Integrand size = 20, antiderivative size = 55 \[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=x \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 p,\frac {3}{2},-\frac {b x^2}{a}\right ) \] Output:
x*(b^2*x^4+2*a*b*x^2+a^2)^p*hypergeom([1/2, -2*p],[3/2],-b*x^2/a)/((1+b*x^ 2/a)^(2*p))
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=x \left (\left (a+b x^2\right )^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 p,\frac {3}{2},-\frac {b x^2}{a}\right ) \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]
Output:
(x*((a + b*x^2)^2)^p*Hypergeometric2F1[1/2, -2*p, 3/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^(2*p)
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1385, 237}
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.
\(\displaystyle \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx\) |
\(\Big \downarrow \) 1385 |
\(\displaystyle \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \int \left (\frac {b x^2}{a}+1\right )^{2 p}dx\) |
\(\Big \downarrow \) 237 |
\(\displaystyle x \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 p,\frac {3}{2},-\frac {b x^2}{a}\right )\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^p,x]
Output:
(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[1/2, -2*p, 3/2, -((b*x^ 2)/a)])/(1 + (b*x^2)/a)^(2*p)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*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)]
\[\int \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}d x\]
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^p,x)
Output:
int((b^2*x^4+2*a*b*x^2+a^2)^p,x)
\[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int { {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \,d x } \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^p,x, algorithm="fricas")
Output:
integral((b^2*x^4 + 2*a*b*x^2 + a^2)^p, x)
\[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{p}\, dx \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**p,x)
Output:
Integral((a**2 + 2*a*b*x**2 + b**2*x**4)**p, x)
\[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int { {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \,d x } \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^p,x, algorithm="maxima")
Output:
integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p, x)
\[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int { {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p} \,d x } \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^p,x, algorithm="giac")
Output:
integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p, x)
Timed out. \[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\int {\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p \,d x \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^p,x)
Output:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^p, x)
\[ \int \left (a^2+2 a b x^2+b^2 x^4\right )^p \, dx=\frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p} x +16 \left (\int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{4 b p \,x^{2}+b \,x^{2}+4 a p +a}d x \right ) a \,p^{2}+4 \left (\int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{4 b p \,x^{2}+b \,x^{2}+4 a p +a}d x \right ) a p}{4 p +1} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^p,x)
Output:
((a**2 + 2*a*b*x**2 + b**2*x**4)**p*x + 16*int((a**2 + 2*a*b*x**2 + b**2*x **4)**p/(4*a*p + a + 4*b*p*x**2 + b*x**2),x)*a*p**2 + 4*int((a**2 + 2*a*b* x**2 + b**2*x**4)**p/(4*a*p + a + 4*b*p*x**2 + b*x**2),x)*a*p)/(4*p + 1)