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

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

[Out]

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

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Rubi [A]  time = 0.382143, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac{\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)}-\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac{\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b} (b c-a d)}-\frac{\sqrt [3]{c} \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} \sqrt [3]{d} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.1417, size = 260, normalized size = 0.9 \[ \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 \sqrt [3]{b} \left (a d - b c\right )} - \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{b} \left (a d - b c\right )} - \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 \sqrt [3]{b} \left (a d - b c\right )} - \frac{\sqrt [3]{c} \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 \sqrt [3]{d} \left (a d - b c\right )} + \frac{\sqrt [3]{c} \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 \sqrt [3]{d} \left (a d - b c\right )} + \frac{\sqrt{3} \sqrt [3]{c} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 \sqrt [3]{d} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.186623, size = 224, normalized size = 0.78 \[ \frac{\frac{\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac{2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}-\frac{\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{d}}+\frac{2 \sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{d}}-\frac{2 \sqrt{3} \sqrt [3]{c} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{\sqrt [3]{d}}}{6 b c-6 a d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 246, normalized size = 0.9 \[{\frac{a}{ \left ( 3\,ad-3\,bc \right ) b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a}{ \left ( 6\,ad-6\,bc \right ) b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{ \left ( 3\,ad-3\,bc \right ) d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{ \left ( 6\,ad-6\,bc \right ) d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\sqrt{3}}{ \left ( 3\,ad-3\,bc \right ) d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a)/(d*x^3+c),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24978, size = 284, normalized size = 0.99 \[ \frac{\sqrt{3}{\left (\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{a}{b}\right )^{\frac{1}{3}} + \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) + \sqrt{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right ) + 6 \, \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} x - \sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (\frac{a}{b}\right )^{\frac{1}{3}}}\right ) + 6 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{c}{d}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )\right )}}{18 \,{\left (b c - a d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 38.6851, size = 342, normalized size = 1.19 \[ \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} d^{4} - 81 a^{2} b c d^{3} + 81 a b^{2} c^{2} d^{2} - 27 b^{3} c^{3} d\right ) + c, \left ( t \mapsto t \log{\left (x + \frac{162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} + \operatorname{RootSum}{\left (t^{3} \left (27 a^{3} b d^{3} - 81 a^{2} b^{2} c d^{2} + 81 a b^{3} c^{2} d - 27 b^{4} c^{3}\right ) - a, \left ( t \mapsto t \log{\left (x + \frac{162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**3*(27*a**3*d**4 - 81*a**2*b*c*d**3 + 81*a*b**2*c**2*d**2 - 27*b**3*c
**3*d) + c, Lambda(_t, _t*log(x + (162*_t**4*a**4*b*d**5 - 648*_t**4*a**3*b**2*c
*d**4 + 972*_t**4*a**2*b**3*c**2*d**3 - 648*_t**4*a*b**4*c**3*d**2 + 162*_t**4*b
**5*c**4*d - 3*_t*a**2*d**2 + 6*_t*a*b*c*d - 3*_t*b**2*c**2)/(a*d + b*c)))) + Ro
otSum(_t**3*(27*a**3*b*d**3 - 81*a**2*b**2*c*d**2 + 81*a*b**3*c**2*d - 27*b**4*c
**3) - a, Lambda(_t, _t*log(x + (162*_t**4*a**4*b*d**5 - 648*_t**4*a**3*b**2*c*d
**4 + 972*_t**4*a**2*b**3*c**2*d**3 - 648*_t**4*a*b**4*c**3*d**2 + 162*_t**4*b**
5*c**4*d - 3*_t*a**2*d**2 + 6*_t*a*b*c*d - 3*_t*b**2*c**2)/(a*d + b*c))))

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GIAC/XCAS [A]  time = 0.227858, size = 375, normalized size = 1.3 \[ \frac{a \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (a b c - a^{2} d\right )}} - \frac{c \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \,{\left (b c^{2} - a c d\right )}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c - \sqrt{3} a b d} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b c d - \sqrt{3} a d^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b^{2} c - a b d\right )}} + \frac{\left (-c d^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \,{\left (b c d - a d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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