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

Optimal. Leaf size=130 \[ -\frac{2^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 (p+1) \sqrt{b^2-4 a c}} \]

[Out]

-(2^(1 + p)*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^3 + c*x^6)^(1 + p)*Hype
rgeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(3*Sqrt[b^2 - 4*a*c]
*(1 + p))

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Rubi [A]  time = 0.0733837, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1352, 624} \[ -\frac{2^{p+1} \left (-\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{2 c x^3+b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 (p+1) \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^3 + c*x^6)^p,x]

[Out]

-(2^(1 + p)*(-((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x^3 + c*x^6)^(1 + p)*Hype
rgeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(3*Sqrt[b^2 - 4*a*c]
*(1 + p))

Rule 1352

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rule 624

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, -Simp[((a + b*x + c*
x^2)^(p + 1)*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q)])/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p
+ 1)), x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rubi steps

\begin{align*} \int x^2 \left (a+b x^3+c x^6\right )^p \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a+b x+c x^2\right )^p \, dx,x,x^3\right )\\ &=-\frac{2^{1+p} \left (-\frac{b-\sqrt{b^2-4 a c}+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{b+\sqrt{b^2-4 a c}+2 c x^3}{2 \sqrt{b^2-4 a c}}\right )}{3 \sqrt{b^2-4 a c} (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.099367, size = 138, normalized size = 1.06 \[ \frac{2^{p-1} \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^3}{\sqrt{b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{-2 c x^3-b+\sqrt{b^2-4 a c}}{2 \sqrt{b^2-4 a c}}\right )}{3 c (p+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-1 + p)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(a + b*x^3 + c*x^6)^p*Hypergeometric2F1[-p, 1 + p, 2 + p, (-b +
Sqrt[b^2 - 4*a*c] - 2*c*x^3)/(2*Sqrt[b^2 - 4*a*c])])/(3*c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/Sqrt[b^2
- 4*a*c])^p)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(c*x**6+b*x**3+a)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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