3.2.17 \(\int x^2 (b+2 c x^3) (b x^3+c x^6)^p \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {1588} \begin {gather*} \frac {\left (b x^3+c x^6\right )^{p+1}}{3 (p+1)} \end {gather*}

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))

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^p \, dx &=\frac {\left (b x^3+c x^6\right )^{1+p}}{3 (1+p)}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 97, normalized size = 4.04 \begin {gather*} \frac {x^3 \left (x^3 \left (b+c x^3\right )\right )^p \left (\frac {c x^3}{b}+1\right )^{-p} \left (2 c (p+1) x^3 \, _2F_1\left (-p,p+2;p+3;-\frac {c x^3}{b}\right )+b (p+2) \, _2F_1\left (-p,p+1;p+2;-\frac {c x^3}{b}\right )\right )}{3 (p+1) (p+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(x^3*(b + c*x^3))^p*(b*(2 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, -((c*x^3)/b)] + 2*c*(1 + p)*x^3*Hyperg
eometric2F1[-p, 2 + p, 3 + p, -((c*x^3)/b)]))/(3*(1 + p)*(2 + p)*(1 + (c*x^3)/b)^p)

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IntegrateAlgebraic [F]  time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (b+2 c x^3\right ) \left (b x^3+c x^6\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][x^2*(b + 2*c*x^3)*(b*x^3 + c*x^6)^p, x]

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fricas [A]  time = 0.77, size = 31, normalized size = 1.29 \begin {gather*} \frac {{\left (c x^{6} + b x^{3}\right )} {\left (c x^{6} + b x^{3}\right )}^{p}}{3 \, {\left (p + 1\right )}} \end {gather*}

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, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.67, size = 22, normalized size = 0.92 \begin {gather*} \frac {{\left (c x^{6} + b x^{3}\right )}^{p + 1}}{3 \, {\left (p + 1\right )}} \end {gather*}

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, algorithm="giac")

[Out]

1/3*(c*x^6 + b*x^3)^(p + 1)/(p + 1)

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maple [A]  time = 0.00, size = 31, normalized size = 1.29 \begin {gather*} \frac {\left (c \,x^{3}+b \right ) x^{3} \left (c \,x^{6}+b \,x^{3}\right )^{p}}{3 p +3} \end {gather*}

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

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maxima [A]  time = 0.59, size = 35, normalized size = 1.46 \begin {gather*} \frac {{\left (c x^{6} + b x^{3}\right )} e^{\left (p \log \left (c x^{3} + b\right ) + 3 \, p \log \relax (x)\right )}}{3 \, {\left (p + 1\right )}} \end {gather*}

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, 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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mupad [B]  time = 2.07, size = 31, normalized size = 1.29 \begin {gather*} \frac {x^3\,\left (c\,x^3+b\right )\,{\left (c\,x^6+b\,x^3\right )}^p}{3\,\left (p+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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