Integrand size = 26, antiderivative size = 367 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {4 (9 A b+a B) (e x)^{3/2} \sqrt {a+b x^2}}{15 e^3}+\frac {8 a (9 A b+a B) \sqrt {e x} \sqrt {a+b x^2}}{15 \sqrt {b} e^2 \left (\sqrt {a}+\sqrt {b} x\right )}+\frac {2 (9 A b+a B) (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 a e^3}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}-\frac {8 a^{5/4} (9 A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{15 b^{3/4} e^{3/2} \sqrt {a+b x^2}}+\frac {4 a^{5/4} (9 A b+a B) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 b^{3/4} e^{3/2} \sqrt {a+b x^2}} \]
2/9*(9*A*b+B*a)*(e*x)^(3/2)*(b*x^2+a)^(3/2)/a/e^3-2*A*(b*x^2+a)^(5/2)/a/e/ (e*x)^(1/2)+4/15*(9*A*b+B*a)*(e*x)^(3/2)*(b*x^2+a)^(1/2)/e^3+8/15*a*(9*A*b +B*a)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/e^2/b^(1/2)/(a^(1/2)+x*b^(1/2))-8/15*a^( 5/4)*(9*A*b+B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))^2)^(1 /2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))*EllipticE(sin(2*arc tan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b^(1/2)) *((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(3/4)/e^(3/2)/(b*x^2+a)^(1/2)+4 /15*a^(5/4)*(9*A*b+B*a)*(cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)) )^2)^(1/2)/cos(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)))*EllipticF(si n(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))*(a^(1/2)+x*b ^(1/2))*((b*x^2+a)/(a^(1/2)+x*b^(1/2))^2)^(1/2)/b^(3/4)/e^(3/2)/(b*x^2+a)^ (1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.23 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {2 x \sqrt {a+b x^2} \left (-\frac {3 A \left (a+b x^2\right )^2}{a}+\frac {(9 A b+a B) x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )}{\sqrt {1+\frac {b x^2}{a}}}\right )}{3 (e x)^{3/2}} \]
(2*x*Sqrt[a + b*x^2]*((-3*A*(a + b*x^2)^2)/a + ((9*A*b + a*B)*x^2*Hypergeo metric2F1[-3/2, 3/4, 7/4, -((b*x^2)/a)])/Sqrt[1 + (b*x^2)/a]))/(3*(e*x)^(3 /2))
Time = 0.45 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {359, 248, 248, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {(a B+9 A b) \int \sqrt {e x} \left (b x^2+a\right )^{3/2}dx}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \int \sqrt {e x} \sqrt {b x^2+a}dx+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 248 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \left (\frac {2}{5} a \int \frac {\sqrt {e x}}{\sqrt {b x^2+a}}dx+\frac {2 (e x)^{3/2} \sqrt {a+b x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \left (\frac {4 a \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{5 e}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\sqrt {a} e \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt {a} e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {b}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {(a B+9 A b) \left (\frac {2}{3} a \left (\frac {4 a \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 b^{3/4} \sqrt {a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}}{\sqrt {b}}\right )}{5 e}+\frac {2 (e x)^{3/2} \sqrt {a+b x^2}}{5 e}\right )+\frac {2 (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 e}\right )}{a e^2}-\frac {2 A \left (a+b x^2\right )^{5/2}}{a e \sqrt {e x}}\) |
(-2*A*(a + b*x^2)^(5/2))/(a*e*Sqrt[e*x]) + ((9*A*b + a*B)*((2*(e*x)^(3/2)* (a + b*x^2)^(3/2))/(9*e) + (2*a*((2*(e*x)^(3/2)*Sqrt[a + b*x^2])/(5*e) + ( 4*a*(-((-((e^2*Sqrt[e*x]*Sqrt[a + b*x^2])/(Sqrt[a]*e + Sqrt[b]*e*x)) + (a^ (1/4)*Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]* e + Sqrt[b]*e*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e ])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2]))/Sqrt[b]) + (a^(1/4)*Sqrt[e]*(Sqrt[a] *e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*El lipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*b^(3/4)* Sqrt[a + b*x^2])))/(5*e)))/3))/(a*e^2)
