Integrand size = 28, antiderivative size = 288 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {4 c \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) e \sqrt {e x} \sqrt {c+d x^2}}{231 d^3}+\frac {2 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) (e x)^{5/2} \sqrt {c+d x^2}}{77 d^2 e}-\frac {2 b (3 b c-10 a d) (e x)^{5/2} \left (c+d x^2\right )^{3/2}}{55 d^2 e}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}-\frac {2 c^{7/4} \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) e^{3/2} \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{13/4} \sqrt {c+d x^2}} \]
-2/55*b*(-10*a*d+3*b*c)*(e*x)^(5/2)*(d*x^2+c)^(3/2)/d^2/e+2/15*b^2*(e*x)^( 9/2)*(d*x^2+c)^(3/2)/d/e^3+2/77*(11*a^2*d^2+b*c*(-10*a*d+3*b*c))*(e*x)^(5/ 2)*(d*x^2+c)^(1/2)/d^2/e+4/231*c*(11*a^2*d^2+b*c*(-10*a*d+3*b*c))*e*(e*x)^ (1/2)*(d*x^2+c)^(1/2)/d^3-2/231*c^(7/4)*(11*a^2*d^2+b*c*(-10*a*d+3*b*c))*e ^(3/2)*(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)/d^(13/4)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 22.35 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.78 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {(e x)^{3/2} \left (\frac {2 \sqrt {x} \left (c+d x^2\right ) \left (55 a^2 d^2 \left (2 c+3 d x^2\right )+10 a b d \left (-10 c^2+6 c d x^2+21 d^2 x^4\right )+b^2 \left (30 c^3-18 c^2 d x^2+14 c d^2 x^4+77 d^3 x^6\right )\right )}{5 d^3}-\frac {4 i c^2 \left (3 b^2 c^2-10 a b c d+11 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d^3}\right )}{231 x^{3/2} \sqrt {c+d x^2}} \]
((e*x)^(3/2)*((2*Sqrt[x]*(c + d*x^2)*(55*a^2*d^2*(2*c + 3*d*x^2) + 10*a*b* d*(-10*c^2 + 6*c*d*x^2 + 21*d^2*x^4) + b^2*(30*c^3 - 18*c^2*d*x^2 + 14*c*d ^2*x^4 + 77*d^3*x^6)))/(5*d^3) - ((4*I)*c^2*(3*b^2*c^2 - 10*a*b*c*d + 11*a ^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[d] ]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]*d^3)))/(231*x^(3/2)*Sqrt[c + d *x^2])
Time = 0.38 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {367, 27, 363, 248, 262, 266, 761}
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.
| \(\displaystyle \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx\) |
| \(\Big \downarrow \) 367 |
| \(\displaystyle \frac {2 \int \frac {3}{2} (e x)^{3/2} \sqrt {d x^2+c} \left (5 a^2 d-b (3 b c-10 a d) x^2\right )dx}{15 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (e x)^{3/2} \sqrt {d x^2+c} \left (5 a^2 d-b (3 b c-10 a d) x^2\right )dx}{5 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {5 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) \int (e x)^{3/2} \sqrt {d x^2+c}dx}{11 d}-\frac {2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{11 d e}}{5 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {5 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) \left (\frac {2}{7} c \int \frac {(e x)^{3/2}}{\sqrt {d x^2+c}}dx+\frac {2 (e x)^{5/2} \sqrt {c+d x^2}}{7 e}\right )}{11 d}-\frac {2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{11 d e}}{5 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {5 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) \left (\frac {2}{7} c \left (\frac {2 e \sqrt {e x} \sqrt {c+d x^2}}{3 d}-\frac {c e^2 \int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx}{3 d}\right )+\frac {2 (e x)^{5/2} \sqrt {c+d x^2}}{7 e}\right )}{11 d}-\frac {2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{11 d e}}{5 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {5 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) \left (\frac {2}{7} c \left (\frac {2 e \sqrt {e x} \sqrt {c+d x^2}}{3 d}-\frac {2 c e \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{3 d}\right )+\frac {2 (e x)^{5/2} \sqrt {c+d x^2}}{7 e}\right )}{11 d}-\frac {2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{11 d e}}{5 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {5 \left (11 a^2 d^2+b c (3 b c-10 a d)\right ) \left (\frac {2}{7} c \left (\frac {2 e \sqrt {e x} \sqrt {c+d x^2}}{3 d}-\frac {c^{3/4} \sqrt {e} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 d^{5/4} \sqrt {c+d x^2}}\right )+\frac {2 (e x)^{5/2} \sqrt {c+d x^2}}{7 e}\right )}{11 d}-\frac {2 b (e x)^{5/2} \left (c+d x^2\right )^{3/2} (3 b c-10 a d)}{11 d e}}{5 d}+\frac {2 b^2 (e x)^{9/2} \left (c+d x^2\right )^{3/2}}{15 d e^3}\) |
(2*b^2*(e*x)^(9/2)*(c + d*x^2)^(3/2))/(15*d*e^3) + ((-2*b*(3*b*c - 10*a*d) *(e*x)^(5/2)*(c + d*x^2)^(3/2))/(11*d*e) + (5*(11*a^2*d^2 + b*c*(3*b*c - 1 0*a*d))*((2*(e*x)^(5/2)*Sqrt[c + d*x^2])/(7*e) + (2*c*((2*e*Sqrt[e*x]*Sqrt [c + d*x^2])/(3*d) - (c^(3/4)*Sqrt[e]*(Sqrt[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)*Sq rt[e*x])/(c^(1/4)*Sqrt[e])], 1/2])/(3*d^(5/4)*Sqrt[c + d*x^2])))/7))/(11*d ))/(5*d)
3.9.22.3.1 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] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, 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[d^2*(e*x)^(m + 3)*((a + b*x^2)^(p + 1)/(b*e^3*(m + 2*p + 5))), x] + Simp[1/(b*(m + 2*p + 5)) Int[(e*x)^m*(a + b*x^2)^p*Simp[b*c^2* (m + 2*p + 5) - d*(a*d*(m + 3) - 2*b*c*(m + 2*p + 5))*x^2, x], x], x] /; Fr eeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + 2*p + 5, 0]
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]
Time = 3.15 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.98
| method | result | size |
| risch | \(\frac {2 \left (77 b^{2} d^{3} x^{6}+210 a b \,d^{3} x^{4}+14 b^{2} c \,d^{2} x^{4}+165 a^{2} d^{3} x^{2}+60 a b c \,d^{2} x^{2}-18 b^{2} c^{2} d \,x^{2}+110 c \,a^{2} d^{2}-100 a b \,c^{2} d +30 b^{2} c^{3}\right ) x \sqrt {d \,x^{2}+c}\, e^{2}}{1155 d^{3} \sqrt {e x}}-\frac {2 c^{2} \left (11 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) e^{2} \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{4} \sqrt {d e \,x^{3}+c e x}\, \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(283\) |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \sqrt {e x}\, \left (\frac {2 b^{2} e \,x^{6} \sqrt {d e \,x^{3}+c e x}}{15}+\frac {2 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) x^{4} \sqrt {d e \,x^{3}+c e x}}{11 d e}+\frac {2 \left (a \left (a d +2 b c \right ) e^{2}-\frac {9 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) c}{11 d}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (a^{2} c \,e^{2}-\frac {5 \left (a \left (a d +2 b c \right ) e^{2}-\frac {9 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) c}{11 d}\right ) c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}-\frac {\left (a^{2} c \,e^{2}-\frac {5 \left (a \left (a d +2 b c \right ) e^{2}-\frac {9 \left (b \left (2 a d +b c \right ) e^{2}-\frac {13 b^{2} e^{2} c}{15}\right ) c}{11 d}\right ) c}{7 d}\right ) c \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{3 d^{2} \sqrt {d e \,x^{3}+c e x}}\right )}{e x \sqrt {d \,x^{2}+c}}\) | \(419\) |
