3.774 \(\int \frac {1}{x^3 (a+b x^2)^2 (c+d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=241 \[ -\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{5/2}}+\frac {(3 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{5/2}}-\frac {d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{2 a^2 c^2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \]

[Out]

1/2*(3*a*d+4*b*c)*arctanh((d*x^2+c)^(1/2)/c^(1/2))/a^3/c^(5/2)-1/2*b^(5/2)*(-7*a*d+4*b*c)*arctanh(b^(1/2)*(d*x
^2+c)^(1/2)/(-a*d+b*c)^(1/2))/a^3/(-a*d+b*c)^(5/2)-1/2*d*(3*a^2*d^2-2*a*b*c*d+2*b^2*c^2)/a^2/c^2/(-a*d+b*c)^2/
(d*x^2+c)^(1/2)-1/2*b*(-a*d+2*b*c)/a^2/c/(-a*d+b*c)/(b*x^2+a)/(d*x^2+c)^(1/2)-1/2/a/c/x^2/(b*x^2+a)/(d*x^2+c)^
(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.34, 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} \[ -\frac {d \left (3 a^2 d^2-2 a b c d+2 b^2 c^2\right )}{2 a^2 c^2 \sqrt {c+d x^2} (b c-a d)^2}-\frac {b^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{5/2}}+\frac {(3 a d+4 b c) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^3 c^{5/2}}-\frac {b (2 b c-a d)}{2 a^2 c \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]

[Out]

-(d*(2*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2))/(2*a^2*c^2*(b*c - a*d)^2*Sqrt[c + d*x^2]) - (b*(2*b*c - a*d))/(2*a^2*
c*(b*c - a*d)*(a + b*x^2)*Sqrt[c + d*x^2]) - 1/(2*a*c*x^2*(a + b*x^2)*Sqrt[c + d*x^2]) + ((4*b*c + 3*a*d)*ArcT
anh[Sqrt[c + d*x^2]/Sqrt[c]])/(2*a^3*c^(5/2)) - (b^(5/2)*(4*b*c - 7*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqr
t[b*c - a*d]])/(2*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^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\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^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 ) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}-\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^2\right )}{2 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 )}{2 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {\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^2\right )}{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 )}{2 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\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^2\right )}{4 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^2\right )}{4 a^3 c^2}\\ &=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{2 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\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^2}\right )}{2 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^2}\right )}{2 a^3 c^2 d}\\ &=-\frac {d \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right )}{2 a^2 c^2 (b c-a d)^2 \sqrt {c+d x^2}}-\frac {b (2 b c-a d)}{2 a^2 c (b c-a d) \left (a+b x^2\right ) \sqrt {c+d x^2}}-\frac {1}{2 a c x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}}+\frac {(4 b c+3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 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^2}}{\sqrt {b c-a d}}\right )}{2 a^3 (b c-a d)^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.13, size = 189, normalized size = 0.78 \[ \frac {b^2 c^2 x^2 \left (a+b x^2\right ) (4 b c-7 a d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \left (d x^2+c\right )}{b c-a d}\right )-(a d-b c) \left (x^2 \left (a+b x^2\right ) \left (3 a^2 d^2+a b c d-4 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d x^2}{c}+1\right )+a c \left (a^2 d+a b \left (d x^2-c\right )-2 b^2 c x^2\right )\right )}{2 a^3 c^2 x^2 \left (a+b x^2\right ) \sqrt {c+d x^2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x]

[Out]

(b^2*c^2*(4*b*c - 7*a*d)*x^2*(a + b*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, (b*(c + d*x^2))/(b*c - a*d)] - (-(b*c
) + a*d)*(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 + 3*a^2*d^2)*x^2*(a + b*x^2)*Hy
pergeometric2F1[-1/2, 1, 1/2, 1 + (d*x^2)/c]))/(2*a^3*c^2*(b*c - a*d)^2*x^2*(a + b*x^2)*Sqrt[c + d*x^2])

________________________________________________________________________________________

fricas [B]  time = 9.22, size = 2554, normalized size = 10.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

