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

Optimal. Leaf size=25 \[ \frac{\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0144127, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\left (a+b x^3+c x^6\right )^{p+1}}{3 (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.62997, size = 19, normalized size = 0.76 \[ \frac{\left (a + b x^{3} + c x^{6}\right )^{p + 1}}{3 \left (p + 1\right )} \]

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+a)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0325372, size = 24, normalized size = 0.96 \[ \frac{\left (a+b x^3+c x^6\right )^{p+1}}{3 p+3} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^3 + c*x^6)^(1 + p)/(3 + 3*p)

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Maple [A]  time = 0.008, size = 24, normalized size = 1. \[{\frac{ \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{1+p}}{3+3\,p}} \]

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+a)^p,x)

[Out]

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

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Maxima [A]  time = 0.849011, size = 45, normalized size = 1.8 \[ \frac{{\left (c x^{6} + b x^{3} + a\right )}{\left (c x^{6} + b x^{3} + a\right )}^{p}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.276342, size = 45, normalized size = 1.8 \[ \frac{{\left (c x^{6} + b x^{3} + a\right )}{\left (c x^{6} + b x^{3} + a\right )}^{p}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280736, size = 92, normalized size = 3.68 \[ \frac{c x^{6} e^{\left (p{\rm ln}\left (c x^{6} + b x^{3} + a\right )\right )} + b x^{3} e^{\left (p{\rm ln}\left (c x^{6} + b x^{3} + a\right )\right )} + a e^{\left (p{\rm ln}\left (c x^{6} + b x^{3} + a\right )\right )}}{3 \,{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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