a*d)^2*x^2*(a + b*x^2)*(c + d*x^2)^(3/2))
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IntegrateAlgebraic [A] time = 1.33, size = 397, normalized size = 1.31 \begin {gather*} \frac {\left (4 b^{9/2} c-9 a b^{7/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2} \sqrt {a d-b c}}{b c-a d}\right )}{2 a^3 (a d-b c)^{7/2}}+\frac {(5 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{7/2}}+\frac {-3 a^4 c^2 d^3-20 a^4 c d^4 x^2-15 a^4 d^5 x^4+9 a^3 b c^3 d^2+41 a^3 b c^2 d^3 x^2+13 a^3 b c d^4 x^4-15 a^3 b d^5 x^6-9 a^2 b^2 c^4 d-9 a^2 b^2 c^3 d^2 x^2+35 a^2 b^2 c^2 d^3 x^4+33 a^2 b^2 c d^4 x^6+3 a b^3 c^5-3 a b^3 c^4 d x^2-15 a b^3 c^3 d^2 x^4-9 a b^3 c^2 d^3 x^6+6 b^4 c^5 x^2+12 b^4 c^4 d x^4+6 b^4 c^3 d^2 x^6}{6 a^2 c^3 x^2 \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (a d-b c)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x]
[Out]
(3*a*b^3*c^5 - 9*a^2*b^2*c^4*d + 9*a^3*b*c^3*d^2 - 3*a^4*c^2*d^3 + 6*b^4*c^5*x^2 - 3*a*b^3*c^4*d*x^2 - 9*a^2*b
^2*c^3*d^2*x^2 + 41*a^3*b*c^2*d^3*x^2 - 20*a^4*c*d^4*x^2 + 12*b^4*c^4*d*x^4 - 15*a*b^3*c^3*d^2*x^4 + 35*a^2*b^
2*c^2*d^3*x^4 + 13*a^3*b*c*d^4*x^4 - 15*a^4*d^5*x^4 + 6*b^4*c^3*d^2*x^6 - 9*a*b^3*c^2*d^3*x^6 + 33*a^2*b^2*c*d
^4*x^6 - 15*a^3*b*d^5*x^6)/(6*a^2*c^3*(-(b*c) + a*d)^3*x^2*(a + b*x^2)*(c + d*x^2)^(3/2)) + ((4*b^(9/2)*c - 9*
a*b^(7/2)*d)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x^2])/(b*c - a*d)])/(2*a^3*(-(b*c) + a*d)^(7/2)) +
((4*b*c + 5*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(7/2))
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fricas [B] time = 29.74, size = 4115, normalized size = 13.54
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="fricas")
[Out]
[1/24*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (
4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x^2)*sqrt(b/(b*c - a*d))*l
og((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d +
a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 6*((4*
b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*
b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*
d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c
^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt
(c) + 2*c)/x^2) - 4*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 -
3*a^2*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^
2*c^3*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4
*b*c^3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 -
a^6*b*c^4*d^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*
x^6 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 -
3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2), -1/24*(12*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^
3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^
3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d
^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*
d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - 3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8
