Integrand size = 28, antiderivative size = 393 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}-\frac {\left (b^2 c^2-2 a b c d+3 a^2 d^2\right ) (e x)^{3/2}}{c^2 d e^3 \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{c^2 d^{3/2} e^2 \left (\sqrt {c}+\sqrt {d} x\right )}-\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}}+\frac {\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 c^{7/4} d^{7/4} e^{3/2} \sqrt {c+d x^2}} \]
-(3*a^2*d^2-2*a*b*c*d+b^2*c^2)*(e*x)^(3/2)/c^2/d/e^3/(d*x^2+c)^(1/2)-2*a^2 /c/e/(e*x)^(1/2)/(d*x^2+c)^(1/2)+(3*a^2*d^2-2*a*b*c*d+3*b^2*c^2)*(e*x)^(1/ 2)*(d*x^2+c)^(1/2)/c^2/d^(3/2)/e^2/(c^(1/2)+x*d^(1/2))-(3*a^2*d^2-2*a*b*c* d+3*b^2*c^2)*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))^2)^(1/2)/ cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2)))*EllipticE(sin(2*arctan( d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2))*(c^(1/2)+x*d^(1/2))*((d *x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/c^(7/4)/d^(7/4)/e^(3/2)/(d*x^2+c)^(1/ 2)+1/2*(3*a^2*d^2-2*a*b*c*d+3*b^2*c^2)*(cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c ^(1/4)/e^(1/2)))^2)^(1/2)/cos(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2) ))*EllipticF(sin(2*arctan(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2))),1/2*2^(1/2 ))*(c^(1/2)+x*d^(1/2))*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/c^(7/4)/d^( 7/4)/e^(3/2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.32 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {x \left (-3 b^2 c^2 x^2+6 a b c d x^2-3 a^2 d \left (2 c+3 d x^2\right )+\left (3 b^2 c^2-2 a b c d+3 a^2 d^2\right ) x^2 \sqrt {1+\frac {d x^2}{c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {d x^2}{c}\right )\right )}{3 c^2 d (e x)^{3/2} \sqrt {c+d x^2}} \]
(x*(-3*b^2*c^2*x^2 + 6*a*b*c*d*x^2 - 3*a^2*d*(2*c + 3*d*x^2) + (3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2)*x^2*Sqrt[1 + (d*x^2)/c]*Hypergeometric2F1[1/2, 3/ 4, 7/4, -((d*x^2)/c)]))/(3*c^2*d*(e*x)^(3/2)*Sqrt[c + d*x^2])
Time = 0.51 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {365, 27, 362, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {e x} \left (b^2 c x^2+a (2 b c-3 a d)\right )}{2 \left (d x^2+c\right )^{3/2}}dx}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {e x} \left (b^2 c x^2+a (2 b c-3 a d)\right )}{\left (d x^2+c\right )^{3/2}}dx}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {-\frac {1}{2} \left (-\frac {3 a^2 d}{c}+2 a b-\frac {3 b^2 c}{d}\right ) \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c d e \sqrt {c+d x^2}}}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {-\frac {\left (-\frac {3 a^2 d}{c}+2 a b-\frac {3 b^2 c}{d}\right ) \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{e}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c d e \sqrt {c+d x^2}}}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {-\frac {\left (-\frac {3 a^2 d}{c}+2 a b-\frac {3 b^2 c}{d}\right ) \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{e}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c d e \sqrt {c+d x^2}}}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (-\frac {3 a^2 d}{c}+2 a b-\frac {3 b^2 c}{d}\right ) \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{e}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c d e \sqrt {c+d x^2}}}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {-\frac {\left (-\frac {3 a^2 d}{c}+2 a b-\frac {3 b^2 c}{d}\right ) \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{e}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c d e \sqrt {c+d x^2}}}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {-\frac {\left (-\frac {3 a^2 d}{c}+2 a b-\frac {3 b^2 c}{d}\right ) \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{e}-\frac {(e x)^{3/2} \left (3 a^2 d^2-2 a b c d+b^2 c^2\right )}{c d e \sqrt {c+d x^2}}}{c e^2}-\frac {2 a^2}{c e \sqrt {e x} \sqrt {c+d x^2}}\) |
