3.598 \(\int x^5 \left (a+b x^3\right )^p \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b x^3\right )^{p+2}}{3 b^2 (p+2)}-\frac{a \left (a+b x^3\right )^{p+1}}{3 b^2 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0658253, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (a+b x^3\right )^{p+2}}{3 b^2 (p+2)}-\frac{a \left (a+b x^3\right )^{p+1}}{3 b^2 (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.1917, size = 37, normalized size = 0.77 \[ - \frac{a \left (a + b x^{3}\right )^{p + 1}}{3 b^{2} \left (p + 1\right )} + \frac{\left (a + b x^{3}\right )^{p + 2}}{3 b^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + b*x**3)**(p + 1)/(3*b**2*(p + 1)) + (a + b*x**3)**(p + 2)/(3*b**2*(p + 2
))

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Mathematica [A]  time = 0.02909, size = 40, normalized size = 0.83 \[ \frac{\left (a+b x^3\right )^{p+1} \left (b (p+1) x^3-a\right )}{3 b^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^3)^p,x]

[Out]

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

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Maple [A]  time = 0.007, size = 42, normalized size = 0.9 \[ -{\frac{ \left ( b{x}^{3}+a \right ) ^{1+p} \left ( -{x}^{3}pb-b{x}^{3}+a \right ) }{3\,{b}^{2} \left ({p}^{2}+3\,p+2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^3+a)^p,x)

[Out]

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

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Maxima [A]  time = 1.45276, size = 63, normalized size = 1.31 \[ \frac{{\left (b^{2}{\left (p + 1\right )} x^{6} + a b p x^{3} - a^{2}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.246074, size = 78, normalized size = 1.62 \[ \frac{{\left ({\left (b^{2} p + b^{2}\right )} x^{6} + a b p x^{3} - a^{2}\right )}{\left (b x^{3} + a\right )}^{p}}{3 \,{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 21.3249, size = 524, normalized size = 10.92 \[ \begin{cases} \frac{a^{p} x^{6}}{6} & \text{for}\: b = 0 \\\frac{a \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{a \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac{1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac{2 a \log{\left (2 \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{a}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{b x^{3} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + x \right )}}{3 a b^{2} + 3 b^{3} x^{3}} + \frac{b x^{3} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac{1}{b}} + 4 x^{2} \right )}}{3 a b^{2} + 3 b^{3} x^{3}} - \frac{2 b x^{3} \log{\left (2 \right )}}{3 a b^{2} + 3 b^{3} x^{3}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + x \right )}}{3 b^{2}} - \frac{a \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac{1}{b}} + 4 x^{2} \right )}}{3 b^{2}} + \frac{x^{3}}{3 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac{a b p x^{3} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac{b^{2} p x^{6} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} + \frac{b^{2} x^{6} \left (a + b x^{3}\right )^{p}}{3 b^{2} p^{2} + 9 b^{2} p + 6 b^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**3+a)**p,x)

[Out]

Piecewise((a**p*x**6/6, Eq(b, 0)), (a*log(-(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x
)/(3*a*b**2 + 3*b**3*x**3) + a*log(4*(-1)**(2/3)*a**(2/3)*(1/b)**(2/3) + 4*(-1)*
*(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*a*b**2 + 3*b**3*x**3) - 2*a*log(2)/(
3*a*b**2 + 3*b**3*x**3) + a/(3*a*b**2 + 3*b**3*x**3) + b*x**3*log(-(-1)**(1/3)*a
**(1/3)*(1/b)**(1/3) + x)/(3*a*b**2 + 3*b**3*x**3) + b*x**3*log(4*(-1)**(2/3)*a*
*(2/3)*(1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*a*b**2
+ 3*b**3*x**3) - 2*b*x**3*log(2)/(3*a*b**2 + 3*b**3*x**3), Eq(p, -2)), (-a*log(-
(-1)**(1/3)*a**(1/3)*(1/b)**(1/3) + x)/(3*b**2) - a*log(4*(-1)**(2/3)*a**(2/3)*(
1/b)**(2/3) + 4*(-1)**(1/3)*a**(1/3)*x*(1/b)**(1/3) + 4*x**2)/(3*b**2) + x**3/(3
*b), Eq(p, -1)), (-a**2*(a + b*x**3)**p/(3*b**2*p**2 + 9*b**2*p + 6*b**2) + a*b*
p*x**3*(a + b*x**3)**p/(3*b**2*p**2 + 9*b**2*p + 6*b**2) + b**2*p*x**6*(a + b*x*
*3)**p/(3*b**2*p**2 + 9*b**2*p + 6*b**2) + b**2*x**6*(a + b*x**3)**p/(3*b**2*p**
2 + 9*b**2*p + 6*b**2), True))

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GIAC/XCAS [A]  time = 0.213444, size = 138, normalized size = 2.88 \[ \frac{{\left (b x^{3} + a\right )}^{2} p e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} -{\left (b x^{3} + a\right )} a p e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} +{\left (b x^{3} + a\right )}^{2} e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )} - 2 \,{\left (b x^{3} + a\right )} a e^{\left (p{\rm ln}\left (b x^{3} + a\right )\right )}}{3 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((b*x^3 + a)^2*p*e^(p*ln(b*x^3 + a)) - (b*x^3 + a)*a*p*e^(p*ln(b*x^3 + a)) +
(b*x^3 + a)^2*e^(p*ln(b*x^3 + a)) - 2*(b*x^3 + a)*a*e^(p*ln(b*x^3 + a)))/((p^2
+ 3*p + 2)*b^2)