3.8.97.3.1 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[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[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 = 3.08 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (-5 b B \,x^{4}-9 A b \,x^{2}-11 B a \,x^{2}+45 A a \right )}{45 e \sqrt {e x}}+\frac {4 a \left (9 A b +B a \right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{15 b \sqrt {b e \,x^{3}+a e x}\, e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(245\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (b e \,x^{2}+a e \right ) a A}{e^{2} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {2 b B \,x^{3} \sqrt {b e \,x^{3}+a e x}}{9 e^{2}}+\frac {2 \left (\frac {b \left (A b +2 B a \right )}{e}-\frac {7 b B a}{9 e}\right ) x \sqrt {b e \,x^{3}+a e x}}{5 b e}+\frac {\left (\frac {a \left (2 A b +B a \right )}{e}+\frac {b a A}{e}-\frac {3 \left (\frac {b \left (A b +2 B a \right )}{e}-\frac {7 b B a}{9 e}\right ) a}{5 b}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b e \,x^{3}+a e x}}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(335\) |
| default | \(\frac {\frac {2 b^{3} B \,x^{6}}{9}+\frac {24 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b}{5}-\frac {12 A \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} b}{5}+\frac {8 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}}{15}-\frac {4 B \sqrt {2}\, \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{3}}{15}+\frac {2 A \,b^{3} x^{4}}{5}+\frac {32 B a \,b^{2} x^{4}}{45}-\frac {8 a A \,b^{2} x^{2}}{5}+\frac {22 B \,a^{2} b \,x^{2}}{45}-2 a^{2} b A}{\sqrt {b \,x^{2}+a}\, b e \sqrt {e x}}\) | \(421\) |
-2/45*(b*x^2+a)^(1/2)*(-5*B*b*x^4-9*A*b*x^2-11*B*a*x^2+45*A*a)/e/(e*x)^(1/ 2)+4/15*a*(9*A*b+B*a)*(-a*b)^(1/2)/b*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^( 1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-x/(-a*b)^(1/2)*b)^(1/2 )/(b*e*x^3+a*e*x)^(1/2)*(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/( -a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^(1/2)/b*EllipticF(((x+(-a*b)^(1/2 )/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2)))/e*((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/ 2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.26 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=-\frac {2 \, {\left (12 \, {\left (B a^{2} + 9 \, A a b\right )} \sqrt {b e} x {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, B b^{2} x^{4} - 45 \, A a b + {\left (11 \, B a b + 9 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {e x}\right )}}{45 \, b e^{2} x} \]
-2/45*(12*(B*a^2 + 9*A*a*b)*sqrt(b*e)*x*weierstrassZeta(-4*a/b, 0, weierst rassPInverse(-4*a/b, 0, x)) - (5*B*b^2*x^4 - 45*A*a*b + (11*B*a*b + 9*A*b^ 2)*x^2)*sqrt(b*x^2 + a)*sqrt(e*x))/(b*e^2*x)
Result contains complex when optimal does not.
Time = 8.32 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.55 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\frac {A a^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {A \sqrt {a} b x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B a^{\frac {3}{2}} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {a} b x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{\frac {3}{2}} \Gamma \left (\frac {11}{4}\right )} \]
A*a**(3/2)*gamma(-1/4)*hyper((-1/2, -1/4), (3/4,), b*x**2*exp_polar(I*pi)/ a)/(2*e**(3/2)*sqrt(x)*gamma(3/4)) + A*sqrt(a)*b*x**(3/2)*gamma(3/4)*hyper ((-1/2, 3/4), (7/4,), b*x**2*exp_polar(I*pi)/a)/(2*e**(3/2)*gamma(7/4)) + B*a**(3/2)*x**(3/2)*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), b*x**2*exp_polar (I*pi)/a)/(2*e**(3/2)*gamma(7/4)) + B*sqrt(a)*b*x**(7/2)*gamma(7/4)*hyper( (-1/2, 7/4), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*e**(3/2)*gamma(11/4))
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\left (e x\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (b\,x^2+a\right )}^{3/2}}{{\left (e\,x\right )}^{3/2}} \,d x \]