3.512 \(\int x^5 \sqrt [3]{a+b x^3} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^3\right )^{7/3}}{7 b^2}-\frac{a \left (a+b x^3\right )^{4/3}}{4 b^2} \]

[Out]

-(a*(a + b*x^3)^(4/3))/(4*b^2) + (a + b*x^3)^(7/3)/(7*b^2)

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Rubi [A]  time = 0.0599424, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a+b x^3\right )^{7/3}}{7 b^2}-\frac{a \left (a+b x^3\right )^{4/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3)^(1/3),x]

[Out]

-(a*(a + b*x^3)^(4/3))/(4*b^2) + (a + b*x^3)^(7/3)/(7*b^2)

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Rubi in Sympy [A]  time = 7.18537, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{3}\right )^{\frac{4}{3}}}{4 b^{2}} + \frac{\left (a + b x^{3}\right )^{\frac{7}{3}}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**3+a)**(1/3),x)

[Out]

-a*(a + b*x**3)**(4/3)/(4*b**2) + (a + b*x**3)**(7/3)/(7*b**2)

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Mathematica [A]  time = 0.0194639, size = 38, normalized size = 1. \[ \frac{\sqrt [3]{a+b x^3} \left (-3 a^2+a b x^3+4 b^2 x^6\right )}{28 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^3)^(1/3),x]

[Out]

((a + b*x^3)^(1/3)*(-3*a^2 + a*b*x^3 + 4*b^2*x^6))/(28*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-4\,b{x}^{3}+3\,a}{28\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^3+a)^(1/3),x)

[Out]

-1/28*(b*x^3+a)^(4/3)*(-4*b*x^3+3*a)/b^2

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Maxima [A]  time = 1.4422, size = 41, normalized size = 1.08 \[ \frac{{\left (b x^{3} + a\right )}^{\frac{7}{3}}}{7 \, b^{2}} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a}{4 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)*x^5,x, algorithm="maxima")

[Out]

1/7*(b*x^3 + a)^(7/3)/b^2 - 1/4*(b*x^3 + a)^(4/3)*a/b^2

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Fricas [A]  time = 0.228234, size = 46, normalized size = 1.21 \[ \frac{{\left (4 \, b^{2} x^{6} + a b x^{3} - 3 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{28 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)*x^5,x, algorithm="fricas")

[Out]

1/28*(4*b^2*x^6 + a*b*x^3 - 3*a^2)*(b*x^3 + a)^(1/3)/b^2

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Sympy [A]  time = 2.11637, size = 63, normalized size = 1.66 \[ \begin{cases} - \frac{3 a^{2} \sqrt [3]{a + b x^{3}}}{28 b^{2}} + \frac{a x^{3} \sqrt [3]{a + b x^{3}}}{28 b} + \frac{x^{6} \sqrt [3]{a + b x^{3}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{6}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**3+a)**(1/3),x)

[Out]

Piecewise((-3*a**2*(a + b*x**3)**(1/3)/(28*b**2) + a*x**3*(a + b*x**3)**(1/3)/(2
8*b) + x**6*(a + b*x**3)**(1/3)/7, Ne(b, 0)), (a**(1/3)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.263729, size = 39, normalized size = 1.03 \[ \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a}{28 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)*x^5,x, algorithm="giac")

[Out]

1/28*(4*(b*x^3 + a)^(7/3) - 7*(b*x^3 + a)^(4/3)*a)/b^2