\(\int x^2 (a+b x^3+c x^6)^p \, dx\) [259]

Optimal result

Integrand size = 18, antiderivative size = 130 \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=-\frac {2^{1+p} \left (-\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-1-p} \left (a+b x^3+c x^6\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 \sqrt {b^2-4 a c} (1+p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1366, 638} \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=-\frac {2^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x^3+c x^6\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {2 c x^3+b+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{3 (p+1) \sqrt {b^2-4 a c}} \]

[In]

Int[x^2*(a + b*x^3 + c*x^6)^p,x]

[Out]

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

Rule 638

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)/(q*(p + 1)*((q - b - 2*c*x)/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/
(2*q)], x]] /; FreeQ[{a, b, c, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[4*p]

Rule 1366

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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {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] (verified)

Time = 0.15 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06 \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\frac {2^{-1+p} \left (b-\sqrt {b^2-4 a c}+2 c x^3\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x^3}{2 \sqrt {b^2-4 a c}}\right )}{3 c (1+p)} \]

[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)

Maple [F]

\[\int x^{2} \left (c \,x^{6}+b \,x^{3}+a \right )^{p}d x\]

[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)

Fricas [F]

\[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2} \,d x } \]

[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)

Giac [F]

\[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int { {\left (c x^{6} + b x^{3} + a\right )}^{p} x^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+b x^3+c x^6\right )^p \, dx=\int x^2\,{\left (c\,x^6+b\,x^3+a\right )}^p \,d x \]

[In]

int(x^2*(a + b*x^3 + c*x^6)^p,x)

[Out]

int(x^2*(a + b*x^3 + c*x^6)^p, x)