3.130 \(\int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx\)

Optimal. Leaf size=41 \[ \frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b (2 p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0612889, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p}{3 b (2 p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.4896, size = 37, normalized size = 0.9 \[ \frac{\left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}}{6 b \left (2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b**2*x**6+2*a*b*x**3+a**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.017983, size = 30, normalized size = 0.73 \[ \frac{\left (a+b x^3\right ) \left (\left (a+b x^3\right )^2\right )^p}{3 (2 b p+b)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]

[Out]

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

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Maple [A]  time = 0.008, size = 40, normalized size = 1. \[{\frac{ \left ( b{x}^{3}+a \right ) \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}}{3\,b \left ( 1+2\,p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b^2*x^6+2*a*b*x^3+a^2)^p,x)

[Out]

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

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Maxima [A]  time = 0.772493, size = 41, normalized size = 1. \[ \frac{{\left (b x^{3} + a\right )}{\left (b x^{3} + a\right )}^{2 \, p}}{3 \, b{\left (2 \, p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.268102, size = 50, normalized size = 1.22 \[ \frac{{\left (b x^{3} + a\right )}{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p}}{3 \,{\left (2 \, b p + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b**2*x**6+2*a*b*x**3+a**2)**p,x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.282747, size = 84, normalized size = 2.05 \[ \frac{b x^{3} e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )} + a e^{\left (p{\rm ln}\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )\right )}}{3 \,{\left (2 \, b p + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x^2,x, algorithm="giac")

[Out]

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