Integrand size = 22, antiderivative size = 58 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {x^2 \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {5}{3}+2 p,\frac {5}{3},-\frac {b x^3}{a}\right )}{2 a} \] Output:
1/2*x^2*(b*x^3+a)*(b^2*x^6+2*a*b*x^3+a^2)^p*hypergeom([1, 5/3+2*p],[5/3],- b*x^3/a)/a
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.88 \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {1}{2} x^2 \left (\left (a+b x^3\right )^2\right )^p \left (1+\frac {b x^3}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-2 p,\frac {5}{3},-\frac {b x^3}{a}\right ) \] Input:
Integrate[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]
Output:
(x^2*((a + b*x^3)^2)^p*Hypergeometric2F1[2/3, -2*p, 5/3, -((b*x^3)/a)])/(2 *(1 + (b*x^3)/a)^(2*p))
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1385, 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.
\(\displaystyle \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx\) |
\(\Big \downarrow \) 1385 |
\(\displaystyle \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \int x \left (\frac {b x^3}{a}+1\right )^{2 p}dx\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {1}{2} x^2 \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \operatorname {Hypergeometric2F1}\left (\frac {2}{3},-2 p,\frac {5}{3},-\frac {b x^3}{a}\right )\) |
Input:
Int[x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]
Output:
(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[2/3, -2*p, 5/3, -((b* x^3)/a)])/(2*(1 + (b*x^3)/a)^(2*p))
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/(1 + 2*c*(x^n/b))^(2* FracPart[p])) Int[u*(1 + 2*c*(x^n/b))^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)]
\[\int x \left (b^{2} x^{6}+2 a \,x^{3} b +a^{2}\right )^{p}d x\]
Input:
int(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x)
Output:
int(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x)
\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x \,d x } \] Input:
integrate(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x, algorithm="fricas")
Output:
integral((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x, x)
\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int x \left (\left (a + b x^{3}\right )^{2}\right )^{p}\, dx \] Input:
integrate(x*(b**2*x**6+2*a*b*x**3+a**2)**p,x)
Output:
Integral(x*((a + b*x**3)**2)**p, x)
\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x \,d x } \] Input:
integrate(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x, algorithm="maxima")
Output:
integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x, x)
\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} x \,d x } \] Input:
integrate(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x, algorithm="giac")
Output:
integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^p*x, x)
Timed out. \[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int x\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^p \,d x \] Input:
int(x*(a^2 + b^2*x^6 + 2*a*b*x^3)^p,x)
Output:
int(x*(a^2 + b^2*x^6 + 2*a*b*x^3)^p, x)
\[ \int x \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p} x^{2}+18 \left (\int \frac {\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p} x}{3 b p \,x^{3}+b \,x^{3}+3 a p +a}d x \right ) a \,p^{2}+6 \left (\int \frac {\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p} x}{3 b p \,x^{3}+b \,x^{3}+3 a p +a}d x \right ) a p}{6 p +2} \] Input:
int(x*(b^2*x^6+2*a*b*x^3+a^2)^p,x)
Output:
((a**2 + 2*a*b*x**3 + b**2*x**6)**p*x**2 + 18*int(((a**2 + 2*a*b*x**3 + b* *2*x**6)**p*x)/(3*a*p + a + 3*b*p*x**3 + b*x**3),x)*a*p**2 + 6*int(((a**2 + 2*a*b*x**3 + b**2*x**6)**p*x)/(3*a*p + a + 3*b*p*x**3 + b*x**3),x)*a*p)/ (2*(3*p + 1))