| default | \(-\frac {2 e \sqrt {e x}\, \left (-77 b^{2} d^{5} x^{9}-210 a b \,d^{5} x^{7}-91 b^{2} c \,d^{4} x^{7}+55 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c^{2} d^{2}-50 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{3} d +15 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{4}-165 a^{2} d^{5} x^{5}-270 a b c \,d^{4} x^{5}+4 b^{2} c^{2} d^{3} x^{5}-275 a^{2} c \,d^{4} x^{3}+40 a b \,c^{2} d^{3} x^{3}-12 b^{2} c^{3} d^{2} x^{3}-110 a^{2} c^{2} d^{3} x +100 a b \,c^{3} d^{2} x -30 b^{2} c^{4} d x \right )}{1155 x \sqrt {d \,x^{2}+c}\, d^{4}}\) | \(448\) |
2/1155*(77*b^2*d^3*x^6+210*a*b*d^3*x^4+14*b^2*c*d^2*x^4+165*a^2*d^3*x^2+60 *a*b*c*d^2*x^2-18*b^2*c^2*d*x^2+110*a^2*c*d^2-100*a*b*c^2*d+30*b^2*c^3)*x* (d*x^2+c)^(1/2)/d^3*e^2/(e*x)^(1/2)-2/231*c^2*(11*a^2*d^2-10*a*b*c*d+3*b^2 *c^2)/d^4*(-c*d)^(1/2)*((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)*EllipticF(((x+(-c*d)^(1/2)/d)/(-c*d)^(1/2)*d)^(1/2),1/2*2^(1/2)) *e^2*(e*x*(d*x^2+c))^(1/2)/(e*x)^(1/2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.58 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=-\frac {2 \, {\left (10 \, {\left (3 \, b^{2} c^{4} - 10 \, a b c^{3} d + 11 \, a^{2} c^{2} d^{2}\right )} \sqrt {d e} e {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (77 \, b^{2} d^{4} e x^{6} + 14 \, {\left (b^{2} c d^{3} + 15 \, a b d^{4}\right )} e x^{4} - 3 \, {\left (6 \, b^{2} c^{2} d^{2} - 20 \, a b c d^{3} - 55 \, a^{2} d^{4}\right )} e x^{2} + 10 \, {\left (3 \, b^{2} c^{3} d - 10 \, a b c^{2} d^{2} + 11 \, a^{2} c d^{3}\right )} e\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{1155 \, d^{4}} \]
-2/1155*(10*(3*b^2*c^4 - 10*a*b*c^3*d + 11*a^2*c^2*d^2)*sqrt(d*e)*e*weiers trassPInverse(-4*c/d, 0, x) - (77*b^2*d^4*e*x^6 + 14*(b^2*c*d^3 + 15*a*b*d ^4)*e*x^4 - 3*(6*b^2*c^2*d^2 - 20*a*b*c*d^3 - 55*a^2*d^4)*e*x^2 + 10*(3*b^ 2*c^3*d - 10*a*b*c^2*d^2 + 11*a^2*c*d^3)*e)*sqrt(d*x^2 + c)*sqrt(e*x))/d^4
Result contains complex when optimal does not.
Time = 11.27 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.52 \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\frac {a^{2} \sqrt {c} e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} + \frac {a b \sqrt {c} e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{\Gamma \left (\frac {13}{4}\right )} + \frac {b^{2} \sqrt {c} e^{\frac {3}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 \Gamma \left (\frac {17}{4}\right )} \]
a**2*sqrt(c)*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), d*x** 2*exp_polar(I*pi)/c)/(2*gamma(9/4)) + a*b*sqrt(c)*e**(3/2)*x**(9/2)*gamma( 9/4)*hyper((-1/2, 9/4), (13/4,), d*x**2*exp_polar(I*pi)/c)/gamma(13/4) + b **2*sqrt(c)*e**(3/2)*x**(13/2)*gamma(13/4)*hyper((-1/2, 13/4), (17/4,), d* x**2*exp_polar(I*pi)/c)/(2*gamma(17/4))
\[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \left (e x\right )^{\frac {3}{2}} \,d x } \]
\[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int { {\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} \left (e x\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (e x)^{3/2} \left (a+b x^2\right )^2 \sqrt {c+d x^2} \, dx=\int {\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c} \,d x \]