Integrand size = 28, antiderivative size = 384 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {(b c-a d)^2 (e x)^{3/2}}{c d^2 e \sqrt {c+d x^2}}+\frac {2 b^2 (e x)^{3/2} \sqrt {c+d x^2}}{5 d^2 e}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e x} \sqrt {c+d x^2}}{5 c d^{5/2} \left (\sqrt {c}+\sqrt {d} x\right )}+\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 c^{3/4} d^{11/4} \sqrt {c+d x^2}}-\frac {\left (21 b^2 c^2-30 a b c d+5 a^2 d^2\right ) \sqrt {e} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{10 c^{3/4} d^{11/4} \sqrt {c+d x^2}} \]
(-a*d+b*c)^2*(e*x)^(3/2)/c/d^2/e/(d*x^2+c)^(1/2)+2/5*b^2*(e*x)^(3/2)*(d*x^ 2+c)^(1/2)/d^2/e-1/5*(5*a^2*d^2-30*a*b*c*d+21*b^2*c^2)*(e*x)^(1/2)*(d*x^2+ c)^(1/2)/c/d^(5/2)/(c^(1/2)+x*d^(1/2))+1/5*(5*a^2*d^2-30*a*b*c*d+21*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*arct an(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))*e^(1/2)*((d*x ^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/c^(3/4)/d^(11/4)/(d*x^2+c)^(1/2)-1/10*( 5*a^2*d^2-30*a*b*c*d+21*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)))*Ell ipticF(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))*e^(1/2)*((d*x^2+c)/(c^(1/2)+x*d^(1/2))^2)^(1/2)/c^(3/4)/d ^(11/4)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.31 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x} \left (x \left (-10 a b c d+5 a^2 d^2+b^2 c \left (7 c+2 d x^2\right )\right )+\left (-21 b^2 c^2+30 a b c d-5 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {c}{d x^2}\right )\right )}{5 c d^2 \sqrt {c+d x^2}} \]
(Sqrt[e*x]*(x*(-10*a*b*c*d + 5*a^2*d^2 + b^2*c*(7*c + 2*d*x^2)) + (-21*b^2 *c^2 + 30*a*b*c*d - 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*Hypergeometric2F1[-1/ 4, 1/2, 3/4, -(c/(d*x^2))]))/(5*c*d^2*Sqrt[c + d*x^2])
Time = 0.48 (sec) , antiderivative size = 382, 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 = {366, 27, 363, 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.
| \(\displaystyle \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {e x} \left (3 b^2 c^2-2 b^2 d x^2 c-6 a b d c+a^2 d^2\right )}{2 \sqrt {d x^2+c}}dx}{c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {e x} \left (3 b^2 c^2-2 b^2 d x^2 c-6 a b d c+a^2 d^2\right )}{\sqrt {d x^2+c}}dx}{2 c d^2}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\frac {1}{5} \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \int \frac {\sqrt {e x}}{\sqrt {d x^2+c}}dx-\frac {4 b^2 c (e x)^{3/2} \sqrt {c+d x^2}}{5 e}}{2 c d^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\frac {2 \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \int \frac {e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{5 e}-\frac {4 b^2 c (e x)^{3/2} \sqrt {c+d x^2}}{5 e}}{2 c d^2}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\frac {2 \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\sqrt {c} e \int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {c} e \sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}-\frac {4 b^2 c (e x)^{3/2} \sqrt {c+d x^2}}{5 e}}{2 c d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\frac {2 \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \left (\frac {\sqrt {c} e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}-\frac {4 b^2 c (e x)^{3/2} \sqrt {c+d x^2}}{5 e}}{2 c d^2}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\frac {2 \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\int \frac {\sqrt {c} e-\sqrt {d} e x}{\sqrt {d x^2+c}}d\sqrt {e x}}{\sqrt {d}}\right )}{5 e}-\frac {4 b^2 c (e x)^{3/2} \sqrt {c+d x^2}}{5 e}}{2 c d^2}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {(e x)^{3/2} (b c-a d)^2}{c d^2 e \sqrt {c+d x^2}}-\frac {\frac {2 \left (5 a^2 d^2-30 a b c d+21 b^2 c^2\right ) \left (\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 d^{3/4} \sqrt {c+d x^2}}-\frac {\frac {\sqrt [4]{c} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{d} \sqrt {c+d x^2}}-\frac {e^2 \sqrt {e x} \sqrt {c+d x^2}}{\sqrt {c} e+\sqrt {d} e x}}{\sqrt {d}}\right )}{5 e}-\frac {4 b^2 c (e x)^{3/2} \sqrt {c+d x^2}}{5 e}}{2 c d^2}\) |
