3.2792 \(\int (c x)^n (a+b x^n)^p \, dx\)

Optimal. Leaf size=52 \[ \frac {(c x)^{n+1} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,p+\frac {1}{n}+2;2+\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.19, 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} \[ \frac {(c x)^{n+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (1+\frac {1}{n},-p;2+\frac {1}{n};-\frac {b x^n}{a}\right )}{c (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^n*(a + b*x^n)^p,x]

[Out]

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

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])

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 58, normalized size = 1.12 \[ \frac {x (c x)^n \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (1+\frac {1}{n},-p;2+\frac {1}{n};-\frac {b x^n}{a}\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^n*(a + b*x^n)^p,x]

[Out]

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

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{n} + a\right )}^{p} \left (c x\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^n + a)^p*(c*x)^n, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Simp
lification assuming x near 0Simplification assuming c near 0Simplification assuming x near 0Simplification ass
uming c near 0Unable to divide, perhaps due to rounding error%%%{-1,[0,0,2,1,0,1,2]%%%}+%%%{-1,[0,0,2,1,0,0,2]
%%%} / %%%{1,[0,0,3,2,1,2,2]%%%}+%%%{2,[0,0,3,2,1,1,2]%%%}+%%%{1,[0,0,3,2,1,0,2]%%%} Error: Bad Argument Value

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{n} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^p*(c*x)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^n*(a + b*x^n)^p,x)

[Out]

int((c*x)^n*(a + b*x^n)^p, x)

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sympy [C]  time = 7.32, size = 48, normalized size = 0.92 \[ \frac {a^{p} c^{n} x x^{n} \Gamma \left (1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {1}{n} \\ 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**p*c**n*x*x**n*gamma(1 + 1/n)*hyper((-p, 1 + 1/n), (2 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(2 + 1/n))

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