3.30.25 \(\int x (a+b (c x)^n)^p \, dx\) [2925]

Optimal. Leaf size=61 \[ \frac {1}{2} x^2 \left (a+b (c x)^n\right )^p \left (1+\frac {b (c x)^n}{a}\right )^{-p} \, _2F_1\left (\frac {2}{n},-p;\frac {2+n}{n};-\frac {b (c x)^n}{a}\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {374, 12, 372, 371} \begin {gather*} \frac {1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {2}{n},-p;\frac {n+2}{n};-\frac {b (c x)^n}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*(c*x)^n)^p,x]

[Out]

(x^2*(a + b*(c*x)^n)^p*Hypergeometric2F1[2/n, -p, (2 + n)/n, -((b*(c*x)^n)/a)])/(2*(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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 374

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

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Mathematica [A]
time = 0.05, size = 61, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 \left (a+b (c x)^n\right )^p \left (1+\frac {b (c x)^n}{a}\right )^{-p} \, _2F_1\left (\frac {2}{n},-p;1+\frac {2}{n};-\frac {b (c x)^n}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*(c*x)^n)^p,x]

[Out]

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

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x \left (a +b \left (c x \right )^{n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p*x, x)

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Fricas [F]
time = 0.39, size = 15, normalized size = 0.25 \begin {gather*} {\rm integral}\left ({\left (\left (c x\right )^{n} b + a\right )}^{p} x, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p*x, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(a + b*(c*x)**n)**p, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p*x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*(c*x)^n)^p,x)

[Out]

int(x*(a + b*(c*x)^n)^p, x)

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