[-1/8*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^6 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^4 + (4*a*b^3*c
^5 - 7*a^2*b^2*c^4*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)) - 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
^6 + (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^4 + (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^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 4*(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^4 + (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^2)*sqrt(d*x^2 + 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^6 + (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^4 + (a^4*b^2*c^6 - 2*a^5*b*c
^5*d + a^6*c^4*d^2)*x^2), -1/8*(4*((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^6 + (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^4 + (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^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) + ((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x
^6 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^2*b^2*c^3*d^2)*x^4 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*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*(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^4 + (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^2)*sqrt(d*x^2 + 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^6 + (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^4 + (a^4*b^2*c^6 - 2*
a^5*b*c^5*d + a^6*c^4*d^2)*x^2), 1/4*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^6 + (4*b^4*c^5 - 3*a*b^3*c^4*d - 7*a^
2*b^2*c^3*d^2)*x^4 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*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)) + ((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^6 + (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^4 + (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^2)*sqrt(c)*log(-(d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(c) + 2*
c)/x^2) - 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
^4 + (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^2)*sqrt(d*x^2 + 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^6 + (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^4 + (a^4*b^2
*c^6 - 2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2), 1/4*(((4*b^4*c^4*d - 7*a*b^3*c^3*d^2)*x^6 + (4*b^4*c^5 - 3*a*b^3*c^4
*d - 7*a^2*b^2*c^3*d^2)*x^4 + (4*a*b^3*c^5 - 7*a^2*b^2*c^4*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)) - 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^6 + (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^4 +
(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^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c))
- 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^4 + (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^2)*sqrt(d*x^2 + 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^6 + (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^4 + (a^4*b^2*c^6 -
2*a^5*b*c^5*d + a^6*c^4*d^2)*x^2)]

________________________________________________________________________________________

giac [A]  time = 0.36, size = 367, normalized size = 1.52 \[ \frac {{\left (4 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{2} + c\right )}^{2} b^{3} c^{2} d - 2 \, {\left (d x^{2} + c\right )} b^{3} c^{3} d - 2 \, {\left (d x^{2} + c\right )}^{2} a b^{2} c d^{2} + 3 \, {\left (d x^{2} + c\right )} a b^{2} c^{2} d^{2} + 3 \, {\left (d x^{2} + c\right )}^{2} a^{2} b d^{3} - 7 \, {\left (d x^{2} + c\right )} a^{2} b c d^{3} + 2 \, a^{2} b c^{2} d^{3} + 3 \, {\left (d x^{2} + c\right )} a^{3} d^{4} - 2 \, a^{3} c d^{4}}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2}\right )} {\left ({\left (d x^{2} + c\right )}^{\frac {5}{2}} b - 2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b c + \sqrt {d x^{2} + c} b c^{2} + {\left (d x^{2} + c\right )}^{\frac {3}{2}} a d - \sqrt {d x^{2} + c} a c d\right )}} - \frac {{\left (4 \, b c + 3 \, a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, a^{3} \sqrt {-c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

1/2*(4*b^4*c - 7*a*b^3*d)*arctan(sqrt(d*x^2 + 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)) - 1/2*(2*(d*x^2 + c)^2*b^3*c^2*d - 2*(d*x^2 + c)*b^3*c^3*d - 2*(d*x^2 + c)^2*a*b^2*c*d
^2 + 3*(d*x^2 + c)*a*b^2*c^2*d^2 + 3*(d*x^2 + c)^2*a^2*b*d^3 - 7*(d*x^2 + c)*a^2*b*c*d^3 + 2*a^2*b*c^2*d^3 + 3
*(d*x^2 + 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^2 + c)^(5/2)*b - 2*(d*x
^2 + c)^(3/2)*b*c + sqrt(d*x^2 + c)*b*c^2 + (d*x^2 + c)^(3/2)*a*d - sqrt(d*x^2 + c)*a*c*d)) - 1/2*(4*b*c + 3*a
*d)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a^3*sqrt(-c)*c^2)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 1778, normalized size = 7.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(3/2),x)

[Out]

-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)^(1/2)-b/a^3*(-a*
b)^(1/2)/(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)^(1/2)*d*x+b^2/
a^3/(a*d-b*c)/(-(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)^(1/2)-3/4*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)^(1/2)+3/4*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)^(1/2)*x+3/4*b^2/a^2*d/(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))-1/2*b^2/a^2/(-a*b)^(1/2)/(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)^(1/2)*d*x-1/2/a^2/c/x^2/(d*x^2+c)^(1/2)-3/2/a^2*d/c^2/(d*x^2+c)^(1
/2)+3/2/a^2*d/c^(5/2)*ln((2*c+2*(d*x^2+c)^(1/2)*c^(1/2))/x)-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)^(1/2)+b/a^3*(-a*b)^(1/2)/(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)^(1/2)*d*x+b^2/a^3/(a*d-b*c)/(-(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)^(1/2)-3/4*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)^(1/2)-3/4*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)^(1/2)*x+3/4*b^2/a^2*d/(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))+1/2*b^2/a^2
/(-a*b)^(1/2)/(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)^(1/2)*d*x
-2*b/a^3/c/(d*x^2+c)^(1/2)+2*b/a^3/c^(3/2)*ln((2*c+2*(d*x^2+c)^(1/2)*c^(1/2))/x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^2+a)^2/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^2*(d*x^2 + c)^(3/2)*x^3), x)

