Integrand size = 28, antiderivative size = 288 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=-\frac {4 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \sqrt {c+d x^2}}{231 d e^3}-\frac {2 \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \sqrt {e x} \left (c+d x^2\right )^{3/2}}{231 c d e^3}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}+\frac {2 b^2 \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e^3}-\frac {4 c^{3/4} \left (3 b^2 c^2-11 a d (6 b c+7 a d)\right ) \left (\sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {c+d x^2}{\left (\sqrt {c}+\sqrt {d} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{231 d^{5/4} e^{5/2} \sqrt {c+d x^2}} \]
-2/3*a^2*(d*x^2+c)^(5/2)/c/e/(e*x)^(3/2)-2/231*(3*b^2*c^2-11*a*d*(7*a*d+6* b*c))*(d*x^2+c)^(3/2)*(e*x)^(1/2)/c/d/e^3+2/11*b^2*(d*x^2+c)^(5/2)*(e*x)^( 1/2)/d/e^3-4/231*(3*b^2*c^2-11*a*d*(7*a*d+6*b*c))*(e*x)^(1/2)*(d*x^2+c)^(1 /2)/d/e^3-4/231*c^(3/4)*(3*b^2*c^2-11*a*d*(7*a*d+6*b*c))*(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^(5/4)/e^(5/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 11.16 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\frac {x^{5/2} \left (\frac {2 \left (c+d x^2\right ) \left (77 a^2 d \left (-c+d x^2\right )+66 a b d x^2 \left (3 c+d x^2\right )+3 b^2 x^2 \left (4 c^2+13 c d x^2+7 d^2 x^4\right )\right )}{d x^{3/2}}+\frac {8 i c \left (-3 b^2 c^2+66 a b c d+77 a^2 d^2\right ) \sqrt {1+\frac {c}{d x^2}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}}}{\sqrt {x}}\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {d}}} d}\right )}{231 (e x)^{5/2} \sqrt {c+d x^2}} \]
(x^(5/2)*((2*(c + d*x^2)*(77*a^2*d*(-c + d*x^2) + 66*a*b*d*x^2*(3*c + d*x^ 2) + 3*b^2*x^2*(4*c^2 + 13*c*d*x^2 + 7*d^2*x^4)))/(d*x^(3/2)) + ((8*I)*c*( -3*b^2*c^2 + 66*a*b*c*d + 77*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*Ar cSinh[Sqrt[(I*Sqrt[c])/Sqrt[d]]/Sqrt[x]], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[d]]* d)))/(231*(e*x)^(5/2)*Sqrt[c + d*x^2])
Time = 0.35 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {365, 27, 363, 248, 248, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle \frac {2 \int \frac {\left (3 b^2 c x^2+a (6 b c+7 a d)\right ) \left (d x^2+c\right )^{3/2}}{2 \sqrt {e x}}dx}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (3 b^2 c x^2+a (6 b c+7 a d)\right ) \left (d x^2+c\right )^{3/2}}{\sqrt {e x}}dx}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {6 b^2 c \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e}-\frac {\left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \int \frac {\left (d x^2+c\right )^{3/2}}{\sqrt {e x}}dx}{11 d}}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {6 b^2 c \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e}-\frac {\left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \left (\frac {6}{7} c \int \frac {\sqrt {d x^2+c}}{\sqrt {e x}}dx+\frac {2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 e}\right )}{11 d}}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {\frac {6 b^2 c \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e}-\frac {\left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \left (\frac {6}{7} c \left (\frac {2}{3} c \int \frac {1}{\sqrt {e x} \sqrt {d x^2+c}}dx+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )+\frac {2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 e}\right )}{11 d}}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {6 b^2 c \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e}-\frac {\left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \left (\frac {6}{7} c \left (\frac {4 c \int \frac {1}{\sqrt {d x^2+c}}d\sqrt {e x}}{3 e}+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )+\frac {2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 e}\right )}{11 d}}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {6 b^2 c \sqrt {e x} \left (c+d x^2\right )^{5/2}}{11 d e}-\frac {\left (3 b^2 c^2-11 a d (7 a d+6 b c)\right ) \left (\frac {6}{7} c \left (\frac {2 c^{3/4} \left (\sqrt {c} e+\sqrt {d} e x\right ) \sqrt {\frac {c e^2+d e^2 x^2}{\left (\sqrt {c} e+\sqrt {d} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{d} e^{3/2} \sqrt {c+d x^2}}+\frac {2 \sqrt {e x} \sqrt {c+d x^2}}{3 e}\right )+\frac {2 \sqrt {e x} \left (c+d x^2\right )^{3/2}}{7 e}\right )}{11 d}}{3 c e^2}-\frac {2 a^2 \left (c+d x^2\right )^{5/2}}{3 c e (e x)^{3/2}}\) |
(-2*a^2*(c + d*x^2)^(5/2))/(3*c*e*(e*x)^(3/2)) + ((6*b^2*c*Sqrt[e*x]*(c + d*x^2)^(5/2))/(11*d*e) - ((3*b^2*c^2 - 11*a*d*(6*b*c + 7*a*d))*((2*Sqrt[e* x]*(c + d*x^2)^(3/2))/(7*e) + (6*c*((2*Sqrt[e*x]*Sqrt[c + d*x^2])/(3*e) + (2*c^(3/4)*(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)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])] , 1/2])/(3*d^(1/4)*e^(3/2)*Sqrt[c + d*x^2])))/7))/(11*d))/(3*c*e^2)
3.9.37.3.1 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] :> 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[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]
Time = 3.08 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(-\frac {2 \sqrt {d \,x^{2}+c}\, \left (-21 b^{2} d^{2} x^{6}-66 a b \,d^{2} x^{4}-39 b^{2} c d \,x^{4}-77 a^{2} d^{2} x^{2}-198 a b c d \,x^{2}-12 b^{2} c^{2} x^{2}+77 a^{2} c d \right )}{231 d x \,e^{2} \sqrt {e x}}+\frac {4 c \left (77 a^{2} d^{2}+66 a b c d -3 b^{2} c^{2}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {e x \left (d \,x^{2}+c \right )}}{231 d^{2} \sqrt {d e \,x^{3}+c e x}\, e^{2} \sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(260\) |
| elliptic | \(\frac {\sqrt {e x \left (d \,x^{2}+c \right )}\, \left (-\frac {2 c \,a^{2} \sqrt {d e \,x^{3}+c e x}}{3 e^{3} x^{2}}+\frac {2 b^{2} d \,x^{4} \sqrt {d e \,x^{3}+c e x}}{11 e^{3}}+\frac {2 \left (\frac {2 b d \left (a d +b c \right )}{e^{2}}-\frac {9 b^{2} d c}{11 e^{2}}\right ) x^{2} \sqrt {d e \,x^{3}+c e x}}{7 d e}+\frac {2 \left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e^{2}}-\frac {5 \left (\frac {2 b d \left (a d +b c \right )}{e^{2}}-\frac {9 b^{2} d c}{11 e^{2}}\right ) c}{7 d}\right ) \sqrt {d e \,x^{3}+c e x}}{3 d e}+\frac {\left (\frac {2 a c \left (a d +b c \right )}{e^{2}}-\frac {d c \,a^{2}}{3 e^{2}}-\frac {\left (\frac {a^{2} d^{2}+4 a b c d +b^{2} c^{2}}{e^{2}}-\frac {5 \left (\frac {2 b d \left (a d +b c \right )}{e^{2}}-\frac {9 b^{2} d c}{11 e^{2}}\right ) c}{7 d}\right ) c}{3 d}\right ) \sqrt {-c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-c d}}{d}\right ) d}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right )}{d \sqrt {d e \,x^{3}+c e x}}\right )}{\sqrt {e x}\, \sqrt {d \,x^{2}+c}}\) | \(403\) |