((b*c - a*d)^2*(e*x)^(3/2))/(c*d^2*e*Sqrt[c + d*x^2]) - ((-4*b^2*c*(e*x)^( 3/2)*Sqrt[c + d*x^2])/(5*e) + (2*(21*b^2*c^2 - 30*a*b*c*d + 5*a^2*d^2)*(-( (-((e^2*Sqrt[e*x]*Sqrt[c + d*x^2])/(Sqrt[c]*e + Sqrt[d]*e*x)) + (c^(1/4)*S qrt[e]*(Sqrt[c]*e + Sqrt[d]*e*x)*Sqrt[(c*e^2 + d*e^2*x^2)/(Sqrt[c]*e + Sqr t[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]*(Sqrt[c]*e + Sq rt[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])))/(5*e))/(2*c*d^2)
3.9.52.3.1 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[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -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 = 4.08 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.83
| method | result | size |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {e \,x^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{2} c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}+\frac {2 b^{2} x \sqrt {d e \,x^{3}+c e x}}{5 d^{2}}+\frac {\left (\frac {b \left (2 a d -b c \right ) e}{d^{2}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) e}{2 d^{2} c}-\frac {3 b^{2} c e}{5 d^{2}}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(319\) |
| risch | \(\frac {2 b^{2} x^{2} \sqrt {d \,x^{2}+c}\, e}{5 d^{2} \sqrt {e x}}+\frac {\left (\frac {\left (10 a b d -8 b^{2} c \right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{d \sqrt {d e \,x^{3}+c e x}}+\left (5 a^{2} d^{2}-10 a b c d +5 b^{2} c^{2}\right ) \left (\frac {x^{2}}{c \sqrt {\left (x^{2}+\frac {c}{d}\right ) d e x}}-\frac {\sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, \left (-\frac {2 \sqrt {-c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {-c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{2 c d \sqrt {d e \,x^{3}+c e x}}\right )\right ) e \sqrt {e x \left (d \,x^{2}+c \right )}}{5 d^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(442\) |
| default | \(-\frac {\sqrt {e x}\, \left (10 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}-60 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d +42 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-5 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2}+30 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d -21 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3}-4 b^{2} c \,d^{2} x^{4}-10 a^{2} d^{3} x^{2}+20 a b c \,d^{2} x^{2}-14 b^{2} c^{2} d \,x^{2}\right )}{10 \sqrt {d \,x^{2}+c}\, d^{3} x c}\) | \(597\) |
(e*x*(d*x^2+c))^(1/2)/e/x*(e*x)^(1/2)/(d*x^2+c)^(1/2)*(1/d^2*e*x^2/c*(a^2* d^2-2*a*b*c*d+b^2*c^2)/((x^2+c/d)*d*e*x)^(1/2)+2/5*b^2/d^2*x*(d*e*x^3+c*e* x)^(1/2)+(b*(2*a*d-b*c)*e/d^2-1/2/d^2/c*(a^2*d^2-2*a*b*c*d+b^2*c^2)*e-3/5* b^2/d^2*c*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 = 156, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (21 \, b^{2} c^{3} - 30 \, a b c^{2} d + 5 \, a^{2} c d^{2} + {\left (21 \, b^{2} c^{2} d - 30 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d e} {\rm weierstrassZeta}\left (-\frac {4 \, c}{d}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right )\right ) + {\left (2 \, b^{2} c d^{2} x^{3} + {\left (7 \, b^{2} c^{2} d - 10 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {e x}}{5 \, {\left (c d^{4} x^{2} + c^{2} d^{3}\right )}} \]
1/5*((21*b^2*c^3 - 30*a*b*c^2*d + 5*a^2*c*d^2 + (21*b^2*c^2*d - 30*a*b*c*d ^2 + 5*a^2*d^3)*x^2)*sqrt(d*e)*weierstrassZeta(-4*c/d, 0, weierstrassPInve rse(-4*c/d, 0, x)) + (2*b^2*c*d^2*x^3 + (7*b^2*c^2*d - 10*a*b*c*d^2 + 5*a^ 2*d^3)*x)*sqrt(d*x^2 + c)*sqrt(e*x))/(c*d^4*x^2 + c^2*d^3)
\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e x} \left (a + b x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} \sqrt {e x}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x}\,{\left (b\,x^2+a\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]