________________________________________________________________________________________

mupad [B]  time = 4.90, size = 4286, normalized size = 17.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^2)^2*(c + d*x^2)^(3/2)),x)

[Out]

(atan((((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*((c + d*x^2)^(1/2)*(512*a^6*b^15*c^18*d^2 - 4608*a^7*b^14*c
^17*d^3 + 17824*a^8*b^13*c^16*d^4 - 38144*a^9*b^12*c^15*d^5 + 47680*a^10*b^11*c^14*d^6 - 31808*a^11*b^10*c^13*
d^7 + 4624*a^12*b^9*c^12*d^8 + 8032*a^13*b^8*c^11*d^9 - 3536*a^14*b^7*c^10*d^10 - 2560*a^15*b^6*c^9*d^11 + 289
6*a^16*b^5*c^8*d^12 - 1056*a^17*b^4*c^7*d^13 + 144*a^18*b^3*c^6*d^14) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4
*b*c)*(128*a^10*b^13*c^19*d^3 - 1216*a^11*b^12*c^18*d^4 + 4800*a^12*b^11*c^17*d^5 - 9792*a^13*b^10*c^16*d^6 +
9216*a^14*b^9*c^15*d^7 + 2688*a^15*b^8*c^14*d^8 - 18816*a^16*b^7*c^13*d^9 + 24960*a^17*b^6*c^12*d^10 - 18048*a
^18*b^5*c^11*d^11 + 7744*a^19*b^4*c^10*d^12 - 1856*a^20*b^3*c^9*d^13 + 192*a^21*b^2*c^8*d^14 - ((-b^5*(a*d - b
*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 4*b*c)*(512*a^12*b^13*c^21*d^2 - 5376*a^13*b^12*c^20*d^3 + 25600*a^14*
b^11*c^19*d^4 - 72960*a^15*b^10*c^18*d^5 + 138240*a^16*b^9*c^17*d^6 - 182784*a^17*b^8*c^16*d^7 + 172032*a^18*b
^7*c^15*d^8 - 115200*a^19*b^6*c^14*d^9 + 53760*a^20*b^5*c^13*d^10 - 16640*a^21*b^4*c^12*d^11 + 3072*a^22*b^3*c
^11*d^12 - 256*a^23*b^2*c^10*d^13))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*
b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*
c^2*d^3 - 5*a^7*b*c*d^4)))*1i)/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c
^2*d^3 - 5*a^7*b*c*d^4)) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*((c + d*x^2)^(1/2)*(512*a^6*b^15*c^18*d
^2 - 4608*a^7*b^14*c^17*d^3 + 17824*a^8*b^13*c^16*d^4 - 38144*a^9*b^12*c^15*d^5 + 47680*a^10*b^11*c^14*d^6 - 3
1808*a^11*b^10*c^13*d^7 + 4624*a^12*b^9*c^12*d^8 + 8032*a^13*b^8*c^11*d^9 - 3536*a^14*b^7*c^10*d^10 - 2560*a^1
5*b^6*c^9*d^11 + 2896*a^16*b^5*c^8*d^12 - 1056*a^17*b^4*c^7*d^13 + 144*a^18*b^3*c^6*d^14) - ((-b^5*(a*d - b*c)
^5)^(1/2)*(7*a*d - 4*b*c)*(128*a^10*b^13*c^19*d^3 - 1216*a^11*b^12*c^18*d^4 + 4800*a^12*b^11*c^17*d^5 - 9792*a
^13*b^10*c^16*d^6 + 9216*a^14*b^9*c^15*d^7 + 2688*a^15*b^8*c^14*d^8 - 18816*a^16*b^7*c^13*d^9 + 24960*a^17*b^6
*c^12*d^10 - 18048*a^18*b^5*c^11*d^11 + 7744*a^19*b^4*c^10*d^12 - 1856*a^20*b^3*c^9*d^13 + 192*a^21*b^2*c^8*d^
14 + ((-b^5*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 4*b*c)*(512*a^12*b^13*c^21*d^2 - 5376*a^13*b^12*c^