+ (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x
^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x^2)*sqrt(b/(b*c - a*d))*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d
^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(2*b^2*c^2 - 3*a*b*c*d + a^2*d^2 + (b^2*c*d - a*b*d^2)*x^2)*sqrt(d*x^2
+ c)*sqrt(b/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2
- 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4
*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2
*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 -
3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*
c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^6 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^
3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2), 1/12*(3*((4*b^5
*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 - 9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^
4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*
x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b*c)) + 3*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3
- 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*
d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6 + a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^
3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6 - 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11
*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) - 2*(3*a^2*b^
3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2*b^3*c^3*d^3 + 11*a^3*b^
2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3*d^3 + 13*a^4*b*c^2*d^4
- 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^3*d^3 - 20*a^5*c^2*d^4)*
x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5)*x^8 + (2*a^3*
b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^6 + (a^3*b^4*c^9 - a^4*b^3*
c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^
7*d^2 - a^7*c^6*d^3)*x^2), 1/12*(3*((4*b^5*c^5*d^2 - 9*a*b^4*c^4*d^3)*x^8 + (8*b^5*c^6*d - 14*a*b^4*c^5*d^2 -
9*a^2*b^3*c^4*d^3)*x^6 + (4*b^5*c^7 - a*b^4*c^6*d - 18*a^2*b^3*c^5*d^2)*x^4 + (4*a*b^4*c^7 - 9*a^2*b^3*c^6*d)*
x^2)*sqrt(-b/(b*c - a*d))*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 + c)*sqrt(-b/(b*c - a*d))/(b*d*x^2 + b
*c)) - 6*((4*b^5*c^4*d^2 - 7*a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 11*a^3*b^2*c*d^5 - 5*a^4*b*d^6)*x^8 + (8*b^5*
c^5*d - 10*a*b^4*c^4*d^2 - 13*a^2*b^3*c^3*d^3 + 19*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 5*a^5*d^6)*x^6 + (4*b^5*c^6
+ a*b^4*c^5*d - 17*a^2*b^3*c^4*d^2 + 5*a^3*b^2*c^3*d^3 + 17*a^4*b*c^2*d^4 - 10*a^5*c*d^5)*x^4 + (4*a*b^4*c^6
- 7*a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 11*a^4*b*c^3*d^3 - 5*a^5*c^2*d^4)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d
*x^2 + c)) - 2*(3*a^2*b^3*c^6 - 9*a^3*b^2*c^5*d + 9*a^4*b*c^4*d^2 - 3*a^5*c^3*d^3 + 3*(2*a*b^4*c^4*d^2 - 3*a^2
*b^3*c^3*d^3 + 11*a^3*b^2*c^2*d^4 - 5*a^4*b*c*d^5)*x^6 + (12*a*b^4*c^5*d - 15*a^2*b^3*c^4*d^2 + 35*a^3*b^2*c^3
