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

Optimal. Leaf size=288 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{c} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.367817, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} (b c-a d)}+\frac{\sqrt [3]{d} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{c} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 65.2044, size = 260, normalized size = 0.9 \[ - \frac{\sqrt [3]{d} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 \sqrt [3]{c} \left (a d - b c\right )} + \frac{\sqrt [3]{d} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{c} \left (a d - b c\right )} - \frac{\sqrt{3} \sqrt [3]{d} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 \sqrt [3]{c} \left (a d - b c\right )} + \frac{\sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{a} \left (a d - b c\right )} - \frac{\sqrt [3]{b} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{a} \left (a d - b c\right )} + \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{a} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.22842, size = 224, normalized size = 0.78 \[ \frac{-\frac{\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}+\frac{2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac{2 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}+\frac{\sqrt [3]{d} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{c}}-\frac{2 \sqrt [3]{d} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{c}}-\frac{2 \sqrt{3} \sqrt [3]{d} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{\sqrt [3]{c}}}{6 a d-6 b c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 222, normalized size = 0.8 \[{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{6\,ad-6\,bc}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{\sqrt{3}}{3\,ad-3\,bc}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{1}{3\,ad-3\,bc}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{1}{6\,ad-6\,bc}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}+{\frac{\sqrt{3}}{3\,ad-3\,bc}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258621, size = 323, normalized size = 1.12 \[ \frac{\sqrt{3}{\left (\sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) + \sqrt{3} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x^{2} - c x \left (-\frac{d}{c}\right )^{\frac{2}{3}} - c \left (-\frac{d}{c}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{d}{c}\right )^{\frac{1}{3}} \log \left (d x + c \left (-\frac{d}{c}\right )^{\frac{2}{3}}\right ) - 6 \, \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b}{a}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b}{a}\right )^{\frac{2}{3}}}\right ) - 6 \, \left (-\frac{d}{c}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} d x - \sqrt{3} c \left (-\frac{d}{c}\right )^{\frac{2}{3}}}{3 \, c \left (-\frac{d}{c}\right )^{\frac{2}{3}}}\right )\right )}}{18 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 17.5527, size = 515, normalized size = 1.79 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{4} d^{3} - 81 a^{3} b c d^{2} + 81 a^{2} b^{2} c^{2} d - 27 a b^{3} c^{3}\right ) - b, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} c d^{3} - 81 a^{2} b c^{2} d^{2} + 81 a b^{2} c^{3} d - 27 b^{3} c^{4}\right ) + d, \left ( t \mapsto t \log{\left (x + \frac{243 t^{5} a^{7} c d^{6} - 1458 t^{5} a^{6} b c^{2} d^{5} + 3645 t^{5} a^{5} b^{2} c^{3} d^{4} - 4860 t^{5} a^{4} b^{3} c^{4} d^{3} + 3645 t^{5} a^{3} b^{4} c^{5} d^{2} - 1458 t^{5} a^{2} b^{5} c^{6} d + 243 t^{5} a b^{6} c^{7} + 9 t^{2} a^{4} d^{4} - 18 t^{2} a^{3} b c d^{3} + 18 t^{2} a^{2} b^{2} c^{2} d^{2} - 18 t^{2} a b^{3} c^{3} d + 9 t^{2} b^{4} c^{4}}{a b d^{2} + b^{2} c d} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**3*(27*a**4*d**3 - 81*a**3*b*c*d**2 + 81*a**2*b**2*c**2*d - 27*a*b**3
*c**3) - b, Lambda(_t, _t*log(x + (243*_t**5*a**7*c*d**6 - 1458*_t**5*a**6*b*c**
2*d**5 + 3645*_t**5*a**5*b**2*c**3*d**4 - 4860*_t**5*a**4*b**3*c**4*d**3 + 3645*
_t**5*a**3*b**4*c**5*d**2 - 1458*_t**5*a**2*b**5*c**6*d + 243*_t**5*a*b**6*c**7
+ 9*_t**2*a**4*d**4 - 18*_t**2*a**3*b*c*d**3 + 18*_t**2*a**2*b**2*c**2*d**2 - 18
*_t**2*a*b**3*c**3*d + 9*_t**2*b**4*c**4)/(a*b*d**2 + b**2*c*d)))) + RootSum(_t*
*3*(27*a**3*c*d**3 - 81*a**2*b*c**2*d**2 + 81*a*b**2*c**3*d - 27*b**3*c**4) + d,
Lambda(_t, _t*log(x + (243*_t**5*a**7*c*d**6 - 1458*_t**5*a**6*b*c**2*d**5 + 36
45*_t**5*a**5*b**2*c**3*d**4 - 4860*_t**5*a**4*b**3*c**4*d**3 + 3645*_t**5*a**3*
b**4*c**5*d**2 - 1458*_t**5*a**2*b**5*c**6*d + 243*_t**5*a*b**6*c**7 + 9*_t**2*a
**4*d**4 - 18*_t**2*a**3*b*c*d**3 + 18*_t**2*a**2*b**2*c**2*d**2 - 18*_t**2*a*b*
*3*c**3*d + 9*_t**2*b**4*c**4)/(a*b*d**2 + b**2*c*d))))

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GIAC/XCAS [A]  time = 0.228643, size = 392, normalized size = 1.36 \[ -\frac{b \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} + \frac{d \left (-\frac{c}{d}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} a b^{2} c - \sqrt{3} a^{2} b d} + \frac{\left (-c d^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c^{2} d - \sqrt{3} a c d^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (a b^{2} c - a^{2} b d\right )}} - \frac{\left (-c d^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c^{2} d - a c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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