20*d^3 + 25600*a^14*b^11*c^19*d^4 - 72960*a^15*b^10*c^18*d^5 + 138240*a^16*b^9*c^17*d^6 - 182784*a^17*b^8*c^16
*d^7 + 172032*a^18*b^7*c^15*d^8 - 115200*a^19*b^6*c^14*d^9 + 53760*a^20*b^5*c^13*d^10 - 16640*a^21*b^4*c^12*d^
11 + 3072*a^22*b^3*c^11*d^12 - 256*a^23*b^2*c^10*d^13))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b
^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c
^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)))*1i)/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^
3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)))/(((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*((c + d*x^2)^(1/2)*
(512*a^6*b^15*c^18*d^2 - 4608*a^7*b^14*c^17*d^3 + 17824*a^8*b^13*c^16*d^4 - 38144*a^9*b^12*c^15*d^5 + 47680*a^
10*b^11*c^14*d^6 - 31808*a^11*b^10*c^13*d^7 + 4624*a^12*b^9*c^12*d^8 + 8032*a^13*b^8*c^11*d^9 - 3536*a^14*b^7*
c^10*d^10 - 2560*a^15*b^6*c^9*d^11 + 2896*a^16*b^5*c^8*d^12 - 1056*a^17*b^4*c^7*d^13 + 144*a^18*b^3*c^6*d^14)
- ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(128*a^10*b^13*c^19*d^3 - 1216*a^11*b^12*c^18*d^4 + 4800*a^12*b^
11*c^17*d^5 - 9792*a^13*b^10*c^16*d^6 + 9216*a^14*b^9*c^15*d^7 + 2688*a^15*b^8*c^14*d^8 - 18816*a^16*b^7*c^13*
d^9 + 24960*a^17*b^6*c^12*d^10 - 18048*a^18*b^5*c^11*d^11 + 7744*a^19*b^4*c^10*d^12 - 1856*a^20*b^3*c^9*d^13 +
192*a^21*b^2*c^8*d^14 + ((-b^5*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 4*b*c)*(512*a^12*b^13*c^21*d^2
- 5376*a^13*b^12*c^20*d^3 + 25600*a^14*b^11*c^19*d^4 - 72960*a^15*b^10*c^18*d^5 + 138240*a^16*b^9*c^17*d^6 -
182784*a^17*b^8*c^16*d^7 + 172032*a^18*b^7*c^15*d^8 - 115200*a^19*b^6*c^14*d^9 + 53760*a^20*b^5*c^13*d^10 - 16
640*a^21*b^4*c^12*d^11 + 3072*a^22*b^3*c^11*d^12 - 256*a^23*b^2*c^10*d^13))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*
b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*
c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^4*c^4*
d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) - ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*((
c + d*x^2)^(1/2)*(512*a^6*b^15*c^18*d^2 - 4608*a^7*b^14*c^17*d^3 + 17824*a^8*b^13*c^16*d^4 - 38144*a^9*b^12*c^
15*d^5 + 47680*a^10*b^11*c^14*d^6 - 31808*a^11*b^10*c^13*d^7 + 4624*a^12*b^9*c^12*d^8 + 8032*a^13*b^8*c^11*d^9
- 3536*a^14*b^7*c^10*d^10 - 2560*a^15*b^6*c^9*d^11 + 2896*a^16*b^5*c^8*d^12 - 1056*a^17*b^4*c^7*d^13 + 144*a^
18*b^3*c^6*d^14) + ((-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*(128*a^10*b^13*c^19*d^3 - 1216*a^11*b^12*c^18*d
^4 + 4800*a^12*b^11*c^17*d^5 - 9792*a^13*b^10*c^16*d^6 + 9216*a^14*b^9*c^15*d^7 + 2688*a^15*b^8*c^14*d^8 - 188
