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

Optimal. Leaf size=419 \[ -\frac{b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}+\frac{2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}-\frac{2 b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} (b c-a d)^3}-\frac{d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac{2 d^{5/3} (4 b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^3}+\frac{b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}+\frac{d x (a d+b c)}{3 a c \left (c+d x^3\right ) (b c-a d)^2} \]

[Out]

(d*(b*c + a*d)*x)/(3*a*c*(b*c - a*d)^2*(c + d*x^3)) + (b*x)/(3*a*(b*c - a*d)*(a
+ b*x^3)*(c + d*x^3)) - (2*b^(5/3)*(b*c - 4*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/
(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(b*c - a*d)^3) - (2*d^(5/3)*(4*b*c - a*d)
*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(3*Sqrt[3]*c^(5/3)*(b*c - a*
d)^3) + (2*b^(5/3)*(b*c - 4*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*(b*c - a*d
)^3) + (2*d^(5/3)*(4*b*c - a*d)*Log[c^(1/3) + d^(1/3)*x])/(9*c^(5/3)*(b*c - a*d)
^3) - (b^(5/3)*(b*c - 4*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*
a^(5/3)*(b*c - a*d)^3) - (d^(5/3)*(4*b*c - a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x
+ d^(2/3)*x^2])/(9*c^(5/3)*(b*c - a*d)^3)

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Rubi [A]  time = 1.07782, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}+\frac{2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}-\frac{2 b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} (b c-a d)^3}-\frac{d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac{2 d^{5/3} (4 b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} (b c-a d)^3}+\frac{b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}+\frac{d x (a d+b c)}{3 a c \left (c+d x^3\right ) (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*c + a*d)*x)/(3*a*c*(b*c - a*d)^2*(c + d*x^3)) + (b*x)/(3*a*(b*c - a*d)*(a
+ b*x^3)*(c + d*x^3)) - (2*b^(5/3)*(b*c - 4*a*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/
(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(b*c - a*d)^3) - (2*d^(5/3)*(4*b*c - a*d)
*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(3*Sqrt[3]*c^(5/3)*(b*c - a*
d)^3) + (2*b^(5/3)*(b*c - 4*a*d)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(5/3)*(b*c - a*d
)^3) + (2*d^(5/3)*(4*b*c - a*d)*Log[c^(1/3) + d^(1/3)*x])/(9*c^(5/3)*(b*c - a*d)
^3) - (b^(5/3)*(b*c - 4*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(9*
a^(5/3)*(b*c - a*d)^3) - (d^(5/3)*(4*b*c - a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x
+ d^(2/3)*x^2])/(9*c^(5/3)*(b*c - a*d)^3)

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Rubi in Sympy [A]  time = 177.573, size = 386, normalized size = 0.92 \[ \frac{d x}{3 c \left (a + b x^{3}\right ) \left (c + d x^{3}\right ) \left (a d - b c\right )} + \frac{2 d^{\frac{5}{3}} \left (a d - 4 b c\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{9 c^{\frac{5}{3}} \left (a d - b c\right )^{3}} - \frac{d^{\frac{5}{3}} \left (a d - 4 b c\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{9 c^{\frac{5}{3}} \left (a d - b c\right )^{3}} - \frac{2 \sqrt{3} d^{\frac{5}{3}} \left (a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{9 c^{\frac{5}{3}} \left (a d - b c\right )^{3}} + \frac{b x \left (a d + b c\right )}{3 a c \left (a + b x^{3}\right ) \left (a d - b c\right )^{2}} + \frac{2 b^{\frac{5}{3}} \left (4 a d - b c\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{5}{3}} \left (a d - b c\right )^{3}} - \frac{b^{\frac{5}{3}} \left (4 a d - b c\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 a^{\frac{5}{3}} \left (a d - b c\right )^{3}} - \frac{2 \sqrt{3} b^{\frac{5}{3}} \left (4 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}} \left (a d - b c\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/(3*c*(a + b*x**3)*(c + d*x**3)*(a*d - b*c)) + 2*d**(5/3)*(a*d - 4*b*c)*log(c
**(1/3) + d**(1/3)*x)/(9*c**(5/3)*(a*d - b*c)**3) - d**(5/3)*(a*d - 4*b*c)*log(c
**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(9*c**(5/3)*(a*d - b*c)**3) - 2*s
qrt(3)*d**(5/3)*(a*d - 4*b*c)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3
))/(9*c**(5/3)*(a*d - b*c)**3) + b*x*(a*d + b*c)/(3*a*c*(a + b*x**3)*(a*d - b*c)
**2) + 2*b**(5/3)*(4*a*d - b*c)*log(a**(1/3) + b**(1/3)*x)/(9*a**(5/3)*(a*d - b*
c)**3) - b**(5/3)*(4*a*d - b*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x*
*2)/(9*a**(5/3)*(a*d - b*c)**3) - 2*sqrt(3)*b**(5/3)*(4*a*d - b*c)*atan(sqrt(3)*
(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(5/3)*(a*d - b*c)**3)

