\(\int (c x)^m (a+b x^n)^p \, dx\) [2789]

Optimal result

Integrand size = 15, antiderivative size = 59 \[ \int (c x)^m \left (a+b x^n\right )^p \, dx=\frac {(c x)^{1+m} \left (a+b x^n\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1+m}{n}+p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a c (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \[ \int (c x)^m \left (a+b x^n\right )^p \, dx=\frac {(c x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{c (m+1)} \]

[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 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])

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

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08 \[ \int (c x)^m \left (a+b x^n\right )^p \, dx=\frac {x (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,1+\frac {1+m}{n},-\frac {b x^n}{a}\right )}{1+m} \]

[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)

Maple [F]

\[\int \left (c x \right )^{m} \left (a +b \,x^{n}\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*x^n + a)^p*(c*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.24 \[ \int (c x)^m \left (a+b x^n\right )^p \, dx=\frac {a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} + p - \frac {1}{n}} c^{m} x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

a**(m/n + 1/n)*a**(-m/n + p - 1/n)*c**m*x**(m + 1)*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))

Maxima [F]

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

[In]

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

[Out]

integrate((b*x^n + a)^p*(c*x)^m, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x^n + a)^p*(c*x)^m, x)

Mupad [F(-1)]

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

[In]

int((c*x)^m*(a + b*x^n)^p,x)

[Out]

int((c*x)^m*(a + b*x^n)^p, x)