16*a^16*b^7*c^13*d^9 + 24960*a^17*b^6*c^12*d^10 - 18048*a^18*b^5*c^11*d^11 + 7744*a^19*b^4*c^10*d^12 - 1856*a^
20*b^3*c^9*d^13 + 192*a^21*b^2*c^8*d^14 - ((-b^5*(a*d - b*c)^5)^(1/2)*(c + d*x^2)^(1/2)*(7*a*d - 4*b*c)*(512*a
^12*b^13*c^21*d^2 - 5376*a^13*b^12*c^20*d^3 + 25600*a^14*b^11*c^19*d^4 - 72960*a^15*b^10*c^18*d^5 + 138240*a^1
6*b^9*c^17*d^6 - 182784*a^17*b^8*c^16*d^7 + 172032*a^18*b^7*c^15*d^8 - 115200*a^19*b^6*c^14*d^9 + 53760*a^20*b
^5*c^13*d^10 - 16640*a^21*b^4*c^12*d^11 + 3072*a^22*b^3*c^11*d^12 - 256*a^23*b^2*c^10*d^13))/(4*(a^8*d^5 - a^3
*b^5*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^3*b^5
*c^5 + 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4))))/(4*(a^8*d^5 - a^3*b^5*c^5
+ 5*a^4*b^4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) + 256*a^4*b^15*c^16*d^3 - 2048*
a^5*b^14*c^15*d^4 + 7216*a^6*b^13*c^14*d^5 - 14672*a^7*b^12*c^13*d^6 + 18424*a^8*b^11*c^12*d^7 - 12992*a^9*b^1
0*c^11*d^8 + 1288*a^10*b^9*c^10*d^9 + 7024*a^11*b^8*c^9*d^10 - 6968*a^12*b^7*c^8*d^11 + 2976*a^13*b^6*c^7*d^12
- 504*a^14*b^5*c^6*d^13))*(-b^5*(a*d - b*c)^5)^(1/2)*(7*a*d - 4*b*c)*1i)/(2*(a^8*d^5 - a^3*b^5*c^5 + 5*a^4*b^
4*c^4*d - 10*a^5*b^3*c^3*d^2 + 10*a^6*b^2*c^2*d^3 - 5*a^7*b*c*d^4)) - (d^3/(b*c^2 - a*c*d) + (d*(c + d*x^2)^2*
(2*b^3*c^2 + 3*a^2*b*d^2 - 2*a*b^2*c*d))/(2*a^2*(b*c^2 - a*c*d)^2) + (d*(c + d*x^2)*(a*d - 2*b*c)*(3*a^2*d^2 +
b^2*c^2 - a*b*c*d))/(2*a^2*(b*c^2 - a*c*d)^2))/(b*(c + d*x^2)^(5/2) + (c + d*x^2)^(1/2)*(b*c^2 - a*c*d) + (c
+ d*x^2)^(3/2)*(a*d - 2*b*c)) - (atan((a^3*b^11*c^21*d^3*(c + d*x^2)^(1/2)*140i - a^14*c^10*d^14*(c + d*x^2)^(
1/2)*27i - a^4*b^10*c^20*d^4*(c + d*x^2)^(1/2)*1015i + a^5*b^9*c^19*d^5*(c + d*x^2)^(1/2)*2996i - a^6*b^8*c^18
*d^6*(c + d*x^2)^(1/2)*4375i + a^7*b^7*c^17*d^7*(c + d*x^2)^(1/2)*2561i + a^8*b^6*c^16*d^8*(c + d*x^2)^(1/2)*1
316i - a^9*b^5*c^15*d^9*(c + d*x^2)^(1/2)*3073i + a^10*b^4*c^14*d^10*(c + d*x^2)^(1/2)*1694i + a^11*b^3*c^13*d
^11*(c + d*x^2)^(1/2)*35i - a^12*b^2*c^12*d^12*(c + d*x^2)^(1/2)*441i + a^13*b*c^11*d^13*(c + d*x^2)^(1/2)*189
i)/(c^5*(c^5)^(1/2)*(c^5*(c^5*(2561*a^7*b^7*d^7 - 4375*a^6*b^8*c*d^6 + 140*a^3*b^11*c^4*d^3 - 1015*a^4*b^10*c^
3*d^4 + 2996*a^5*b^9*c^2*d^5) - 441*a^12*b^2*d^12 + 35*a^11*b^3*c*d^11 + 1316*a^8*b^6*c^4*d^8 - 3073*a^9*b^5*c
^3*d^9 + 1694*a^10*b^4*c^2*d^10) - 27*a^14*c^3*d^14 + 189*a^13*b*c^4*d^13)))*(3*a*d + 4*b*c)*1i)/(2*a^3*(c^5)^
(1/2))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**2+a)**2/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/(x**3*(a + b*x**2)**2*(c + d*x**2)**(3/2)), x)

________________________________________________________________________________________