Integrand size = 24, antiderivative size = 326 \[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=\frac {2 A e \sqrt {e x} \sqrt {a+c x^2}}{3 c}+\frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {6 a B e^2 x \sqrt {a+c x^2}}{5 c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {6 a^{5/4} B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {a^{3/4} \left (9 \sqrt {a} B+5 A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
2/5*B*(e*x)^(3/2)*(c*x^2+a)^(1/2)/c-6/5*a*B*e^2*x*(c*x^2+a)^(1/2)/c^(3/2)/ (a^(1/2)+x*c^(1/2))/(e*x)^(1/2)+2/3*A*e*(e*x)^(1/2)*(c*x^2+a)^(1/2)/c+6/5* a^(5/4)*B*e^2*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arcta n(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4) )),1/2*2^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a)/(a^(1/2)+x*c^(1/2)) ^2)^(1/2)/c^(7/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)-1/15*a^(3/4)*e^2*(cos(2*arct an(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4) ))*EllipticF(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*2^(1/2))*(9*B*a^(1 /2)+5*A*c^(1/2))*(a^(1/2)+x*c^(1/2))*x^(1/2)*((c*x^2+a)/(a^(1/2)+x*c^(1/2) )^2)^(1/2)/c^(7/4)/(e*x)^(1/2)/(c*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 = 118, normalized size of antiderivative = 0.36 \[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=\frac {2 e \sqrt {e x} \left ((5 A+3 B x) \left (a+c x^2\right )-5 a A \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )-3 a B x \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{15 c \sqrt {a+c x^2}} \]
(2*e*Sqrt[e*x]*((5*A + 3*B*x)*(a + c*x^2) - 5*a*A*Sqrt[1 + (c*x^2)/a]*Hype rgeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] - 3*a*B*x*Sqrt[1 + (c*x^2)/a]*H ypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/a)]))/(15*c*Sqrt[a + c*x^2])
Time = 0.42 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.95, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {552, 27, 552, 27, 556, 555, 1512, 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.
| \(\displaystyle \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {2 e \int \frac {\sqrt {e x} (3 a B-5 A c x)}{2 \sqrt {c x^2+a}}dx}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \int \frac {\sqrt {e x} (3 a B-5 A c x)}{\sqrt {c x^2+a}}dx}{5 c}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (-\frac {2 e \int -\frac {a c (5 A+9 B x)}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{3 c}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {1}{3} a e \int \frac {5 A+9 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {a e \sqrt {x} \int \frac {5 A+9 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{3 \sqrt {e x}}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {2 a e \sqrt {x} \int \frac {5 A+9 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{3 \sqrt {e x}}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {2 a e \sqrt {x} \left (\left (\frac {9 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {9 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 \sqrt {e x}}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {2 a e \sqrt {x} \left (\left (\frac {9 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}-\frac {9 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 \sqrt {e x}}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {2 a e \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {9 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {9 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 \sqrt {e x}}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {2 B (e x)^{3/2} \sqrt {a+c x^2}}{5 c}-\frac {e \left (\frac {2 a e \sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\frac {9 \sqrt {a} B}{\sqrt {c}}+5 A\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}-\frac {9 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{3 \sqrt {e x}}-\frac {10}{3} A \sqrt {e x} \sqrt {a+c x^2}\right )}{5 c}\) |
(2*B*(e*x)^(3/2)*Sqrt[a + c*x^2])/(5*c) - (e*((-10*A*Sqrt[e*x]*Sqrt[a + c* x^2])/3 + (2*a*e*Sqrt[x]*((-9*B*(-((Sqrt[x]*Sqrt[a + c*x^2])/(Sqrt[a] + Sq rt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqr t[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(c^(1/4)*S qrt[a + c*x^2])))/Sqrt[c] + ((5*A + (9*Sqrt[a]*B)/Sqrt[c])*(Sqrt[a] + Sqrt [c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/ 4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^2])))/(3*Sqrt[ e*x])))/(5*c)
3.5.60.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e /(b*(m + 2*p + 2)) Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + c*x^4], x], x, Sqrt[x]], x] /; Free Q[{a, c, f, g}, x]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symb ol] :> Simp[Sqrt[x]/Sqrt[e*x] Int[(c + d*x)/(Sqrt[x]*Sqrt[a + b*x^2]), x] , x] /; FreeQ[{a, b, c, d, e}, x]
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[((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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Time = 0.24 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {e \sqrt {e x}\, \left (5 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a +18 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-9 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 B \,c^{2} x^{4}-10 A \,c^{2} x^{3}-6 a B c \,x^{2}-10 a A c x \right )}{15 x \sqrt {c \,x^{2}+a}\, c^{2}}\) | \(316\) |
| risch | \(\frac {2 \left (3 B x +5 A \right ) x \sqrt {c \,x^{2}+a}\, e^{2}}{15 c \sqrt {e x}}-\frac {a \left (\frac {5 A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c \sqrt {c e \,x^{3}+a e x}}+\frac {9 B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c e \,x^{3}+a e x}}\right ) e^{2} \sqrt {\left (c \,x^{2}+a \right ) e x}}{15 c \sqrt {e x}\, \sqrt {c \,x^{2}+a}}\) | \(341\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {2 B e x \sqrt {c e \,x^{3}+a e x}}{5 c}+\frac {2 A e \sqrt {c e \,x^{3}+a e x}}{3 c}-\frac {A \,e^{2} a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{3 c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {3 B \,e^{2} a \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{5 c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(356\) |
-1/15*e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^2*(5*A*((c*x+(-a*c)^(1/2))/(-a*c)^ (1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^( 1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2 ))*(-a*c)^(1/2)*a+18*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((- c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticE( ((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2-9*B*((c*x+(-a*c)^ (1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2 )*(-x/(-a*c)^(1/2)*c)^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1 /2),1/2*2^(1/2))*a^2-6*B*c^2*x^4-10*A*c^2*x^3-6*a*B*c*x^2-10*a*A*c*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.25 \[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {c e} A a e {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) - 9 \, \sqrt {c e} B a e {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) - {\left (3 \, B c e x + 5 \, A c e\right )} \sqrt {c x^{2} + a} \sqrt {e x}\right )}}{15 \, c^{2}} \]
-2/15*(5*sqrt(c*e)*A*a*e*weierstrassPInverse(-4*a/c, 0, x) - 9*sqrt(c*e)*B *a*e*weierstrassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) - (3*B* c*e*x + 5*A*c*e)*sqrt(c*x^2 + a)*sqrt(e*x))/c^2
Result contains complex when optimal does not.
Time = 4.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.29 \[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=\frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {11}{4}\right )} \]
A*e**(3/2)*x**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), c*x**2*exp_polar( I*pi)/a)/(2*sqrt(a)*gamma(9/4)) + B*e**(3/2)*x**(7/2)*gamma(7/4)*hyper((1/ 2, 7/4), (11/4,), c*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(11/4))
\[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}}}{\sqrt {c x^{2} + a}} \,d x } \]
\[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}}}{\sqrt {c x^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{3/2} (A+B x)}{\sqrt {a+c x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,\left (A+B\,x\right )}{\sqrt {c\,x^2+a}} \,d x \]