3.120 \(\int \frac{1}{x^5 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\)

Optimal. Leaf size=318 \[ \frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{d^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{7/3} (b c-a d)}-\frac{1}{4 a c x^4} \]

[Out]

-1/(4*a*c*x^4) + (b*c + a*d)/(a^2*c^2*x) - (b^(7/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*
x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)*(b*c - a*d)) + (d^(7/3)*ArcTan[(c^(1/3)
- 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(7/3)*(b*c - a*d)) - (b^(7/3)*Log[
a^(1/3) + b^(1/3)*x])/(3*a^(7/3)*(b*c - a*d)) + (d^(7/3)*Log[c^(1/3) + d^(1/3)*x
])/(3*c^(7/3)*(b*c - a*d)) + (b^(7/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(6*a^(7/3)*(b*c - a*d)) - (d^(7/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/
3)*x^2])/(6*c^(7/3)*(b*c - a*d))

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Rubi [A]  time = 1.0002, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{b^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3} (b c-a d)}-\frac{b^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3} (b c-a d)}-\frac{b^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{7/3} (b c-a d)}+\frac{a d+b c}{a^2 c^2 x}-\frac{d^{7/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{7/3} (b c-a d)}+\frac{d^{7/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{7/3} (b c-a d)}+\frac{d^{7/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{7/3} (b c-a d)}-\frac{1}{4 a c x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^3)*(c + d*x^3)),x]

[Out]

-1/(4*a*c*x^4) + (b*c + a*d)/(a^2*c^2*x) - (b^(7/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*
x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(7/3)*(b*c - a*d)) + (d^(7/3)*ArcTan[(c^(1/3)
- 2*d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(7/3)*(b*c - a*d)) - (b^(7/3)*Log[
a^(1/3) + b^(1/3)*x])/(3*a^(7/3)*(b*c - a*d)) + (d^(7/3)*Log[c^(1/3) + d^(1/3)*x
])/(3*c^(7/3)*(b*c - a*d)) + (b^(7/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(6*a^(7/3)*(b*c - a*d)) - (d^(7/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/
3)*x^2])/(6*c^(7/3)*(b*c - a*d))

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Rubi in Sympy [A]  time = 178.47, size = 286, normalized size = 0.9 \[ - \frac{d^{\frac{7}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{7}{3}} \left (a d - b c\right )} + \frac{d^{\frac{7}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{7}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} d^{\frac{7}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{7}{3}} \left (a d - b c\right )} - \frac{1}{4 a c x^{4}} + \frac{a d + b c}{a^{2} c^{2} x} + \frac{b^{\frac{7}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{7}{3}} \left (a d - b c\right )} - \frac{b^{\frac{7}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{7}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} b^{\frac{7}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{7}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**3+a)/(d*x**3+c),x)

[Out]

-d**(7/3)*log(c**(1/3) + d**(1/3)*x)/(3*c**(7/3)*(a*d - b*c)) + d**(7/3)*log(c**
(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(7/3)*(a*d - b*c)) - sqrt(3)*
d**(7/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(7/3)*(a*d -
b*c)) - 1/(4*a*c*x**4) + (a*d + b*c)/(a**2*c**2*x) + b**(7/3)*log(a**(1/3) + b*
*(1/3)*x)/(3*a**(7/3)*(a*d - b*c)) - b**(7/3)*log(a**(2/3) - a**(1/3)*b**(1/3)*x
+ b**(2/3)*x**2)/(6*a**(7/3)*(a*d - b*c)) + sqrt(3)*b**(7/3)*atan(sqrt(3)*(a**(
1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(7/3)*(a*d - b*c))

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Mathematica [A]  time = 0.3825, size = 282, normalized size = 0.89 \[ \frac{\frac{4 b^{7/3} x^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{7/3}}+\frac{4 \sqrt{3} b^{7/3} x^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{7/3}}-\frac{2 b^{7/3} x^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{7/3}}-\frac{12 b^2 x^3}{a^2}+\frac{3 b}{a}-\frac{4 d^{7/3} x^4 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{7/3}}-\frac{4 \sqrt{3} d^{7/3} x^4 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{7/3}}+\frac{2 d^{7/3} x^4 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{7/3}}+\frac{12 d^2 x^3}{c^2}-\frac{3 d}{c}}{12 x^4 (a d-b c)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^3)*(c + d*x^3)),x]

