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

Optimal. Leaf size=58 \[ \frac{x^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (1,2 p+\frac{5}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a} \]

[Out]

(x^2*(a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[1, 5/3 + 2*p, 5/3, -((b*x^3)/a)])/(2*a)

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Rubi [A]  time = 0.0151618, antiderivative size = 60, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1356, 364} \[ \frac{1}{2} x^2 \left (\frac{b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[2/3, -2*p, 5/3, -((b*x^3)/a)])/(2*(1 + (b*x^3)/a)^(2*p))

Rule 1356

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a
+ b*x^n + c*x^(2*n))^FracPart[p])/(1 + (2*c*x^n)/b)^(2*FracPart[p]), Int[(d*x)^m*(1 + (2*c*x^n)/b)^(2*p), x],
x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[2*p]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx &=\left (\left (1+\frac{b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p\right ) \int x \left (1+\frac{b x^3}{a}\right )^{2 p} \, dx\\ &=\frac{1}{2} x^2 \left (1+\frac{b x^3}{a}\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right )\\ \end{align*}

Mathematica [A]  time = 0.0061913, size = 51, normalized size = 0.88 \[ \frac{1}{2} x^2 \left (\left (a+b x^3\right )^2\right )^p \left (\frac{b x^3}{a}+1\right )^{-2 p} \, _2F_1\left (\frac{2}{3},-2 p;\frac{5}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*((a + b*x^3)^2)^p*Hypergeometric2F1[2/3, -2*p, 5/3, -((b*x^3)/a)])/(2*(1 + (b*x^3)/a)^(2*p))

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \begin{align*} \int x \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*((a + b*x**3)**2)**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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