3.334 \(\int \frac{x^4}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=136 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}-\frac{x^2}{3 b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.150278, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} \sqrt [3]{a} b^{5/3}}-\frac{x^2}{3 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^3)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 29.693, size = 126, normalized size = 0.93 \[ - \frac{x^{2}}{3 b \left (a + b x^{3}\right )} - \frac{2 \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}}} + \frac{\log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}}} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 \sqrt [3]{a} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a)**2,x)

[Out]

-x**2/(3*b*(a + b*x**3)) - 2*log(a**(1/3) + b**(1/3)*x)/(9*a**(1/3)*b**(5/3)) +
log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(9*a**(1/3)*b**(5/3)) - 2*sq
rt(3)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(1/3)*b**(5/3))

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Mathematica [A]  time = 0.163187, size = 119, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac{3 b^{2/3} x^2}{a+b x^3}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{a}}}{9 b^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^3)^2,x]

[Out]

((-3*b^(2/3)*x^2)/(a + b*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sq
rt[3]])/a^(1/3) - (2*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + Log[a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2]/a^(1/3))/(9*b^(5/3))

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Maple [A]  time = 0.01, size = 108, normalized size = 0.8 \[ -{\frac{{x}^{2}}{3\,b \left ( b{x}^{3}+a \right ) }}-{\frac{2}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{9\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a)^2,x)

[Out]

-1/3*x^2/b/(b*x^3+a)-2/9/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/9/b^2/(a/b)^(1/3)*l
n(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/9/b^2*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225993, size = 201, normalized size = 1.48 \[ -\frac{\sqrt{3}{\left (3 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} x^{2} + \sqrt{3}{\left (b x^{3} + a\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left (b x^{3} + a\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left (b x^{3} + a\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{27 \,{\left (b^{2} x^{3} + a b\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*(3*sqrt(3)*(-a*b^2)^(1/3)*x^2 + sqrt(3)*(b*x^3 + a)*log((-a*b^2)^(
1/3)*b*x^2 - a*b + (-a*b^2)^(2/3)*x) - 2*sqrt(3)*(b*x^3 + a)*log(a*b + (-a*b^2)^
(2/3)*x) + 6*(b*x^3 + a)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(-a*b^2)^(2/3)*x)/
(a*b)))/((b^2*x^3 + a*b)*(-a*b^2)^(1/3))

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Sympy [A]  time = 1.60661, size = 44, normalized size = 0.32 \[ - \frac{x^{2}}{3 a b + 3 b^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a b^{5} + 8, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a b^{3}}{4} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a)**2,x)

[Out]

-x**2/(3*a*b + 3*b**2*x**3) + RootSum(729*_t**3*a*b**5 + 8, Lambda(_t, _t*log(81
*_t**2*a*b**3/4 + x)))

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GIAC/XCAS [A]  time = 0.226157, size = 178, normalized size = 1.31 \[ -\frac{x^{2}}{3 \,{\left (b x^{3} + a\right )} b} - \frac{2 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a b} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a b^{3}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{9 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

-1/3*x^2/((b*x^3 + a)*b) - 2/9*(-a/b)^(2/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b) - 2/
9*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(
a*b^3) + 1/9*(-a*b^2)^(2/3)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^3)