*d^3 + 13*a^4*b*c^2*d^4 - 15*a^5*c*d^5)*x^4 + (6*a*b^4*c^6 - 3*a^2*b^3*c^5*d - 9*a^3*b^2*c^4*d^2 + 41*a^4*b*c^
3*d^3 - 20*a^5*c^2*d^4)*x^2)*sqrt(d*x^2 + c))/((a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*
b*c^4*d^5)*x^8 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x^6 +
(a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^4 + (a^4*b^3*c^9 - 3*a^
5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^2)]
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giac [A] time = 0.42, size = 505, normalized size = 1.66 \begin {gather*} \frac {{\left (4 \, b^{5} c - 9 \, a b^{4} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{4} c^{3} d - 2 \, \sqrt {d x^{2} + c} b^{4} c^{4} d - 3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{2} + 4 \, \sqrt {d x^{2} + c} a b^{3} c^{3} d^{2} + 3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{3} - 6 \, \sqrt {d x^{2} + c} a^{2} b^{2} c^{2} d^{3} - {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{3} b d^{4} + 4 \, \sqrt {d x^{2} + c} a^{3} b c d^{4} - \sqrt {d x^{2} + c} a^{4} d^{5}}{2 \, {\left (a^{2} b^{3} c^{6} - 3 \, a^{3} b^{2} c^{5} d + 3 \, a^{4} b c^{4} d^{2} - a^{5} c^{3} d^{3}\right )} {\left ({\left (d x^{2} + c\right )}^{2} b - 2 \, {\left (d x^{2} + c\right )} b c + b c^{2} + {\left (d x^{2} + c\right )} a d - a c d\right )}} - \frac {12 \, {\left (d x^{2} + c\right )} b c d^{3} + b c^{2} d^{3} - 6 \, {\left (d x^{2} + c\right )} a d^{4} - a c d^{4}}{3 \, {\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{3} c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, b c + 5 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="giac")
[Out]
1/2*(4*b^5*c - 9*a*b^4*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a
^5*b*c*d^2 - a^6*d^3)*sqrt(-b^2*c + a*b*d)) - 1/2*(2*(d*x^2 + c)^(3/2)*b^4*c^3*d - 2*sqrt(d*x^2 + c)*b^4*c^4*d
- 3*(d*x^2 + c)^(3/2)*a*b^3*c^2*d^2 + 4*sqrt(d*x^2 + c)*a*b^3*c^3*d^2 + 3*(d*x^2 + c)^(3/2)*a^2*b^2*c*d^3 - 6
*sqrt(d*x^2 + c)*a^2*b^2*c^2*d^3 - (d*x^2 + c)^(3/2)*a^3*b*d^4 + 4*sqrt(d*x^2 + c)*a^3*b*c*d^4 - sqrt(d*x^2 +
c)*a^4*d^5)/((a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*((d*x^2 + c)^2*b - 2*(d*x^2 + c)*
b*c + b*c^2 + (d*x^2 + c)*a*d - a*c*d)) - 1/3*(12*(d*x^2 + c)*b*c*d^3 + b*c^2*d^3 - 6*(d*x^2 + c)*a*d^4 - a*c*
d^4)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^3*c^3*d^3)*(d*x^2 + c)^(3/2)) - 1/2*(4*b*c + 5*a*d)*arcta
n(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^3)
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maple [B] time = 0.02, size = 2980, normalized size = 9.80 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(5/2),x)
[Out]
-5/4*b^3/a^2*d/(a*d-b*c)^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-2*(a*d-b*c)/b+2*(-(
a*d-b*c)/b)^(1/2)*((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2))/(x+(-a*b)^
(1/2)/b))-5/4*b^3/a^2*d/(a*d-b*c)^3/(-(a*d-b*c)/b)^(1/2)*ln((2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-2*(a*d-b*c)
