3.42 \(\int \frac{1}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^3} \, dx\)

Optimal. Leaf size=315 \[ \frac{3 d x (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{8 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}+\frac{d x \left (-2 a^2 d^2-13 a b c d+3 b^2 c^2\right )}{8 a^2 c \left (c+d x^2\right )^2 (b c-a d)^3}-\frac{3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^5}+\frac{b x (3 b c-11 a d)}{8 a^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{3 b^{5/2} \left (21 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^5}+\frac{b x}{4 a \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

(d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2)
+ (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/
(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c + a*d)*(b^2*c^2 - 6*
a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^
2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^
5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/
(8*c^(5/2)*(b*c - a*d)^5)

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Rubi [A]  time = 1.03743, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 d x (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right )}{8 a^2 c^2 \left (c+d x^2\right ) (b c-a d)^4}+\frac{d x \left (-2 a^2 d^2-13 a b c d+3 b^2 c^2\right )}{8 a^2 c \left (c+d x^2\right )^2 (b c-a d)^3}-\frac{3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 c^{5/2} (b c-a d)^5}+\frac{b x (3 b c-11 a d)}{8 a^2 \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{3 b^{5/2} \left (21 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} (b c-a d)^5}+\frac{b x}{4 a \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(3*b^2*c^2 - 13*a*b*c*d - 2*a^2*d^2)*x)/(8*a^2*c*(b*c - a*d)^3*(c + d*x^2)^2)
+ (b*x)/(4*a*(b*c - a*d)*(a + b*x^2)^2*(c + d*x^2)^2) + (b*(3*b*c - 11*a*d)*x)/
(8*a^2*(b*c - a*d)^2*(a + b*x^2)*(c + d*x^2)^2) + (3*d*(b*c + a*d)*(b^2*c^2 - 6*
a*b*c*d + a^2*d^2)*x)/(8*a^2*c^2*(b*c - a*d)^4*(c + d*x^2)) + (3*b^(5/2)*(b^2*c^
2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*(b*c - a*d)^
5) - (3*d^(5/2)*(21*b^2*c^2 - 6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/
(8*c^(5/2)*(b*c - a*d)^5)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 1.71351, size = 233, normalized size = 0.74 \[ \frac{1}{8} \left (\frac{x (b c-a d) \left (\frac{3 b^4 c}{a^2 \left (a+b x^2\right )}+\frac{b^3 \left (-17 a d+2 b c-15 b d x^2\right )}{a \left (a+b x^2\right )^2}-\frac{d^3 \left (-2 a d+17 b c+15 b d x^2\right )}{c \left (c+d x^2\right )^2}+\frac{3 a d^4}{c^2 \left (c+d x^2\right )}\right )-\frac{3 d^{5/2} \left (a^2 d^2-6 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2}}}{(b c-a d)^5}-\frac{3 b^{5/2} \left (21 a^2 d^2-6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} (a d-b c)^5}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-3*b^(5/2)*(b^2*c^2 - 6*a*b*c*d + 21*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^
(5/2)*(-(b*c) + a*d)^5) + ((b*c - a*d)*x*((3*b^4*c)/(a^2*(a + b*x^2)) + (3*a*d^4
)/(c^2*(c + d*x^2)) + (b^3*(2*b*c - 17*a*d - 15*b*d*x^2))/(a*(a + b*x^2)^2) - (d
^3*(17*b*c - 2*a*d + 15*b*d*x^2))/(c*(c + d*x^2)^2)) - (3*d^(5/2)*(21*b^2*c^2 -
6*a*b*c*d + a^2*d^2)*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/c^(5/2))/(b*c - a*d)^5)/8

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Maple [A]  time = 0.024, size = 568, normalized size = 1.8 \[{\frac{3\,{d}^{6}{x}^{3}{a}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}{c}^{2}}}-{\frac{9\,{d}^{5}{x}^{3}ab}{4\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}c}}+{\frac{15\,{d}^{4}{x}^{3}{b}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,{d}^{5}x{a}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}c}}-{\frac{11\,{d}^{4}xab}{4\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{17\,{d}^{3}cx{b}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{d}^{5}{a}^{2}}{8\, \left ( ad-bc \right ) ^{5}{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{9\,a{d}^{4}b}{4\, \left ( ad-bc \right ) ^{5}c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{63\,{d}^{3}{b}^{2}}{8\, \left ( ad-bc \right ) ^{5}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{15\,{b}^{4}{x}^{3}{d}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,{b}^{5}{x}^{3}cd}{4\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{3\,{b}^{6}{x}^{3}{c}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}-{\frac{17\,{b}^{3}ax{d}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{b}^{4}xcd}{4\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{b}^{5}x{c}^{2}}{8\, \left ( ad-bc \right ) ^{5} \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{63\,{d}^{2}{b}^{3}}{8\, \left ( ad-bc \right ) ^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{b}^{4}cd}{4\, \left ( ad-bc \right ) ^{5}a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,{b}^{5}{c}^{2}}{8\, \left ( ad-bc \right ) ^{5}{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*d^6/(a*d-b*c)^5/(d*x^2+c)^2/c^2*x^3*a^2-9/4*d^5/(a*d-b*c)^5/(d*x^2+c)^2/c*x^
3*a*b+15/8*d^4/(a*d-b*c)^5/(d*x^2+c)^2*x^3*b^2+5/8*d^5/(a*d-b*c)^5/(d*x^2+c)^2/c
*x*a^2-11/4*d^4/(a*d-b*c)^5/(d*x^2+c)^2*x*a*b+17/8*d^3/(a*d-b*c)^5/(d*x^2+c)^2*c
*x*b^2+3/8*d^5/(a*d-b*c)^5/c^2/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a^2-9/4*d^4/(
a*d-b*c)^5/c/(c*d)^(1/2)*arctan(x*d/(c*d)^(1/2))*a*b+63/8*d^3/(a*d-b*c)^5/(c*d)^
(1/2)*arctan(x*d/(c*d)^(1/2))*b^2-15/8*b^4/(a*d-b*c)^5/(b*x^2+a)^2*x^3*d^2+9/4*b
^5/(a*d-b*c)^5/(b*x^2+a)^2/a*x^3*c*d-3/8*b^6/(a*d-b*c)^5/(b*x^2+a)^2/a^2*x^3*c^2
-17/8*b^3/(a*d-b*c)^5/(b*x^2+a)^2*a*x*d^2+11/4*b^4/(a*d-b*c)^5/(b*x^2+a)^2*x*c*d
-5/8*b^5/(a*d-b*c)^5/(b*x^2+a)^2/a*x*c^2-63/8*b^3/(a*d-b*c)^5/(a*b)^(1/2)*arctan
(x*b/(a*b)^(1/2))*d^2+9/4*b^4/(a*d-b*c)^5/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*
c*d-3/8*b^5/(a*d-b*c)^5/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 13.6178, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(6*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2
*(6*b^6*c^5*d - 31*a*b^5*c^4*d^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^
4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^6*c^6 - 8*a*b^5*c^5*d - 29*a^2*b^4*c^4*d
^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 - 3*(a^2*b^4*c^6 - 6*a^
3*b^3*c^5*d + 21*a^4*b^2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c
^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c
^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3
+ 21*a^4*b^2*c^2*d^4)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2
+ 21*a^4*b^2*c^3*d^3)*x^2)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2
+ a)) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*
d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^
2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*
d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 +
15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt(-d/c)*log((d*x^2 +
2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*
b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c
^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*d^3 + 5*a^8*b*c^5*d^4
- a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*
a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d -
4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 -
a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d - 9*a^4*b^5*c^7*d^2 + 25*a^5*b
^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^
7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^
4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2), 1/16*(6*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3
+ 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a*b^5*c^4*d^2 - 9*a^2
*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d^6)*x^5 + 2*(3*b^
6*c^6 - 8*a*b^5*c^5*d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4 + 8*a^5*b*c*d^5
- 3*a^6*d^6)*x^3 - 6*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a
^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15
*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a
^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^
3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*d^5)*x^2)*sqrt(d/c)*arc
tan(d*x/(c*sqrt(d/c))) - 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^4*b^2*c^4*d^2 +
(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^5*d - 5*a*b
^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6 - 2*a*b^5*c
^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 + 21*a^4*b^2*c^2*d^4)*x^4 + 2*(a*b
^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x^2)*sqrt(-b
/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + 2*(5*a*b^5*c^6 - 22*a^2*b^
4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*d^4 - 5*a^6*c*d
