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

Optimal. Leaf size=24 \[ \frac{\left (b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

[Out]

(b*x^3 + c*x^6)^(1 + p)/(3*(1 + p))

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Rubi [A]  time = 0.0126921, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{\left (b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(b*x^3 + c*x^6)^(1 + p)/(3*(1 + p))

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Rubi in Sympy [A]  time = 5.01885, size = 17, normalized size = 0.71 \[ \frac{\left (b x^{3} + c x^{6}\right )^{p + 1}}{3 \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x**3 + c*x**6)**(p + 1)/(3*(p + 1))

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Mathematica [A]  time = 0.0454302, size = 24, normalized size = 1. \[ \frac{\left (x^3 \left (b+c x^3\right )\right )^{p+1}}{3 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(b + c*x^3))^(1 + p)/(3*(1 + p))

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Maple [A]  time = 0.005, size = 31, normalized size = 1.3 \[{\frac{{x}^{3} \left ( c{x}^{3}+b \right ) \left ( c{x}^{6}+b{x}^{3} \right ) ^{p}}{3+3\,p}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.922656, size = 47, normalized size = 1.96 \[ \frac{{\left (c x^{6} + b x^{3}\right )} e^{\left (p \log \left (c x^{3} + b\right ) + 3 \, p \log \left (x\right )\right )}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c*x^6 + b*x^3)*e^(p*log(c*x^3 + b) + 3*p*log(x))/(p + 1)

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Fricas [A]  time = 0.280533, size = 42, normalized size = 1.75 \[ \frac{{\left (c x^{6} + b x^{3}\right )}{\left (c x^{6} + b x^{3}\right )}^{p}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270885, size = 65, normalized size = 2.71 \[ \frac{c x^{6} e^{\left (p{\rm ln}\left (c x^{6} + b x^{3}\right )\right )} + b x^{3} e^{\left (p{\rm ln}\left (c x^{6} + b x^{3}\right )\right )}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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