(-2*a^2)/(c*e*Sqrt[e*x]*Sqrt[c + d*x^2]) + (-(((b^2*c^2 - 2*a*b*c*d + 3*a^ 2*d^2)*(e*x)^(3/2))/(c*d*e*Sqrt[c + d*x^2])) - ((2*a*b - (3*b^2*c)/d - (3* a^2*d)/c)*(-((-((e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(Sqrt[c]*e + Sqrt[d]*e*x)) + (c^(1/4)*Sqrt[e]*(Sqrt[c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sq rt[c]*e + Sqrt[d]*e*x)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)* Sqrt[e])], 1/2])/(d^(1/4)*Sqrt[c + d*x^2]))/Sqrt[d]) + (c^(1/4)*Sqrt[e]*(S qrt[c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sqrt[c]*e + Sqrt[d]*e*x) ^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(2*d^ (3/4)*Sqrt[c + d*x^2])))/e)/(c*e^2)
3.9.54.3.1 Defintions of rubi rules used
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_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Time = 3.84 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.83
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 \left (d e \,x^{2}+c e \right ) a^{2}}{e^{2} c^{2} \sqrt {x \left (d e \,x^{2}+c e \right )}}-\frac {x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d e \,c^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {\left (\frac {b^{2}}{e d}+\frac {d \,a^{2}}{c^{2} e}+\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 d \,c^{2} e}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(327\) |
| risch | \(-\frac {2 a^{2} \sqrt {d \,x^{2}+c}}{c^{2} e \sqrt {e x}}+\frac {\left (\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d^{2} \sqrt {d e \,x^{3}+c e x}}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )}{d}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{c^{2} e \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(448\) |
| default | \(\frac {6 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-4 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +6 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-3 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+2 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -3 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-6 a^{2} d^{3} x^{2}+4 a b c \,d^{2} x^{2}-2 b^{2} c^{2} d \,x^{2}-4 c \,a^{2} d^{2}}{2 \sqrt {d \,x^{2}+c}\, d^{2} e \sqrt {e x}\, c^{2}}\) | \(594\) |
(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)*(-2*(d*e*x^2+c*e)/e^2/c^ 2*a^2/(x*(d*e*x^2+c*e))^(1/2)-1/d/e*x^2/c^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)/(( x^2+c/d)*d*e*x)^(1/2)+(b^2/e/d+d/c^2/e*a^2+1/2/d/c^2*(a^2*d^2-2*a*b*c*d+b^ 2*c^2)/e)*(-c*d)^(1/2)/d*((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-2*(x- (-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)/(d*e*x^3+c *e*x)^(1/2)*(-2*(-c*d)^(1/2)/d*EllipticE(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)* d)^(1/2),1/2*2^(1/2))+(-c*d)^(1/2)/d*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^ (1/2)*d)^(1/2),1/2*2^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.42 \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left ({\left (3 \, b^{2} c^{2} d - 2 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{3} + {\left (3 \, b^{2} c^{3} - 2 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (2 \, a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + 3 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{c^{2} d^{3} e^{2} x^{3} + c^{3} d^{2} e^{2} x} \]
-(((3*b^2*c^2*d - 2*a*b*c*d^2 + 3*a^2*d^3)*x^3 + (3*b^2*c^3 - 2*a*b*c^2*d + 3*a^2*c*d^2)*x)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPInverse (-4*c/d, 0, x)) + (2*a^2*c*d^2 + (b^2*c^2*d - 2*a*b*c*d^2 + 3*a^2*d^3)*x^2 )*sqrt(d*x^2 + c)*sqrt(e*x))/(c^2*d^3*e^2*x^3 + c^3*d^2*e^2*x)
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{\left (e x\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} \left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2}{(e x)^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2}{{\left (e\,x\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]