^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*d^3
+ 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*a^4*
b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8 + 2*
(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5 + 4*a
^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d - 9*a^4*b^5*c
^7*d^2 + 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b*c^3
*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d^2 -
5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2), 1/16*(6*(b^6*c^4*d^2 -
6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + 2*(6*b^6*c^5*d - 31*a*b^
5*c^4*d^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d^5 - 6*a^5*b*d
^6)*x^5 + 2*(3*b^6*c^6 - 8*a*b^5*c^5*d - 29*a^2*b^4*c^4*d^2 + 29*a^4*b^2*c^2*d^4
+ 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + 6*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d + 21*a^4*b^
2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8 + 2*(b^6*c^
5*d - 5*a*b^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6 + (b^6*c^6
- 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 + 21*a^4*b^2*c^2*d^4)*x
^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^2*c^3*d^3)*x
^2)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 3*(21*a^4*b^2*c^4*d^2 - 6*a^5*b*c^3*d^
3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^8 + 2*(
21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^6)*x^6 + (21
*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*c*d^5 + a^6*
d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*d^4 + a^6*c*
d^5)*x^2)*sqrt(-d/c)*log((d*x^2 + 2*c*x*sqrt(-d/c) - c)/(d*x^2 + c)) + 2*(5*a*b^
5*c^6 - 22*a^2*b^4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^
2*d^4 - 5*a^6*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10
*a^7*b^2*c^6*d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*
c^6*d^3 + 10*a^4*b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*
c^2*d^7)*x^8 + 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 - 5*a^6*
b^3*c^4*d^5 + 4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^
8*d - 9*a^4*b^5*c^7*d^2 + 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^
4*d^5 + a^8*b*c^3*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*
a^5*b^4*c^7*d^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2), 1/8*(
3*(b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 6*a^3*b^3*c*d^5 - a^4*b^2*d^6)*x^7 + (6*b^6*c
^5*d - 31*a*b^5*c^4*d^2 - 9*a^2*b^4*c^3*d^3 + 9*a^3*b^3*c^2*d^4 + 31*a^4*b^2*c*d
^5 - 6*a^5*b*d^6)*x^5 + (3*b^6*c^6 - 8*a*b^5*c^5*d - 29*a^2*b^4*c^4*d^2 + 29*a^4
*b^2*c^2*d^4 + 8*a^5*b*c*d^5 - 3*a^6*d^6)*x^3 + 3*(a^2*b^4*c^6 - 6*a^3*b^3*c^5*d
+ 21*a^4*b^2*c^4*d^2 + (b^6*c^4*d^2 - 6*a*b^5*c^3*d^3 + 21*a^2*b^4*c^2*d^4)*x^8
+ 2*(b^6*c^5*d - 5*a*b^5*c^4*d^2 + 15*a^2*b^4*c^3*d^3 + 21*a^3*b^3*c^2*d^4)*x^6
+ (b^6*c^6 - 2*a*b^5*c^5*d - 2*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 + 21*a^4*b^
2*c^2*d^4)*x^4 + 2*(a*b^5*c^6 - 5*a^2*b^4*c^5*d + 15*a^3*b^3*c^4*d^2 + 21*a^4*b^
2*c^3*d^3)*x^2)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))) - 3*(21*a^4*b^2*c^4*d^2 - 6*
a^5*b*c^3*d^3 + a^6*c^2*d^4 + (21*a^2*b^4*c^2*d^4 - 6*a^3*b^3*c*d^5 + a^4*b^2*d^
6)*x^8 + 2*(21*a^2*b^4*c^3*d^3 + 15*a^3*b^3*c^2*d^4 - 5*a^4*b^2*c*d^5 + a^5*b*d^
6)*x^6 + (21*a^2*b^4*c^4*d^2 + 78*a^3*b^3*c^3*d^3 - 2*a^4*b^2*c^2*d^4 - 2*a^5*b*
c*d^5 + a^6*d^6)*x^4 + 2*(21*a^3*b^3*c^4*d^2 + 15*a^4*b^2*c^3*d^3 - 5*a^5*b*c^2*
d^4 + a^6*c*d^5)*x^2)*sqrt(d/c)*arctan(d*x/(c*sqrt(d/c))) + (5*a*b^5*c^6 - 22*a^
2*b^4*c^5*d + 17*a^3*b^3*c^4*d^2 - 17*a^4*b^2*c^3*d^3 + 22*a^5*b*c^2*d^4 - 5*a^6
*c*d^5)*x)/(a^4*b^5*c^9 - 5*a^5*b^4*c^8*d + 10*a^6*b^3*c^7*d^2 - 10*a^7*b^2*c^6*
d^3 + 5*a^8*b*c^5*d^4 - a^9*c^4*d^5 + (a^2*b^7*c^7*d^2 - 5*a^3*b^6*c^6*d^3 + 10*
a^4*b^5*c^5*d^4 - 10*a^5*b^4*c^4*d^5 + 5*a^6*b^3*c^3*d^6 - a^7*b^2*c^2*d^7)*x^8
+ 2*(a^2*b^7*c^8*d - 4*a^3*b^6*c^7*d^2 + 5*a^4*b^5*c^6*d^3 - 5*a^6*b^3*c^4*d^5 +
4*a^7*b^2*c^3*d^6 - a^8*b*c^2*d^7)*x^6 + (a^2*b^7*c^9 - a^3*b^6*c^8*d - 9*a^4*b
^5*c^7*d^2 + 25*a^5*b^4*c^6*d^3 - 25*a^6*b^3*c^5*d^4 + 9*a^7*b^2*c^4*d^5 + a^8*b
*c^3*d^6 - a^9*c^2*d^7)*x^4 + 2*(a^3*b^6*c^9 - 4*a^4*b^5*c^8*d + 5*a^5*b^4*c^7*d
^2 - 5*a^7*b^2*c^5*d^4 + 4*a^8*b*c^4*d^5 - a^9*c^3*d^6)*x^2)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.494045, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done