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

Optimal. Leaf size=301 \[ \frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac{b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} (b c-a d)}-\frac{d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)}+\frac{d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}-\frac{d^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{5/3} (b c-a d)}-\frac{1}{2 a c x^2} \]

[Out]

-1/(2*a*c*x^2) + (b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*a^(5/3)*(b*c - a*d)) - (d^(5/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^
(1/3))])/(Sqrt[3]*c^(5/3)*(b*c - a*d)) - (b^(5/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a
^(5/3)*(b*c - a*d)) + (d^(5/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(5/3)*(b*c - a*d))
+ (b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(5/3)*(b*c - a*
d)) - (d^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(5/3)*(b*c -
a*d))

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Rubi [A]  time = 0.634966, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac{b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} (b c-a d)}-\frac{d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)}+\frac{d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}-\frac{d^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{5/3} (b c-a d)}-\frac{1}{2 a c x^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*a*c*x^2) + (b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*a^(5/3)*(b*c - a*d)) - (d^(5/3)*ArcTan[(c^(1/3) - 2*d^(1/3)*x)/(Sqrt[3]*c^
(1/3))])/(Sqrt[3]*c^(5/3)*(b*c - a*d)) - (b^(5/3)*Log[a^(1/3) + b^(1/3)*x])/(3*a
^(5/3)*(b*c - a*d)) + (d^(5/3)*Log[c^(1/3) + d^(1/3)*x])/(3*c^(5/3)*(b*c - a*d))
+ (b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(5/3)*(b*c - a*
d)) - (d^(5/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(5/3)*(b*c -
a*d))

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Rubi in Sympy [A]  time = 103.149, size = 270, normalized size = 0.9 \[ - \frac{d^{\frac{5}{3}} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{5}{3}} \left (a d - b c\right )} + \frac{d^{\frac{5}{3}} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{5}{3}} \left (a d - b c\right )} + \frac{\sqrt{3} d^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{5}{3}} \left (a d - b c\right )} - \frac{1}{2 a c x^{2}} + \frac{b^{\frac{5}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} \left (a d - b c\right )} - \frac{b^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} \left (a d - b c\right )} - \frac{\sqrt{3} b^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d**(5/3)*log(c**(1/3) + d**(1/3)*x)/(3*c**(5/3)*(a*d - b*c)) + d**(5/3)*log(c**
(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(5/3)*(a*d - b*c)) + sqrt(3)*
d**(5/3)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(5/3)*(a*d -
b*c)) - 1/(2*a*c*x**2) + b**(5/3)*log(a**(1/3) + b**(1/3)*x)/(3*a**(5/3)*(a*d -
b*c)) - b**(5/3)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(5/3
)*(a*d - b*c)) - sqrt(3)*b**(5/3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**
(1/3))/(3*a**(5/3)*(a*d - b*c))

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Mathematica [A]  time = 0.374834, size = 259, normalized size = 0.86 \[ \frac{\frac{2 b^{5/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac{b^{5/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}-\frac{2 \sqrt{3} b^{5/3} x^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{5/3}}+\frac{3 b}{a}-\frac{2 d^{5/3} x^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3}}+\frac{d^{5/3} x^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3}}+\frac{2 \sqrt{3} d^{5/3} x^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{c^{5/3}}-\frac{3 d}{c}}{6 x^2 (a d-b c)} \]

Antiderivative was successfully verified.

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

[Out]

((3*b)/a - (3*d)/c - (2*Sqrt[3]*b^(5/3)*x^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]])/a^(5/3) + (2*Sqrt[3]*d^(5/3)*x^2*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqr
t[3]])/c^(5/3) + (2*b^(5/3)*x^2*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) - (2*d^(5/3)*x
^2*Log[c^(1/3) + d^(1/3)*x])/c^(5/3) - (b^(5/3)*x^2*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/a^(5/3) + (d^(5/3)*x^2*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(
2/3)*x^2])/c^(5/3))/(6*(-(b*c) + a*d)*x^2)

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Maple [A]  time = 0.013, size = 257, normalized size = 0.9 \[{\frac{b}{3\,a \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{6\,a \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}}{3\,a \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{2\,ac{x}^{2}}}-{\frac{d}{3\,c \left ( ad-bc \right ) }\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{d}{6\,c \left ( ad-bc \right ) }\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{d\sqrt{3}}{3\,c \left ( ad-bc \right ) }\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/a*b/(a*d-b*c)/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/a*b/(a*d-b*c)/(a/b)^(2/3)*ln
(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/3/a*b/(a*d-b*c)/(a/b)^(2/3)*3^(1/2)*arctan(1/3
*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/2/a/c/x^2-1/3/c*d/(a*d-b*c)/(c/d)^(2/3)*ln(x+(c/
d)^(1/3))+1/6/c*d/(a*d-b*c)/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))-1/3/c*
d/(a*d-b*c)/(c/d)^(2/3)*3^(1/2)*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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.12364, size = 447, normalized size = 1.49 \[ \frac{\sqrt{3}{\left (\sqrt{3} b c x^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + \sqrt{3} a d x^{2} \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d^{2} x^{2} + c d x \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} + c^{2} \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b c x^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} a d x^{2} \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \log \left (d x - c \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}\right ) + 6 \, b c x^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 6 \, a d x^{2} \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} d x + \sqrt{3} c \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}{3 \, c \left (-\frac{d^{2}}{c^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (b c - a d\right )}\right )}}{18 \,{\left (a b c^{2} - a^{2} c d\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(sqrt(3)*b*c*x^2*(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)*a*d*x^2*(-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*c*x^2*(b^2/a^2)^(1/3)*log
(b*x + a*(b^2/a^2)^(1/3)) - 2*sqrt(3)*a*d*x^2*(-d^2/c^2)^(1/3)*log(d*x - c*(-d^2
/c^2)^(1/3)) + 6*b*c*x^2*(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*a*d*x^2*(-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*
c - a*d))/((a*b*c^2 - a^2*c*d)*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/x**3/(b*x**3+a)/(d*x**3+c),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.225098, size = 417, normalized size = 1.39 \[ \frac{b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a^{2} b c - a^{3} d\right )}} - \frac{d^{2} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a^{2} b c - \sqrt{3} a^{3} d} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{3} - \sqrt{3} a c^{2} d} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a^{2} b c - a^{3} d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}} d{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{3} - a c^{2} d\right )}} - \frac{1}{2 \, a c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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