Integrand size = 20, antiderivative size = 53 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {x \left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3}+2 p,\frac {4}{3},-\frac {b x^3}{a}\right )}{a} \] Output:
x*(b*x^3+a)*(b^2*x^6+2*a*b*x^3+a^2)^p*hypergeom([1, 4/3+2*p],[4/3],-b*x^3/ a)/a
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 3.98 \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {4^{-p} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}\right )^{-2 p} \left (\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}\right )^{-2 p} \left (\left (a+b x^3\right )^2\right )^p \operatorname {AppellF1}\left (1+2 p,-2 p,-2 p,2 (1+p),-\frac {(-1)^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}},\frac {i+\sqrt {3}-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}}{3 i+\sqrt {3}}\right )}{\sqrt [3]{b} (1+2 p)} \] Input:
Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]
Output:
(((-1)^(2/3)*a^(1/3) + b^(1/3)*x)*((a + b*x^3)^2)^p*AppellF1[1 + 2*p, -2*p , -2*p, 2*(1 + p), -(((-1)^(2/3)*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/((1 + ( -1)^(1/3))*a^(1/3))), (I + Sqrt[3] - ((2*I)*b^(1/3)*x)/a^(1/3))/(3*I + Sqr t[3])])/(4^p*b^(1/3)*(1 + 2*p)*((a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1 )^(1/3))*a^(1/3)))^(2*p)*((I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I + Sqrt[3]))^( 2*p))
Time = 0.18 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1385, 778}
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.
\(\displaystyle \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx\) |
\(\Big \downarrow \) 1385 |
\(\displaystyle \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \int \left (\frac {b x^3}{a}+1\right )^{2 p}dx\) |
\(\Big \downarrow \) 778 |
\(\displaystyle x \left (\frac {b x^3}{a}+1\right )^{-2 p} \left (a^2+2 a b x^3+b^2 x^6\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-2 p,\frac {4}{3},-\frac {b x^3}{a}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^p,x]
Output:
(x*(a^2 + 2*a*b*x^3 + b^2*x^6)^p*Hypergeometric2F1[1/3, -2*p, 4/3, -((b*x^ 3)/a)])/(1 + (b*x^3)/a)^(2*p)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 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)]
\[\int \left (b^{2} x^{6}+2 a \,x^{3} b +a^{2}\right )^{p}d x\]
Input:
int((b^2*x^6+2*a*b*x^3+a^2)^p,x)
Output:
int((b^2*x^6+2*a*b*x^3+a^2)^p,x)
\[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} \,d x } \] Input:
integrate((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)
\[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{p}\, dx \] Input:
integrate((b**2*x**6+2*a*b*x**3+a**2)**p,x)
Output:
Integral((a**2 + 2*a*b*x**3 + b**2*x**6)**p, x)
\[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} \,d x } \] Input:
integrate((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)
\[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int { {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{p} \,d x } \] Input:
integrate((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)
Timed out. \[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\int {\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^p \,d x \] Input:
int((a^2 + b^2*x^6 + 2*a*b*x^3)^p,x)
Output:
int((a^2 + b^2*x^6 + 2*a*b*x^3)^p, x)
\[ \int \left (a^2+2 a b x^3+b^2 x^6\right )^p \, dx=\frac {\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p} x +36 \left (\int \frac {\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p}}{6 b p \,x^{3}+b \,x^{3}+6 a p +a}d x \right ) a \,p^{2}+6 \left (\int \frac {\left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )^{p}}{6 b p \,x^{3}+b \,x^{3}+6 a p +a}d x \right ) a p}{6 p +1} \] Input:
int((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 + 36*int((a**2 + 2*a*b*x**3 + b**2*x **6)**p/(6*a*p + a + 6*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/(6*a*p + a + 6*b*p*x**3 + b*x**3),x)*a*p)/(6*p + 1)