3.348 \(\int \frac{x^4}{\left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=158 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{x^2}{6 b \left (a+b x^3\right )^2} \]

[Out]

-x^2/(6*b*(a + b*x^3)^2) + x^2/(9*a*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)
*x)/(Sqrt[3]*a^(1/3))]/(9*Sqrt[3]*a^(4/3)*b^(5/3)) - Log[a^(1/3) + b^(1/3)*x]/(2
7*a^(4/3)*b^(5/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(54*a^(4/3)*
b^(5/3))

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Rubi [A]  time = 0.1716, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{x^2}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^3)^3,x]

[Out]

-x^2/(6*b*(a + b*x^3)^2) + x^2/(9*a*b*(a + b*x^3)) - ArcTan[(a^(1/3) - 2*b^(1/3)
*x)/(Sqrt[3]*a^(1/3))]/(9*Sqrt[3]*a^(4/3)*b^(5/3)) - Log[a^(1/3) + b^(1/3)*x]/(2
7*a^(4/3)*b^(5/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(54*a^(4/3)*
b^(5/3))

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Rubi in Sympy [A]  time = 38.0752, size = 139, normalized size = 0.88 \[ - \frac{x^{2}}{6 b \left (a + b x^{3}\right )^{2}} + \frac{x^{2}}{9 a b \left (a + b x^{3}\right )} - \frac{\log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{4}{3}} b^{\frac{5}{3}}} + \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{4}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{4}{3}} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a)**3,x)

[Out]

-x**2/(6*b*(a + b*x**3)**2) + x**2/(9*a*b*(a + b*x**3)) - log(a**(1/3) + b**(1/3
)*x)/(27*a**(4/3)*b**(5/3)) + log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2
)/(54*a**(4/3)*b**(5/3)) - sqrt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a*
*(1/3))/(27*a**(4/3)*b**(5/3))

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Mathematica [A]  time = 0.205564, size = 141, normalized size = 0.89 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}+\frac{6 b^{2/3} x^2}{a^2+a b x^3}-\frac{9 b^{2/3} x^2}{\left (a+b x^3\right )^2}}{54 b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^3)^3,x]

[Out]

((-9*b^(2/3)*x^2)/(a + b*x^3)^2 + (6*b^(2/3)*x^2)/(a^2 + a*b*x^3) - (2*Sqrt[3]*A
rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(4/3) - (2*Log[a^(1/3) + b^(1/3)*x
])/a^(4/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/a^(4/3))/(54*b^(5/3)
)

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Maple [A]  time = 0.013, size = 127, normalized size = 0.8 \[{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{{x}^{5}}{9\,a}}-{\frac{{x}^{2}}{18\,b}} \right ) }-{\frac{1}{27\,a{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{54\,a{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{27\,a{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a)^3,x)

[Out]

(1/9*x^5/a-1/18*x^2/b)/(b*x^3+a)^2-1/27/b^2/a/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/54
/b^2/a/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+1/27/b^2/a*3^(1/2)/(a/b)^(1
/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(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^4/(b*x^3 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241312, size = 278, normalized size = 1.76 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) - 3 \, \sqrt{3}{\left (2 \, b x^{5} - a x^{2}\right )} \left (-a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^3,x, algorithm="fricas")

[Out]

-1/162*sqrt(3)*(sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log((-a*b^2)^(1/3)*b*x^2 - a
*b + (-a*b^2)^(2/3)*x) - 2*sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)*log(a*b + (-a*b^2
)^(2/3)*x) + 6*(b^2*x^6 + 2*a*b*x^3 + a^2)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*
(-a*b^2)^(2/3)*x)/(a*b)) - 3*sqrt(3)*(2*b*x^5 - a*x^2)*(-a*b^2)^(1/3))/((a*b^3*x
^6 + 2*a^2*b^2*x^3 + a^3*b)*(-a*b^2)^(1/3))

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Sympy [A]  time = 2.48766, size = 70, normalized size = 0.44 \[ \frac{- a x^{2} + 2 b x^{5}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{4} b^{5} + 1, \left ( t \mapsto t \log{\left (729 t^{2} a^{3} b^{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a)**3,x)

[Out]

(-a*x**2 + 2*b*x**5)/(18*a**3*b + 36*a**2*b**2*x**3 + 18*a*b**3*x**6) + RootSum(
19683*_t**3*a**4*b**5 + 1, Lambda(_t, _t*log(729*_t**2*a**3*b**3 + x)))

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GIAC/XCAS [A]  time = 0.224805, size = 196, normalized size = 1.24 \[ -\frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac{2 \, b x^{5} - a x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a b} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^3,x, algorithm="giac")

[Out]

-1/27*(-a/b)^(2/3)*ln(abs(x - (-a/b)^(1/3)))/(a^2*b) - 1/27*sqrt(3)*(-a*b^2)^(2/
3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^3) + 1/18*(2*b*x
^5 - a*x^2)/((b*x^3 + a)^2*a*b) + 1/54*(-a*b^2)^(2/3)*ln(x^2 + x*(-a/b)^(1/3) +
(-a/b)^(2/3))/(a^2*b^3)