/b+2*(-(a*d-b*c)/b)^(1/2)*((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2))/(x
-(-a*b)^(1/2)/b))+1/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)/(x-(-a*b)^(1/2)/b)/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)
*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)-1/4*b^2/a^2/(-a*b)^(1/2)/(a*d-b*c)/(x+(-a*b)^(1/2)/b)/((x+(-a*b)^(1
/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)-2*b/a^3/c^2/(d*x^2+c)^(1/2)+2*b/a^3/c^(5/2
)*ln((2*c+2*(d*x^2+c)^(1/2)*c^(1/2))/x)-1/2/a^2/c/x^2/(d*x^2+c)^(3/2)-5/6/a^2*d/c^2/(d*x^2+c)^(3/2)-5/2/a^2*d/
c^3/(d*x^2+c)^(1/2)+5/2/a^2*d/c^(7/2)*ln((2*c+2*(d*x^2+c)^(1/2)*c^(1/2))/x)-1/3*b^2/a^3/(a*d-b*c)/((x+(-a*b)^(
1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)+b^3/a^3/(a*d-b*c)^2/((x+(-a*b)^(1/2)/b)^2
*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)-1/3*b^2/a^3/(a*d-b*c)/((x-(-a*b)^(1/2)/b)^2*d+2*(-
a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)+b^3/a^3/(a*d-b*c)^2/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2
)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)-2/3*b/a^3/c/(d*x^2+c)^(3/2)-b^2/a^3/(a*d-b*c)^2*(-a*b)^(1/2)/c/((x
-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*d*x-5/12*b^2/a^2*d/(a*d-b*c)^2/(
(x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)+5/4*b^3/a^2*d/(a*d-b*c)^3/((x-
(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)-b^3/a^3/(a*d-b*c)^2/(-(a*d-b*c)/b
)^(1/2)*ln((-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-2*(a*d-b*c)/b+2*(-(a*d-b*c)/b)^(1/2)*((x+(-a*b)^(1/2)/b)^2*
d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2))/(x+(-a*b)^(1/2)/b))-5/12*b^2/a^2*d/(a*d-b*c)^2/((x
+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)+5/4*b^3/a^2*d/(a*d-b*c)^3/((x+(-
a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)-b^3/a^3/(a*d-b*c)^2/(-(a*d-b*c)/b)^
(1/2)*ln((2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-2*(a*d-b*c)/b+2*(-(a*d-b*c)/b)^(1/2)*((x-(-a*b)^(1/2)/b)^2*d+2
*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2))/(x-(-a*b)^(1/2)/b))-5/12*b^2/a/(-a*b)^(1/2)*d^2/(a*d-
b*c)^2/c/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)*x-5/6*b^2/a/(-a*b)^(
1/2)*d^2/(a*d-b*c)^2/c^2/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*x+5/
4*b^3/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^3/c/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)
/b)^(1/2)*x+1/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b*c)/c/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b)/b
*d-(a*d-b*c)/b)^(3/2)*x+2/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b*c)/c^2/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-(-a
*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*x-1/3*b/a^3*(-a*b)^(1/2)*d/(a*d-b*c)/c/((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1
/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)*x-2/3*b/a^3*(-a*b)^(1/2)*d/(a*d-b*c)/c^2/((x+(-a*b)^(1/2)/b)^2*d