| default | \(\frac {\frac {2 b^{2} d^{4} x^{8}}{11}+\frac {4 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c \,d^{2} x}{3}+\frac {8 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} d x}{7}-\frac {4 \sqrt {2}\, \sqrt {-c d}\, \sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {\frac {-d x +\sqrt {-c d}}{\sqrt {-c d}}}\, \sqrt {-\frac {x d}{\sqrt {-c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {-c d}}{\sqrt {-c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{3} x}{77}+\frac {4 a b \,d^{4} x^{6}}{7}+\frac {40 b^{2} c \,d^{3} x^{6}}{77}+\frac {2 a^{2} d^{4} x^{4}}{3}+\frac {16 c a b \,x^{4} d^{3}}{7}+\frac {34 b^{2} c^{2} d^{2} x^{4}}{77}+\frac {12 a b \,c^{2} d^{2} x^{2}}{7}+\frac {8 b^{2} c^{3} d \,x^{2}}{77}-\frac {2 a^{2} c^{2} d^{2}}{3}}{\sqrt {d \,x^{2}+c}\, x \,d^{2} e^{2} \sqrt {e x}}\) | \(415\) |
-2/231*(d*x^2+c)^(1/2)*(-21*b^2*d^2*x^6-66*a*b*d^2*x^4-39*b^2*c*d*x^4-77*a ^2*d^2*x^2-198*a*b*c*d*x^2-12*b^2*c^2*x^2+77*a^2*c*d)/d/x/e^2/(e*x)^(1/2)+ 4/231*c*(77*a^2*d^2+66*a*b*c*d-3*b^2*c^2)/d^2*(-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 = 146, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=-\frac {2 \, {\left (4 \, {\left (3 \, b^{2} c^{3} - 66 \, a b c^{2} d - 77 \, a^{2} c d^{2}\right )} \sqrt {d e} x^{2} {\rm weierstrassPInverse}\left (-\frac {4 \, c}{d}, 0, x\right ) - {\left (21 \, b^{2} d^{3} x^{6} - 77 \, a^{2} c d^{2} + 3 \, {\left (13 \, b^{2} c d^{2} + 22 \, a b d^{3}\right )} x^{4} + {\left (12 \, b^{2} c^{2} d + 198 \, a b c d^{2} + 77 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {e x}\right )}}{231 \, d^{2} e^{3} x^{2}} \]
-2/231*(4*(3*b^2*c^3 - 66*a*b*c^2*d - 77*a^2*c*d^2)*sqrt(d*e)*x^2*weierstr assPInverse(-4*c/d, 0, x) - (21*b^2*d^3*x^6 - 77*a^2*c*d^2 + 3*(13*b^2*c*d ^2 + 22*a*b*d^3)*x^4 + (12*b^2*c^2*d + 198*a*b*c*d^2 + 77*a^2*d^3)*x^2)*sq rt(d*x^2 + c)*sqrt(e*x))/(d^2*e^3*x^2)
Result contains complex when optimal does not.
Time = 16.30 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\frac {a^{2} c^{\frac {3}{2}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {1}{4}\right )} + \frac {a^{2} \sqrt {c} d \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {a b c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {a b \sqrt {c} d x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {b^{2} \sqrt {c} d x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {d x^{2} e^{i \pi }}{c}} \right )}}{2 e^{\frac {5}{2}} \Gamma \left (\frac {13}{4}\right )} \]
a**2*c**(3/2)*gamma(-3/4)*hyper((-3/4, -1/2), (1/4,), d*x**2*exp_polar(I*p i)/c)/(2*e**(5/2)*x**(3/2)*gamma(1/4)) + a**2*sqrt(c)*d*sqrt(x)*gamma(1/4) *hyper((-1/2, 1/4), (5/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(5/2)*gamma(5/ 4)) + a*b*c**(3/2)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), d*x**2*ex p_polar(I*pi)/c)/(e**(5/2)*gamma(5/4)) + a*b*sqrt(c)*d*x**(5/2)*gamma(5/4) *hyper((-1/2, 5/4), (9/4,), d*x**2*exp_polar(I*pi)/c)/(e**(5/2)*gamma(9/4) ) + b**2*c**(3/2)*x**(5/2)*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), d*x**2*ex p_polar(I*pi)/c)/(2*e**(5/2)*gamma(9/4)) + b**2*sqrt(c)*d*x**(9/2)*gamma(9 /4)*hyper((-1/2, 9/4), (13/4,), d*x**2*exp_polar(I*pi)/c)/(2*e**(5/2)*gamm a(13/4))
\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{(e x)^{5/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{{\left (e\,x\right )}^{5/2}} \,d x \]