[Out]

((3*b)/a - (3*d)/c - (12*b^2*x^3)/a^2 + (12*d^2*x^3)/c^2 + (4*Sqrt[3]*b^(7/3)*x^
4*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(7/3) - (4*Sqrt[3]*d^(7/3)*x^4*
ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/c^(7/3) + (4*b^(7/3)*x^4*Log[a^(1/3
) + b^(1/3)*x])/a^(7/3) - (4*d^(7/3)*x^4*Log[c^(1/3) + d^(1/3)*x])/c^(7/3) - (2*
b^(7/3)*x^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(7/3) + (2*d^(7/3)
*x^4*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(7/3))/(12*(-(b*c) + a*d)
*x^4)

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Maple [A]  time = 0.016, size = 291, normalized size = 0.9 \[ -{\frac{1}{4\,ac{x}^{4}}}+{\frac{d}{a{c}^{2}x}}+{\frac{b}{{a}^{2}cx}}+{\frac{{b}^{2}}{3\,{a}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}}{6\,{a}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{{d}^{2}}{3\,{c}^{2} \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{d}^{2}}{6\,{c}^{2} \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{{d}^{2}\sqrt{3}}{3\,{c}^{2} \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

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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)*(d*x^3 + c)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.1535, size = 470, normalized size = 1.48 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} b^{2} c^{2} x^{4} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) + 2 \, \sqrt{3} a^{2} d^{2} x^{4} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (-\frac{d}{c}\right )^{\frac{2}{3}} - c \left (-\frac{d}{c}\right )^{\frac{1}{3}}\right ) - 4 \, \sqrt{3} b^{2} c^{2} x^{4} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 4 \, \sqrt{3} a^{2} d^{2} x^{4} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (-\frac{d}{c}\right )^{\frac{2}{3}}\right ) - 12 \, b^{2} c^{2} x^{4} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 12 \, a^{2} d^{2} x^{4} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (-\frac{d}{c}\right )^{\frac{2}{3}}}{3 \, c \left (-\frac{d}{c}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (a b c^{2} - a^{2} c d - 4 \,{\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{3}\right )}\right )}}{36 \,{\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^5),x, algorithm="fricas")

[Out]

1/36*sqrt(3)*(2*sqrt(3)*b^2*c^2*x^4*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*
(b/a)^(1/3)) + 2*sqrt(3)*a^2*d^2*x^4*(-d/c)^(1/3)*log(d*x^2 - c*x*(-d/c)^(2/3) -
c*(-d/c)^(1/3)) - 4*sqrt(3)*b^2*c^2*x^4*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) -
4*sqrt(3)*a^2*d^2*x^4*(-d/c)^(1/3)*log(d*x + c*(-d/c)^(2/3)) - 12*b^2*c^2*x^4*(b
/a)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(b/a)^(2/3))/(a*(b/a)^(2/3))) -
12*a^2*d^2*x^4*(-d/c)^(1/3)*arctan(-1/3*(2*sqrt(3)*d*x - sqrt(3)*c*(-d/c)^(2/3)
)/(c*(-d/c)^(2/3))) - 3*sqrt(3)*(a*b*c^2 - a^2*c*d - 4*(b^2*c^2 - a^2*d^2)*x^3))
/((a^2*b*c^3 - a^3*c^2*d)*x^4)

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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/x**5/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228953, size = 443, normalized size = 1.39 \[ -\frac{b^{3} \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{3} b c - a^{4} d\right )}} + \frac{d^{3} \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{4} - a c^{3} d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{3} b c - \sqrt{3} a^{4} d} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} d \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{4} - \sqrt{3} a c^{3} d} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{3} b c - a^{4} d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}} d{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{4} - a c^{3} d\right )}} + \frac{4 \, b c x^{3} + 4 \, a d x^{3} - a c}{4 \, a^{2} c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)*(d*x^3 + c)*x^5),x, algorithm="giac")

[Out]

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