-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*x+b^2/a^3/(a*d-b*c)^2*(-a*b)^(1/2)/c/((x+(-a*b)^(1/2
)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*d*x+5/12*b^2/a/(-a*b)^(1/2)*d^2/(a*d-b*c)^2/
c/((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)*x+5/6*b^2/a/(-a*b)^(1/2)*d^
2/(a*d-b*c)^2/c^2/((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*x-5/4*b^3/a
/(-a*b)^(1/2)*d^2/(a*d-b*c)^3/c/((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/
2)*x-1/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b*c)/c/((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b)/b*d-(a*d
-b*c)/b)^(3/2)*x-2/3*b^2/a^2/(-a*b)^(1/2)*d/(a*d-b*c)/c^2/((x+(-a*b)^(1/2)/b)^2*d-2*(-a*b)^(1/2)*(x+(-a*b)^(1/
2)/b)/b*d-(a*d-b*c)/b)^(1/2)*x+1/3*b/a^3*(-a*b)^(1/2)*d/(a*d-b*c)/c/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*b)^(1/2)*(x-
(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(3/2)*x+2/3*b/a^3*(-a*b)^(1/2)*d/(a*d-b*c)/c^2/((x-(-a*b)^(1/2)/b)^2*d+2*(-a*
b)^(1/2)*(x-(-a*b)^(1/2)/b)/b*d-(a*d-b*c)/b)^(1/2)*x
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {5}{2}} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(5/2),x, algorithm="maxima")
[Out]
integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(5/2)*x^3), x)
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mupad [B] time = 6.58, size = 5800, normalized size = 19.08
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(5/2)),x)
[Out]
((5*d^3*(c + d*x^2)*(a*d - 2*b*c))/(3*(b*c^2 - a*c*d)^2) - d^3/(3*(b*c^2 - a*c*d)) + (d*(c + d*x^2)^2*(15*a^4*
d^4 + 6*b^4*c^4 + 64*a^2*b^2*c^2*d^2 - 12*a*b^3*c^3*d - 58*a^3*b*c*d^3))/(6*a^2*(b*c^2 - a*c*d)^3) + (d*(c + d
*x^2)^3*(a*d - 2*b*c)*(b^3*c^2 + 5*a^2*b*d^2 - a*b^2*c*d))/(2*a^2*(b*c^2 - a*c*d)^3))/(b*(c + d*x^2)^(7/2) + (
c + d*x^2)^(3/2)*(b*c^2 - a*c*d) + (c + d*x^2)^(5/2)*(a*d - 2*b*c)) - (atan((a^19*c^15*d^19*(c + d*x^2)^(1/2)*
125i + a^3*b^16*c^31*d^3*(c + d*x^2)^(1/2)*420i - a^4*b^15*c^30*d^4*(c + d*x^2)^(1/2)*4515i + a^5*b^14*c^29*d^
5*(c + d*x^2)^(1/2)*20916i - a^6*b^13*c^28*d^6*(c + d*x^2)^(1/2)*52836i + a^7*b^12*c^27*d^7*(c + d*x^2)^(1/2)*
71070i - a^8*b^11*c^26*d^8*(c + d*x^2)^(1/2)*19530i - a^9*b^10*c^25*d^9*(c + d*x^2)^(1/2)*107740i + a^10*b^9*c
^24*d^10*(c + d*x^2)^(1/2)*212608i - a^11*b^8*c^23*d^11*(c + d*x^2)^(1/2)*184563i + a^12*b^7*c^22*d^12*(c + d*
x^2)^(1/2)*40965i + a^13*b^6*c^21*d^13*(c + d*x^2)^(1/2)*91560i - a^14*b^5*c^20*d^14*(c + d*x^2)^(1/2)*126720i
+ a^15*b^4*c^19*d^15*(c + d*x^2)^(1/2)*87276i - a^16*b^3*c^18*d^16*(c + d*x^2)^(1/2)*37776i + a^17*b^2*c^17*d
^17*(c + d*x^2)^(1/2)*10440i - a^18*b*c^16*d^18*(c + d*x^2)^(1/2)*1700i)/(c^7*(c^7)^(1/2)*(c^7*(c^7*(212608*a^
10*b^9*d^10 - 107740*a^9*b^10*c*d^9 + 420*a^3*b^16*c^7*d^3 - 4515*a^4*b^15*c^6*d^4 + 20916*a^5*b^14*c^5*d^5 -
52836*a^6*b^13*c^4*d^6 + 71070*a^7*b^12*c^3*d^7 - 19530*a^8*b^11*c^2*d^8) + 10440*a^17*b^2*d^17 - 37776*a^16*b
^3*c*d^16 - 184563*a^11*b^8*c^6*d^11 + 40965*a^12*b^7*c^5*d^12 + 91560*a^13*b^6*c^4*d^13 - 126720*a^14*b^5*c^3
