3.90 \(\int \frac{x^3}{\sqrt{a^2+2 a b x^3+b^2 x^6}} \, dx\)

Optimal. Leaf size=235 \[ \frac{x \left (a+b x^3\right )}{b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{a} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{a} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

(x*(a + b*x^3))/(b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (a^(1/3)*(a + b*x^3)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(4/3)*Sqrt[a^2 + 2*a*b*
x^3 + b^2*x^6]) - (a^(1/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(4/3)*Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6]) + (a^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(6*b^(4/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.263548, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{x \left (a+b x^3\right )}{b \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\sqrt [3]{a} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{a} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\sqrt [3]{a} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

(x*(a + b*x^3))/(b*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (a^(1/3)*(a + b*x^3)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(4/3)*Sqrt[a^2 + 2*a*b*
x^3 + b^2*x^6]) - (a^(1/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(4/3)*Sqrt
[a^2 + 2*a*b*x^3 + b^2*x^6]) + (a^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(6*b^(4/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{\left (a + b x^{3}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/((b*x**3+a)**2)**(1/2),x)

[Out]

Integral(x**3/sqrt((a + b*x**3)**2), x)

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Mathematica [A]  time = 0.0575173, size = 128, normalized size = 0.54 \[ \frac{\left (a+b x^3\right ) \left (\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 \sqrt [3]{b} x\right )}{6 b^{4/3} \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6],x]

[Out]

((a + b*x^3)*(6*b^(1/3)*x + 2*Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))
/Sqrt[3]] - 2*a^(1/3)*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]))/(6*b^(4/3)*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.01, size = 110, normalized size = 0.5 \[{\frac{b{x}^{3}+a}{6\,{b}^{2}} \left ( 6\,xb \left ({\frac{a}{b}} \right ) ^{2/3}+2\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}a-2\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) a+\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/((b*x^3+a)^2)^(1/2),x)

[Out]

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

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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/sqrt((b*x^3 + a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274634, size = 157, normalized size = 0.67 \[ -\frac{\sqrt{3}{\left (\sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 6 \, \sqrt{3} x + 6 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} x + \sqrt{3} \left (-\frac{a}{b}\right )^{\frac{1}{3}}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )\right )}}{18 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/sqrt((b*x^3 + a)^2),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)) - 2
*sqrt(3)*(-a/b)^(1/3)*log(x - (-a/b)^(1/3)) - 6*sqrt(3)*x + 6*(-a/b)^(1/3)*arcta
n(1/3*(2*sqrt(3)*x + sqrt(3)*(-a/b)^(1/3))/(-a/b)^(1/3)))/b

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Sympy [A]  time = 1.32502, size = 22, normalized size = 0.09 \[ \operatorname{RootSum}{\left (27 t^{3} b^{4} + a, \left ( t \mapsto t \log{\left (- 3 t b + x \right )} \right )\right )} + \frac{x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/((b*x**3+a)**2)**(1/2),x)

[Out]

RootSum(27*_t**3*b**4 + a, Lambda(_t, _t*log(-3*_t*b + x))) + x/b

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GIAC/XCAS [A]  time = 0.316725, size = 193, normalized size = 0.82 \[ \frac{\left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right ){\rm sign}\left (b x^{3} + a\right )}{3 \, b} + \frac{x{\rm sign}\left (b x^{3} + a\right )}{b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right ){\rm sign}\left (b x^{3} + a\right )}{3 \, b^{2}} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right ){\rm sign}\left (b x^{3} + a\right )}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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