3.2789 \(\int (c x)^m (a+b x^n)^p \, dx\)

Optimal. Leaf size=59 \[ \frac{(c x)^{m+1} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,\frac{m+1}{n}+p+1;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a c (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0234132, antiderivative size = 67, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac{(c x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*(a + b*x^n)^p,x]

[Out]

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

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

Mathematica [A]  time = 0.0161758, size = 64, normalized size = 1.08 \[ \frac{x (c x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m*(a + b*x^n)^p,x]

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^p*(c*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^n + a)^p*(c*x)^m, x)

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Sympy [C]  time = 9.99296, size = 58, normalized size = 0.98 \begin{align*} \frac{a^{p} c^{m} x x^{m} \Gamma \left (\frac{m}{n} + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m}{n} + \frac{1}{n} \\ \frac{m}{n} + 1 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**p*c**m*x*x**m*gamma(m/n + 1/n)*hyper((-p, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(
m/n + 1 + 1/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^p*(c*x)^m, x)