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Mathematica [A]  time = 1.63819, size = 381, normalized size = 0.91 \[ \frac{1}{9} \left (\frac{b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3} (a d-b c)^3}+\frac{2 b^{5/3} (4 a d-b c) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3} (a d-b c)^3}+\frac{2 \sqrt{3} b^{5/3} (b c-4 a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3} (a d-b c)^3}+\frac{3 b^2 x}{a \left (a+b x^3\right ) (b c-a d)^2}+\frac{d^{5/3} (a d-4 b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3} (b c-a d)^3}+\frac{2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3} (b c-a d)^3}+\frac{2 \sqrt{3} d^{5/3} (a d-4 b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{5/3} (b c-a d)^3}+\frac{3 d^2 x}{c \left (c+d x^3\right ) (b c-a d)^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((3*b^2*x)/(a*(b*c - a*d)^2*(a + b*x^3)) + (3*d^2*x)/(c*(b*c - a*d)^2*(c + d*x^3
)) + (2*Sqrt[3]*b^(5/3)*(b*c - 4*a*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]
])/(a^(5/3)*(-(b*c) + a*d)^3) + (2*Sqrt[3]*d^(5/3)*(-4*b*c + a*d)*ArcTan[(1 - (2
*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/(c^(5/3)*(b*c - a*d)^3) + (2*b^(5/3)*(-(b*c) + 4*
a*d)*Log[a^(1/3) + b^(1/3)*x])/(a^(5/3)*(-(b*c) + a*d)^3) + (2*d^(5/3)*(4*b*c -
a*d)*Log[c^(1/3) + d^(1/3)*x])/(c^(5/3)*(b*c - a*d)^3) + (b^(5/3)*(b*c - 4*a*d)*
Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(a^(5/3)*(-(b*c) + a*d)^3) + (d^
(5/3)*(-4*b*c + a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(c^(5/3)*(b
*c - a*d)^3))/9

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Maple [A]  time = 0.023, size = 606, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^2/(a*d-b*c)^3*x/(b*x^3+a)*d-1/3*b^3/(a*d-b*c)^3*x/a/(b*x^3+a)*c+8/9*b/(a*d
-b*c)^3/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-2/9*b^2/(a*d-b*c)^3/a/(a/b)^(2/3)*ln(x+(
a/b)^(1/3))*c-4/9*b/(a*d-b*c)^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*d+
1/9*b^2/(a*d-b*c)^3/a/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c+8/9*b/(a*d
-b*c)^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d-2/9*b^2/(a
*d-b*c)^3/a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c+1/3*d^
3/(a*d-b*c)^3/c*x/(d*x^3+c)*a-1/3*d^2/(a*d-b*c)^3*x/(d*x^3+c)*b+2/9*d^2/(a*d-b*c
)^3/c/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a-8/9*d/(a*d-b*c)^3/(c/d)^(2/3)*ln(x+(c/d)^(
1/3))*b-1/9*d^2/(a*d-b*c)^3/c/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*a+4/
9*d/(a*d-b*c)^3/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*b+2/9*d^2/(a*d-b*c
)^3/c/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a-8/9*d/(a*d-b
*c)^3/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*b

