Integrand size = 28, antiderivative size = 65 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=-\frac {2 \left (1+\frac {b x^2}{a}\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-2 p,\frac {3}{4},-\frac {b x^2}{a}\right )}{d \sqrt {d x}} \] Output:
-2*(b^2*x^4+2*a*b*x^2+a^2)^p*hypergeom([-1/4, -2*p],[3/4],-b*x^2/a)/d/(d*x )^(1/2)/((1+b*x^2/a)^(2*p))
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=-\frac {2 x \left (\left (a+b x^2\right )^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-2 p,\frac {3}{4},-\frac {b x^2}{a}\right )}{(d x)^{3/2}} \] Input:
Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(3/2),x]
Output:
(-2*x*((a + b*x^2)^2)^p*Hypergeometric2F1[-1/4, -2*p, 3/4, -((b*x^2)/a)])/ ((d*x)^(3/2)*(1 + (b*x^2)/a)^(2*p))
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1385, 278}
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.
\(\displaystyle \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1385 |
\(\displaystyle \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \int \frac {\left (\frac {b x^2}{a}+1\right )^{2 p}}{(d x)^{3/2}}dx\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {2 \left (\frac {b x^2}{a}+1\right )^{-2 p} \left (a^2+2 a b x^2+b^2 x^4\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},-2 p,\frac {3}{4},-\frac {b x^2}{a}\right )}{d \sqrt {d x}}\) |
Input:
Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^p/(d*x)^(3/2),x]
Output:
(-2*(a^2 + 2*a*b*x^2 + b^2*x^4)^p*Hypergeometric2F1[-1/4, -2*p, 3/4, -((b* x^2)/a)])/(d*Sqrt[d*x]*(1 + (b*x^2)/a)^(2*p))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[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)]
\[\int \frac {\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{\left (d x \right )^{\frac {3}{2}}}d x\]
Input:
int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x)
Output:
int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x)
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(d*x)*(b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d^2*x^2), x)
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int \frac {\left (\left (a + b x^{2}\right )^{2}\right )^{p}}{\left (d x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b**2*x**4+2*a*b*x**2+a**2)**p/(d*x)**(3/2),x)
Output:
Integral(((a + b*x**2)**2)**p/(d*x)**(3/2), x)
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x, algorithm="maxima")
Output:
integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2), x)
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int { \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{p}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x, algorithm="giac")
Output:
integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^p/(d*x)^(3/2), x)
Timed out. \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\int \frac {{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^p}{{\left (d\,x\right )}^{3/2}} \,d x \] Input:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^p/(d*x)^(3/2),x)
Output:
int((a^2 + b^2*x^4 + 2*a*b*x^2)^p/(d*x)^(3/2), x)
\[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^p}{(d x)^{3/2}} \, dx=\frac {2 \sqrt {d}\, \left (\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}+32 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{8 b p \,x^{4}-b \,x^{4}+8 a p \,x^{2}-a \,x^{2}}d x \right ) a \,p^{2}-4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{p}}{8 b p \,x^{4}-b \,x^{4}+8 a p \,x^{2}-a \,x^{2}}d x \right ) a p \right )}{\sqrt {x}\, d^{2} \left (8 p -1\right )} \] Input:
int((b^2*x^4+2*a*b*x^2+a^2)^p/(d*x)^(3/2),x)
Output:
(2*sqrt(d)*((a**2 + 2*a*b*x**2 + b**2*x**4)**p + 32*sqrt(x)*int((sqrt(x)*( a**2 + 2*a*b*x**2 + b**2*x**4)**p)/(8*a*p*x**2 - a*x**2 + 8*b*p*x**4 - b*x **4),x)*a*p**2 - 4*sqrt(x)*int((sqrt(x)*(a**2 + 2*a*b*x**2 + b**2*x**4)**p )/(8*a*p*x**2 - a*x**2 + 8*b*p*x**4 - b*x**4),x)*a*p))/(sqrt(x)*d**2*(8*p - 1))