*d^14 + 87276*a^15*b^4*c^2*d^15) + 125*a^19*c^5*d^19 - 1700*a^18*b*c^6*d^18)))*(5*a*d + 4*b*c)*1i)/(2*a^3*(c^7
)^(1/2)) + (atan((((-b^7*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(512*a^6*b^20*c^26*d^2 - 6656*a^7*b^19*c^25*d
^3 + 38560*a^8*b^18*c^24*d^4 - 129920*a^9*b^17*c^23*d^5 + 275920*a^10*b^16*c^22*d^6 - 363440*a^11*b^15*c^21*d^
7 + 235312*a^12*b^14*c^20*d^8 + 85360*a^13*b^13*c^19*d^9 - 316400*a^14*b^12*c^18*d^10 + 205840*a^15*b^11*c^17*
d^11 + 152384*a^16*b^10*c^16*d^12 - 430816*a^17*b^9*c^15*d^13 + 444080*a^18*b^8*c^14*d^14 - 281680*a^19*b^7*c^
13*d^15 + 118640*a^20*b^6*c^12*d^16 - 32656*a^21*b^5*c^11*d^17 + 5360*a^22*b^4*c^10*d^18 - 400*a^23*b^3*c^9*d^
19) + ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c)*(128*a^10*b^18*c^28*d^3 - 1792*a^11*b^17*c^27*d^4 + 10624*a^
12*b^16*c^26*d^5 - 33280*a^13*b^15*c^25*d^6 + 47936*a^14*b^14*c^24*d^7 + 40448*a^15*b^13*c^23*d^8 - 368896*a^1
6*b^12*c^22*d^9 + 948992*a^17*b^11*c^21*d^10 - 1531200*a^18*b^10*c^20*d^11 + 1754368*a^19*b^9*c^19*d^12 - 1485
440*a^20*b^8*c^18*d^13 + 939008*a^21*b^7*c^17*d^14 - 439616*a^22*b^6*c^16*d^15 + 148480*a^23*b^5*c^15*d^16 - 3
4304*a^24*b^4*c^14*d^17 + 4864*a^25*b^3*c^13*d^18 - 320*a^26*b^2*c^12*d^19 - ((-b^7*(a*d - b*c)^7)^(1/2)*(c +
d*x^2)^(1/2)*(9*a*d - 4*b*c)*(512*a^12*b^18*c^31*d^2 - 7936*a^13*b^17*c^30*d^3 + 57600*a^14*b^16*c^29*d^4 - 25
9840*a^15*b^15*c^28*d^5 + 815360*a^16*b^14*c^27*d^6 - 1886976*a^17*b^13*c^26*d^7 + 3331328*a^18*b^12*c^25*d^8
- 4576000*a^19*b^11*c^24*d^9 + 4942080*a^20*b^10*c^23*d^10 - 4209920*a^21*b^9*c^22*d^11 + 2818816*a^22*b^8*c^2
1*d^12 - 1467648*a^23*b^7*c^20*d^13 + 582400*a^24*b^6*c^19*d^14 - 170240*a^25*b^5*c^18*d^15 + 34560*a^26*b^4*c
^17*d^16 - 4352*a^27*b^3*c^16*d^17 + 256*a^28*b^2*c^15*d^18))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 2
1*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6))))/(4*(a^10*
d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^
2*c^2*d^5 - 7*a^9*b*c*d^6)))*(9*a*d - 4*b*c)*1i)/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5
*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)) + ((-b^7*(a*d - b*c)^7)^
(1/2)*((c + d*x^2)^(1/2)*(512*a^6*b^20*c^26*d^2 - 6656*a^7*b^19*c^25*d^3 + 38560*a^8*b^18*c^24*d^4 - 129920*a^
9*b^17*c^23*d^5 + 275920*a^10*b^16*c^22*d^6 - 363440*a^11*b^15*c^21*d^7 + 235312*a^12*b^14*c^20*d^8 + 85360*a^
13*b^13*c^19*d^9 - 316400*a^14*b^12*c^18*d^10 + 205840*a^15*b^11*c^17*d^11 + 152384*a^16*b^10*c^16*d^12 - 4308
16*a^17*b^9*c^15*d^13 + 444080*a^18*b^8*c^14*d^14 - 281680*a^19*b^7*c^13*d^15 + 118640*a^20*b^6*c^12*d^16 - 32
656*a^21*b^5*c^11*d^17 + 5360*a^22*b^4*c^10*d^18 - 400*a^23*b^3*c^9*d^19) - ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d
- 4*b*c)*(128*a^10*b^18*c^28*d^3 - 1792*a^11*b^17*c^27*d^4 + 10624*a^12*b^16*c^26*d^5 - 33280*a^13*b^15*c^25*
d^6 + 47936*a^14*b^14*c^24*d^7 + 40448*a^15*b^13*c^23*d^8 - 368896*a^16*b^12*c^22*d^9 + 948992*a^17*b^11*c^21*
