3.2924 \(\int x^2 (a+b (c x)^n)^p \, dx\)

Optimal. Leaf size=61 \[ \frac {1}{3} x^3 \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{n},-p;\frac {n+3}{n};-\frac {b (c x)^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {367, 12, 365, 364} \[ \frac {1}{3} x^3 \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{n},-p;\frac {n+3}{n};-\frac {b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*(c*x)^n)^p,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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])

Rule 365

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

Rule 367

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Dist[1/c, Subst[Int[((d*x)/c)^m*(a
+ b*x^n)^p, x], x, c*x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 61, normalized size = 1.00 \[ \frac {1}{3} x^3 \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {3}{n},-p;1+\frac {3}{n};-\frac {b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*(c*x)^n)^p,x]

[Out]

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

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fricas [F]  time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c x\right )^{n} b + a\right )}^{p} x^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p*x^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c x\right )^{n} b + a\right )}^{p} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p*x^2, x)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int x^{2} \left (b \left (c x \right )^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\left (c x\right )^{n} b + a\right )}^{p} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p*x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x^2\,{\left (a+b\,{\left (c\,x\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*(c*x)^n)^p,x)

[Out]

int(x^2*(a + b*(c*x)^n)^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*(c*x)**n)**p, x)

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