3.1977 \(\int \frac{x^4}{\left (a+\frac{b}{x^3}\right )^2} \, dx\)

Optimal. Leaf size=157 \[ \frac{4 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{11/3}}-\frac{8 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{11/3}}-\frac{8 b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{11/3}}-\frac{4 b x^2}{3 a^3}+\frac{8 x^5}{15 a^2}-\frac{x^8}{3 a \left (a x^3+b\right )} \]

[Out]

(-4*b*x^2)/(3*a^3) + (8*x^5)/(15*a^2) - x^8/(3*a*(b + a*x^3)) - (8*b^(5/3)*ArcTa
n[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(11/3)) - (8*b^(5/3)*
Log[b^(1/3) + a^(1/3)*x])/(9*a^(11/3)) + (4*b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3
)*x + a^(2/3)*x^2])/(9*a^(11/3))

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Rubi [A]  time = 0.239825, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692 \[ \frac{4 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{9 a^{11/3}}-\frac{8 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{11/3}}-\frac{8 b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{11/3}}-\frac{4 b x^2}{3 a^3}+\frac{8 x^5}{15 a^2}-\frac{x^8}{3 a \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x^3)^2,x]

[Out]

(-4*b*x^2)/(3*a^3) + (8*x^5)/(15*a^2) - x^8/(3*a*(b + a*x^3)) - (8*b^(5/3)*ArcTa
n[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(3*Sqrt[3]*a^(11/3)) - (8*b^(5/3)*
Log[b^(1/3) + a^(1/3)*x])/(9*a^(11/3)) + (4*b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3
)*x + a^(2/3)*x^2])/(9*a^(11/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{x^{8}}{3 a \left (a x^{3} + b\right )} + \frac{8 x^{5}}{15 a^{2}} - \frac{8 b \int x\, dx}{3 a^{3}} - \frac{8 b^{\frac{5}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{9 a^{\frac{11}{3}}} + \frac{4 b^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{9 a^{\frac{11}{3}}} - \frac{8 \sqrt{3} b^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{9 a^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**8/(3*a*(a*x**3 + b)) + 8*x**5/(15*a**2) - 8*b*Integral(x, x)/(3*a**3) - 8*b*
*(5/3)*log(a**(1/3)*x + b**(1/3))/(9*a**(11/3)) + 4*b**(5/3)*log(a**(2/3)*x**2 -
a**(1/3)*b**(1/3)*x + b**(2/3))/(9*a**(11/3)) - 8*sqrt(3)*b**(5/3)*atan(sqrt(3)
*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(9*a**(11/3))

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Mathematica [A]  time = 0.187062, size = 144, normalized size = 0.92 \[ \frac{20 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-\frac{15 a^{2/3} b^2 x^2}{a x^3+b}-45 a^{2/3} b x^2+9 a^{5/3} x^5-40 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-40 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{45 a^{11/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b/x^3)^2,x]

[Out]

(-45*a^(2/3)*b*x^2 + 9*a^(5/3)*x^5 - (15*a^(2/3)*b^2*x^2)/(b + a*x^3) - 40*Sqrt[
3]*b^(5/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] - 40*b^(5/3)*Log[b^(1/3)
+ a^(1/3)*x] + 20*b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(45*a^
(11/3))

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Maple [A]  time = 0.012, size = 137, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,{a}^{2}}}-{\frac{b{x}^{2}}{{a}^{3}}}-{\frac{{b}^{2}{x}^{2}}{3\,{a}^{3} \left ( a{x}^{3}+b \right ) }}-{\frac{8\,{b}^{2}}{9\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{4\,{b}^{2}}{9\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{8\,{b}^{2}\sqrt{3}}{9\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^3)^2,x)

[Out]

1/5*x^5/a^2-b*x^2/a^3-1/3/a^3*b^2*x^2/(a*x^3+b)-8/9/a^4*b^2/(b/a)^(1/3)*ln(x+(b/
a)^(1/3))+4/9/a^4*b^2/(b/a)^(1/3)*ln(x^2-x*(b/a)^(1/3)+(b/a)^(2/3))+8/9/a^4*b^2*
3^(1/2)/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(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/(a + b/x^3)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228273, size = 288, normalized size = 1.83 \[ -\frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (a b x^{3} + b^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 40 \, \sqrt{3}{\left (a b x^{3} + b^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 120 \,{\left (a b x^{3} + b^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (3 \, a^{2} x^{8} - 12 \, a b x^{5} - 20 \, b^{2} x^{2}\right )}\right )}}{135 \,{\left (a^{4} x^{3} + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/135*sqrt(3)*(20*sqrt(3)*(a*b*x^3 + b^2)*(-b^2/a^2)^(1/3)*log(b*x^2 - a*x*(-b^
2/a^2)^(2/3) - b*(-b^2/a^2)^(1/3)) - 40*sqrt(3)*(a*b*x^3 + b^2)*(-b^2/a^2)^(1/3)
*log(b*x + a*(-b^2/a^2)^(2/3)) - 120*(a*b*x^3 + b^2)*(-b^2/a^2)^(1/3)*arctan(-1/
3*(2*sqrt(3)*b*x - sqrt(3)*a*(-b^2/a^2)^(2/3))/(a*(-b^2/a^2)^(2/3))) - 3*sqrt(3)
*(3*a^2*x^8 - 12*a*b*x^5 - 20*b^2*x^2))/(a^4*x^3 + a^3*b)

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Sympy [A]  time = 1.92913, size = 70, normalized size = 0.45 \[ - \frac{b^{2} x^{2}}{3 a^{4} x^{3} + 3 a^{3} b} + \operatorname{RootSum}{\left (729 t^{3} a^{11} + 512 b^{5}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{7}}{64 b^{3}} + x \right )} \right )\right )} + \frac{x^{5}}{5 a^{2}} - \frac{b x^{2}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x**3)**2,x)

[Out]

-b**2*x**2/(3*a**4*x**3 + 3*a**3*b) + RootSum(729*_t**3*a**11 + 512*b**5, Lambda
(_t, _t*log(81*_t**2*a**7/(64*b**3) + x))) + x**5/(5*a**2) - b*x**2/a**3

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GIAC/XCAS [A]  time = 0.244994, size = 204, normalized size = 1.3 \[ -\frac{b^{2} x^{2}}{3 \,{\left (a x^{3} + b\right )} a^{3}} - \frac{8 \, b \left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} - \frac{8 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{5}} + \frac{4 \, \left (-a^{2} b\right )^{\frac{2}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{9 \, a^{5}} + \frac{a^{8} x^{5} - 5 \, a^{7} b x^{2}}{5 \, a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b^2*x^2/((a*x^3 + b)*a^3) - 8/9*b*(-b/a)^(2/3)*ln(abs(x - (-b/a)^(1/3)))/a^
3 - 8/9*sqrt(3)*(-a^2*b)^(2/3)*b*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^
(1/3))/a^5 + 4/9*(-a^2*b)^(2/3)*b*ln(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^5 +
1/5*(a^8*x^5 - 5*a^7*b*x^2)/a^10