d^10 - 1531200*a^18*b^10*c^20*d^11 + 1754368*a^19*b^9*c^19*d^12 - 1485440*a^20*b^8*c^18*d^13 + 939008*a^21*b^7
*c^17*d^14 - 439616*a^22*b^6*c^16*d^15 + 148480*a^23*b^5*c^15*d^16 - 34304*a^24*b^4*c^14*d^17 + 4864*a^25*b^3*
c^13*d^18 - 320*a^26*b^2*c^12*d^19 + ((-b^7*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(9*a*d - 4*b*c)*(512*a^12*b
^18*c^31*d^2 - 7936*a^13*b^17*c^30*d^3 + 57600*a^14*b^16*c^29*d^4 - 259840*a^15*b^15*c^28*d^5 + 815360*a^16*b^
14*c^27*d^6 - 1886976*a^17*b^13*c^26*d^7 + 3331328*a^18*b^12*c^25*d^8 - 4576000*a^19*b^11*c^24*d^9 + 4942080*a
^20*b^10*c^23*d^10 - 4209920*a^21*b^9*c^22*d^11 + 2818816*a^22*b^8*c^21*d^12 - 1467648*a^23*b^7*c^20*d^13 + 58
2400*a^24*b^6*c^19*d^14 - 170240*a^25*b^5*c^18*d^15 + 34560*a^26*b^4*c^17*d^16 - 4352*a^27*b^3*c^16*d^17 + 256
*a^28*b^2*c^15*d^18))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 -
35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6))))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21
*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)))*(9*a*d - 4*
b*c)*1i)/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c
^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)))/(256*a^4*b^20*c^23*d^3 - 2944*a^5*b^19*c^22*d^4 + 16048*a^6*b^1
8*c^21*d^5 - 55160*a^7*b^17*c^20*d^6 + 130000*a^8*b^16*c^19*d^7 - 206112*a^9*b^15*c^18*d^8 + 182808*a^10*b^14*
c^17*d^9 + 23664*a^11*b^13*c^16*d^10 - 332160*a^12*b^12*c^15*d^11 + 519200*a^13*b^11*c^14*d^12 - 460544*a^14*b
^10*c^13*d^13 + 260936*a^15*b^9*c^12*d^14 - 93712*a^16*b^8*c^11*d^15 + 19520*a^17*b^7*c^10*d^16 - 1800*a^18*b^
6*c^9*d^17 - ((-b^7*(a*d - b*c)^7)^(1/2)*((c + d*x^2)^(1/2)*(512*a^6*b^20*c^26*d^2 - 6656*a^7*b^19*c^25*d^3 +
38560*a^8*b^18*c^24*d^4 - 129920*a^9*b^17*c^23*d^5 + 275920*a^10*b^16*c^22*d^6 - 363440*a^11*b^15*c^21*d^7 + 2
35312*a^12*b^14*c^20*d^8 + 85360*a^13*b^13*c^19*d^9 - 316400*a^14*b^12*c^18*d^10 + 205840*a^15*b^11*c^17*d^11
+ 152384*a^16*b^10*c^16*d^12 - 430816*a^17*b^9*c^15*d^13 + 444080*a^18*b^8*c^14*d^14 - 281680*a^19*b^7*c^13*d^
15 + 118640*a^20*b^6*c^12*d^16 - 32656*a^21*b^5*c^11*d^17 + 5360*a^22*b^4*c^10*d^18 - 400*a^23*b^3*c^9*d^19) +
((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c)*(128*a^10*b^18*c^28*d^3 - 1792*a^11*b^17*c^27*d^4 + 10624*a^12*b^
16*c^26*d^5 - 33280*a^13*b^15*c^25*d^6 + 47936*a^14*b^14*c^24*d^7 + 40448*a^15*b^13*c^23*d^8 - 368896*a^16*b^1
2*c^22*d^9 + 948992*a^17*b^11*c^21*d^10 - 1531200*a^18*b^10*c^20*d^11 + 1754368*a^19*b^9*c^19*d^12 - 1485440*a
^20*b^8*c^18*d^13 + 939008*a^21*b^7*c^17*d^14 - 439616*a^22*b^6*c^16*d^15 + 148480*a^23*b^5*c^15*d^16 - 34304*
a^24*b^4*c^14*d^17 + 4864*a^25*b^3*c^13*d^18 - 320*a^26*b^2*c^12*d^19 - ((-b^7*(a*d - b*c)^7)^(1/2)*(c + d*x^2
)^(1/2)*(9*a*d - 4*b*c)*(512*a^12*b^18*c^31*d^2 - 7936*a^13*b^17*c^30*d^3 + 57600*a^14*b^16*c^29*d^4 - 259840*
a^15*b^15*c^28*d^5 + 815360*a^16*b^14*c^27*d^6 - 1886976*a^17*b^13*c^26*d^7 + 3331328*a^18*b^12*c^25*d^8 - 457