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 77.3424, size = 1250, normalized size = 2.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*sqrt(3)*(sqrt(3)*((b^3*c^2*d - 4*a*b^2*c*d^2)*x^6 + a*b^2*c^3 - 4*a^2*b*c^
2*d + (b^3*c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d^2)*x^3)*(b^2/a^2)^(1/3)*log(b^2*x^2
- a*b*x*(b^2/a^2)^(1/3) + a^2*(b^2/a^2)^(2/3)) + sqrt(3)*((4*a*b^2*c*d^2 - a^2*
b*d^3)*x^6 + 4*a^2*b*c^2*d - a^3*c*d^2 + (4*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^
3)*x^3)*(d^2/c^2)^(1/3)*log(d^2*x^2 - c*d*x*(d^2/c^2)^(1/3) + c^2*(d^2/c^2)^(2/3
)) - 2*sqrt(3)*((b^3*c^2*d - 4*a*b^2*c*d^2)*x^6 + a*b^2*c^3 - 4*a^2*b*c^2*d + (b
^3*c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d^2)*x^3)*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^
2)^(1/3)) - 2*sqrt(3)*((4*a*b^2*c*d^2 - a^2*b*d^3)*x^6 + 4*a^2*b*c^2*d - a^3*c*d
^2 + (4*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3)*(d^2/c^2)^(1/3)*log(d*x + c*
(d^2/c^2)^(1/3)) + 6*((b^3*c^2*d - 4*a*b^2*c*d^2)*x^6 + a*b^2*c^3 - 4*a^2*b*c^2*
d + (b^3*c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d^2)*x^3)*(b^2/a^2)^(1/3)*arctan(-1/3*(
2*sqrt(3)*b*x - sqrt(3)*a*(b^2/a^2)^(1/3))/(a*(b^2/a^2)^(1/3))) + 6*((4*a*b^2*c*
d^2 - a^2*b*d^3)*x^6 + 4*a^2*b*c^2*d - a^3*c*d^2 + (4*a*b^2*c^2*d + 3*a^2*b*c*d^
2 - a^3*d^3)*x^3)*(d^2/c^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*d*x - sqrt(3)*c*(d^2/c^
2)^(1/3))/(c*(d^2/c^2)^(1/3))) - 3*sqrt(3)*((b^3*c^2*d - a^2*b*d^3)*x^4 + (b^3*c
^3 - a*b^2*c^2*d + a^2*b*c*d^2 - a^3*d^3)*x))/(a^2*b^3*c^5 - 3*a^3*b^2*c^4*d + 3
*a^4*b*c^3*d^2 - a^5*c^2*d^3 + (a*b^4*c^4*d - 3*a^2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*
d^3 - a^4*b*c*d^4)*x^6 + (a*b^4*c^5 - 2*a^2*b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*
d^4)*x^3)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.230403, size = 896, normalized size = 2.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(b^3*c - 4*a*b^2*d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b^3*c^3 - 3
*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 2/9*(4*b*c*d^2 - a*d^3)*(-c/d)^(1/3)
*ln(abs(x - (-c/d)^(1/3)))/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*
d^3) + 2/3*((-a*b^2)^(1/3)*b^2*c - 4*(-a*b^2)^(1/3)*a*b*d)*arctan(1/3*sqrt(3)*(2
*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^2*b^3*c^3 - 3*sqrt(3)*a^3*b^2*c^2*d
+ 3*sqrt(3)*a^4*b*c*d^2 - sqrt(3)*a^5*d^3) + 2/3*(4*(-c*d^2)^(1/3)*b*c*d - (-c*d
^2)^(1/3)*a*d^2)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*
b^3*c^5 - 3*sqrt(3)*a*b^2*c^4*d + 3*sqrt(3)*a^2*b*c^3*d^2 - sqrt(3)*a^3*c^2*d^3)
+ 1/9*((-a*b^2)^(1/3)*b^2*c - 4*(-a*b^2)^(1/3)*a*b*d)*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) + 1/9*(
4*(-c*d^2)^(1/3)*b*c*d - (-c*d^2)^(1/3)*a*d^2)*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^
(2/3))/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) + 1/3*(b^2*c*d*
x^4 + a*b*d^2*x^4 + b^2*c^2*x + a^2*d^2*x)/((b*d*x^6 + b*c*x^3 + a*d*x^3 + a*c)*
(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2))