Integrand size = 24, antiderivative size = 342 \[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {e x} (a B+3 A c x)}{6 a^2 c \sqrt {a+c x^2}}-\frac {A e x \sqrt {a+c x^2}}{2 a^2 \sqrt {c} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {A e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} c^{3/4} \sqrt {e x} \sqrt {a+c x^2}}+\frac {\left (\sqrt {a} B-3 A \sqrt {c}\right ) e \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{7/4} c^{5/4} \sqrt {e x} \sqrt {a+c x^2}} \]
-1/3*(-A*c*x+B*a)*(e*x)^(1/2)/a/c/(c*x^2+a)^(3/2)+1/6*(3*A*c*x+B*a)*(e*x)^ (1/2)/a^2/c/(c*x^2+a)^(1/2)-1/2*A*e*x*(c*x^2+a)^(1/2)/a^2/c^(1/2)/(a^(1/2) +x*c^(1/2))/(e*x)^(1/2)+1/2*A*e*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2) ^(1/2)/cos(2*arctan(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)/a^(7/4)/c^(3/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)+1/ 12*e*(cos(2*arctan(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))*(B*a^(1/2)-3*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)/a^(7/4)/c^(5/4)/(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 = 10.10 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {e x} \left (-a^2 B+5 a A c x+a B c x^2+3 A c^2 x^3+a B \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )-A c x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{6 a^2 c \left (a+c x^2\right )^{3/2}} \]
(Sqrt[e*x]*(-(a^2*B) + 5*a*A*c*x + a*B*c*x^2 + 3*A*c^2*x^3 + a*B*(a + c*x^ 2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] - A* c*x*(a + c*x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c* x^2)/a)]))/(6*a^2*c*(a + c*x^2)^(3/2))
Time = 0.45 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {550, 27, 551, 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.
| \(\displaystyle \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 550 |
| \(\displaystyle \frac {e \int \frac {a B+3 A c x}{2 \sqrt {e x} \left (c x^2+a\right )^{3/2}}dx}{3 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {a B+3 A c x}{\sqrt {e x} \left (c x^2+a\right )^{3/2}}dx}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}-\frac {\int -\frac {a B-3 A c x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{a}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {\int \frac {a B-3 A c x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{2 a}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \int \frac {a B-3 A c x}{\sqrt {x} \sqrt {c x^2+a}}dx}{2 a \sqrt {e x}}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \int \frac {a B-3 A c x}{\sqrt {c x^2+a}}d\sqrt {x}}{a \sqrt {e x}}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \left (\sqrt {a} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+3 \sqrt {a} A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \left (\sqrt {a} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+3 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \left (3 A \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (\sqrt {a} B-3 A \sqrt {c}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {a+c x^2}}+3 A \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (a B+3 A c x)}{a e \sqrt {a+c x^2}}\right )}{6 a c}-\frac {\sqrt {e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
-1/3*(Sqrt[e*x]*(a*B - A*c*x))/(a*c*(a + c*x^2)^(3/2)) + (e*((Sqrt[e*x]*(a *B + 3*A*c*x))/(a*e*Sqrt[a + c*x^2]) + (Sqrt[x]*(3*A*Sqrt[c]*(-((Sqrt[x]*S qrt[a + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sq rt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x ])/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^2])) + (a^(1/4)*(Sqrt[a]*B - 3*A* Sqrt[c])*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*E llipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[a + c* x^2])))/(a*Sqrt[e*x])))/(6*a*c)
3.5.80.3.1 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[(e*x)^m*(a*d - b*c*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x ] - Simp[e/(2*a*b*(p + 1)) Int[(e*x)^(m - 1)*(a*d*m - b*c*(m + 2*p + 3)*x )*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && LtQ[0, m, 1]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[(-(e*x)^(m + 1))*(c + d*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1) )), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(c*(m + 2*p + 3) + d*(m + 2*p + 4)*x)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p , -1] && LtQ[m, 0]
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 = 0.10 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (\frac {A x}{3 a \,c^{2}}-\frac {B}{3 c^{3}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (-\frac {A x}{4 c \,a^{2}}-\frac {B}{12 a \,c^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {e B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 c^{2} a \sqrt {c e \,x^{3}+a e x}}-\frac {e A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a^{2} c \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(397\) |
| default | \(\frac {\left (3 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}-6 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a \,c^{2} x^{2}+B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+3 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c -6 A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2} c +B \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+6 A \,c^{3} x^{4}+2 a B \,c^{2} x^{3}+10 a A \,c^{2} x^{2}-2 a^{2} B c x \right ) \sqrt {e x}}{12 a^{2} x \,c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(596\) |
1/e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)*((c*x^2+a)*e*x)^(1/2)*((1/3/a/c^2*A*x-1/ 3*B/c^3)*(c*e*x^3+a*e*x)^(1/2)/(x^2+a/c)^2-2*x*e*c*(-1/4*A/c/a^2*x-1/12*B/ a/c^2)/((x^2+a/c)*x*e*c)^(1/2)+1/12/c^2/a*e*B*(-a*c)^(1/2)*((x+(-a*c)^(1/2 )/c)/(-a*c)^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(- x/(-a*c)^(1/2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*EllipticF(((x+(-a*c)^(1/2)/c )/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))-1/4/a^2*e*A*(-a*c)^(1/2)/c*((x+(-a*c) ^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2)*(-2*(x-(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/ 2)*(-x/(-a*c)^(1/2)*c)^(1/2)/(c*e*x^3+a*e*x)^(1/2)*(-2*(-a*c)^(1/2)/c*Elli pticE(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))+(-a*c)^(1/2)/ c*EllipticF(((x+(-a*c)^(1/2)/c)/(-a*c)^(1/2)*c)^(1/2),1/2*2^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.50 \[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (B a c^{2} x^{4} + 2 \, B a^{2} c x^{2} + B a^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 3 \, {\left (A c^{3} x^{4} + 2 \, A a c^{2} x^{2} + A a^{2} c\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3 \, A c^{3} x^{3} + B a c^{2} x^{2} + 5 \, A a c^{2} x - B a^{2} c\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} \]
1/6*((B*a*c^2*x^4 + 2*B*a^2*c*x^2 + B*a^3)*sqrt(c*e)*weierstrassPInverse(- 4*a/c, 0, x) + 3*(A*c^3*x^4 + 2*A*a*c^2*x^2 + A*a^2*c)*sqrt(c*e)*weierstra ssZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (3*A*c^3*x^3 + B*a* c^2*x^2 + 5*A*a*c^2*x - B*a^2*c)*sqrt(c*x^2 + a)*sqrt(e*x))/(a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)
Result contains complex when optimal does not.
Time = 43.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {A \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {7}{4}\right )} + \frac {B \sqrt {e} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} \]
A*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((3/4, 5/2), (7/4,), c*x**2*exp_polar(I *pi)/a)/(2*a**(5/2)*gamma(7/4)) + B*sqrt(e)*x**(5/2)*gamma(5/4)*hyper((5/4 , 5/2), (9/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*gamma(9/4))
\[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {e\,x}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]