6000*a^19*b^11*c^24*d^9 + 4942080*a^20*b^10*c^23*d^10 - 4209920*a^21*b^9*c^22*d^11 + 2818816*a^22*b^8*c^21*d^1
2 - 1467648*a^23*b^7*c^20*d^13 + 582400*a^24*b^6*c^19*d^14 - 170240*a^25*b^5*c^18*d^15 + 34560*a^26*b^4*c^17*d
^16 - 4352*a^27*b^3*c^16*d^17 + 256*a^28*b^2*c^15*d^18))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5
*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6))))/(4*(a^10*d^7 -
a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2
*d^5 - 7*a^9*b*c*d^6)))*(9*a*d - 4*b*c))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 3
5*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)) + ((-b^7*(a*d - b*c)^7)^(1/2)*((
c + d*x^2)^(1/2)*(512*a^6*b^20*c^26*d^2 - 6656*a^7*b^19*c^25*d^3 + 38560*a^8*b^18*c^24*d^4 - 129920*a^9*b^17*c
^23*d^5 + 275920*a^10*b^16*c^22*d^6 - 363440*a^11*b^15*c^21*d^7 + 235312*a^12*b^14*c^20*d^8 + 85360*a^13*b^13*
c^19*d^9 - 316400*a^14*b^12*c^18*d^10 + 205840*a^15*b^11*c^17*d^11 + 152384*a^16*b^10*c^16*d^12 - 430816*a^17*
b^9*c^15*d^13 + 444080*a^18*b^8*c^14*d^14 - 281680*a^19*b^7*c^13*d^15 + 118640*a^20*b^6*c^12*d^16 - 32656*a^21
*b^5*c^11*d^17 + 5360*a^22*b^4*c^10*d^18 - 400*a^23*b^3*c^9*d^19) - ((-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c
)*(128*a^10*b^18*c^28*d^3 - 1792*a^11*b^17*c^27*d^4 + 10624*a^12*b^16*c^26*d^5 - 33280*a^13*b^15*c^25*d^6 + 47
936*a^14*b^14*c^24*d^7 + 40448*a^15*b^13*c^23*d^8 - 368896*a^16*b^12*c^22*d^9 + 948992*a^17*b^11*c^21*d^10 - 1
531200*a^18*b^10*c^20*d^11 + 1754368*a^19*b^9*c^19*d^12 - 1485440*a^20*b^8*c^18*d^13 + 939008*a^21*b^7*c^17*d^
14 - 439616*a^22*b^6*c^16*d^15 + 148480*a^23*b^5*c^15*d^16 - 34304*a^24*b^4*c^14*d^17 + 4864*a^25*b^3*c^13*d^1
8 - 320*a^26*b^2*c^12*d^19 + ((-b^7*(a*d - b*c)^7)^(1/2)*(c + d*x^2)^(1/2)*(9*a*d - 4*b*c)*(512*a^12*b^18*c^31
*d^2 - 7936*a^13*b^17*c^30*d^3 + 57600*a^14*b^16*c^29*d^4 - 259840*a^15*b^15*c^28*d^5 + 815360*a^16*b^14*c^27*
d^6 - 1886976*a^17*b^13*c^26*d^7 + 3331328*a^18*b^12*c^25*d^8 - 4576000*a^19*b^11*c^24*d^9 + 4942080*a^20*b^10
*c^23*d^10 - 4209920*a^21*b^9*c^22*d^11 + 2818816*a^22*b^8*c^21*d^12 - 1467648*a^23*b^7*c^20*d^13 + 582400*a^2
4*b^6*c^19*d^14 - 170240*a^25*b^5*c^18*d^15 + 34560*a^26*b^4*c^17*d^16 - 4352*a^27*b^3*c^16*d^17 + 256*a^28*b^
2*c^15*d^18))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*
b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6))))/(4*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5
*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6)))*(9*a*d - 4*b*c))/(4
*(a^10*d^7 - a^3*b^7*c^7 + 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21
*a^8*b^2*c^2*d^5 - 7*a^9*b*c*d^6))))*(-b^7*(a*d - b*c)^7)^(1/2)*(9*a*d - 4*b*c)*1i)/(2*(a^10*d^7 - a^3*b^7*c^7
+ 7*a^4*b^6*c^6*d - 21*a^5*b^5*c^5*d^2 + 35*a^6*b^4*c^4*d^3 - 35*a^7*b^3*c^3*d^4 + 21*a^8*b^2*c^2*d^5 - 7*a^9
*b*c*d^6))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**(5/2),x)
[Out]
Integral(1/(x**3*(a + b*x**2)**2*(c + d*x**2)**(5/2)), x)
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