Integrand size = 13, antiderivative size = 61 \[ \int x \left (a+b (c x)^n\right )^p \, dx=\frac {1}{2} x^2 \left (a+b (c x)^n\right )^p \left (1+\frac {b (c x)^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {2+n}{n},-\frac {b (c x)^n}{a}\right ) \] Output:
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)
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int x \left (a+b (c x)^n\right )^p \, dx=\frac {1}{2} x^2 \left (a+b (c x)^n\right )^p \left (1+\frac {b (c x)^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,1+\frac {2}{n},-\frac {b (c x)^n}{a}\right ) \] Input:
Integrate[x*(a + b*(c*x)^n)^p,x]
Output:
(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)
Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {891, 27, 889, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \left (a+b (c x)^n\right )^p \, dx\) |
\(\Big \downarrow \) 891 |
\(\displaystyle \frac {\int x \left (b (c x)^n+a\right )^pd(c x)}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int c x \left (b (c x)^n+a\right )^pd(c x)}{c^2}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {\left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \int c x \left (\frac {b (c x)^n}{a}+1\right )^pd(c x)}{c^2}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2}{n},-p,\frac {n+2}{n},-\frac {b (c x)^n}{a}\right )\) |
Input:
Int[x*(a + b*(c*x)^n)^p,x]
Output:
(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)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[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])
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Simp[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]
\[\int x \left (a +b \left (c x \right )^{n}\right )^{p}d x\]
Input:
int(x*(a+b*(c*x)^n)^p,x)
Output:
int(x*(a+b*(c*x)^n)^p,x)
\[ \int x \left (a+b (c x)^n\right )^p \, dx=\int { {\left (\left (c x\right )^{n} b + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*(c*x)^n)^p,x, algorithm="fricas")
Output:
integral(((c*x)^n*b + a)^p*x, x)
\[ \int x \left (a+b (c x)^n\right )^p \, dx=\int x \left (a + b \left (c x\right )^{n}\right )^{p}\, dx \] Input:
integrate(x*(a+b*(c*x)**n)**p,x)
Output:
Integral(x*(a + b*(c*x)**n)**p, x)
\[ \int x \left (a+b (c x)^n\right )^p \, dx=\int { {\left (\left (c x\right )^{n} b + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*(c*x)^n)^p,x, algorithm="maxima")
Output:
integrate(((c*x)^n*b + a)^p*x, x)
\[ \int x \left (a+b (c x)^n\right )^p \, dx=\int { {\left (\left (c x\right )^{n} b + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*(c*x)^n)^p,x, algorithm="giac")
Output:
integrate(((c*x)^n*b + a)^p*x, x)
Timed out. \[ \int x \left (a+b (c x)^n\right )^p \, dx=\int x\,{\left (a+b\,{\left (c\,x\right )}^n\right )}^p \,d x \] Input:
int(x*(a + b*(c*x)^n)^p,x)
Output:
int(x*(a + b*(c*x)^n)^p, x)
\[ \int x \left (a+b (c x)^n\right )^p \, dx=\frac {\left (x^{n} c^{n} b +a \right )^{p} x^{2}+\left (\int \frac {\left (x^{n} c^{n} b +a \right )^{p} x}{x^{n} c^{n} b n p +2 x^{n} c^{n} b +a n p +2 a}d x \right ) a \,n^{2} p^{2}+2 \left (\int \frac {\left (x^{n} c^{n} b +a \right )^{p} x}{x^{n} c^{n} b n p +2 x^{n} c^{n} b +a n p +2 a}d x \right ) a n p}{n p +2} \] Input:
int(x*(a+b*(c*x)^n)^p,x)
Output:
((x**n*c**n*b + a)**p*x**2 + int(((x**n*c**n*b + a)**p*x)/(x**n*c**n*b*n*p + 2*x**n*c**n*b + a*n*p + 2*a),x)*a*n**2*p**2 + 2*int(((x**n*c**n*b + a)* *p*x)/(x**n*c**n*b*n*p + 2*x**n*c**n*b + a*n*p + 2*a),x)*a*n*p)/(n*p + 2)