Integrand size = 24, antiderivative size = 341 \[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}+\frac {e \sqrt {e x} (A+3 B x)}{6 a c \sqrt {a+c x^2}}-\frac {B e^2 x \sqrt {a+c x^2}}{2 a c^{3/2} \sqrt {e x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {B e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}}-\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) e^2 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{12 a^{5/4} c^{7/4} \sqrt {e x} \sqrt {a+c x^2}} \]
-1/3*e*(B*x+A)*(e*x)^(1/2)/c/(c*x^2+a)^(3/2)+1/6*e*(3*B*x+A)*(e*x)^(1/2)/a /c/(c*x^2+a)^(1/2)-1/2*B*e^2*x*(c*x^2+a)^(1/2)/a/c^(3/2)/(a^(1/2)+x*c^(1/2 ))/(e*x)^(1/2)+1/2*B*e^2*(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^(3/4)/c^(7/4)/(e*x)^(1/2)/(c*x^2+a)^(1/2)-1/12*e^2* (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) )*(3*B*a^(1/2)-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^(5/4)/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 = 10.07 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.40 \[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {e \sqrt {e x} \left (a A-a B x-A c x^2-3 B c x^3-A \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^2}{a}\right )+B x \left (a+c x^2\right ) \sqrt {1+\frac {c x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {c x^2}{a}\right )\right )}{6 a c \left (a+c x^2\right )^{3/2}} \]
-1/6*(e*Sqrt[e*x]*(a*A - a*B*x - A*c*x^2 - 3*B*c*x^3 - A*(a + c*x^2)*Sqrt[ 1 + (c*x^2)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((c*x^2)/a)] + B*x*(a + c *x^2)*Sqrt[1 + (c*x^2)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((c*x^2)/a)])) /(a*c*(a + c*x^2)^(3/2))
Time = 0.44 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {549, 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.
| \(\displaystyle \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {e^2 \int \frac {A+3 B x}{2 \sqrt {e x} \left (c x^2+a\right )^{3/2}}dx}{3 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int \frac {A+3 B x}{\sqrt {e x} \left (c x^2+a\right )^{3/2}}dx}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 551 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}-\frac {\int -\frac {A-3 B x}{2 \sqrt {e x} \sqrt {c x^2+a}}dx}{a}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {\int \frac {A-3 B x}{\sqrt {e x} \sqrt {c x^2+a}}dx}{2 a}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 556 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {x} \int \frac {A-3 B x}{\sqrt {x} \sqrt {c x^2+a}}dx}{2 a \sqrt {e x}}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 555 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {x} \int \frac {A-3 B x}{\sqrt {c x^2+a}}d\sqrt {x}}{a \sqrt {e x}}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {x} \left (\left (A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {3 \sqrt {a} B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {a} \sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {x} \left (\left (A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^2+a}}d\sqrt {x}+\frac {3 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {x} \left (\frac {3 B \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+a}}d\sqrt {x}}{\sqrt {c}}+\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {e^2 \left (\frac {\sqrt {x} \left (\frac {\left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \left (A-\frac {3 \sqrt {a} B}{\sqrt {c}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^2}}+\frac {3 B \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^2}}-\frac {\sqrt {x} \sqrt {a+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{a \sqrt {e x}}+\frac {\sqrt {e x} (A+3 B x)}{a e \sqrt {a+c x^2}}\right )}{6 c}-\frac {e \sqrt {e x} (A+B x)}{3 c \left (a+c x^2\right )^{3/2}}\) |
-1/3*(e*Sqrt[e*x]*(A + B*x))/(c*(a + c*x^2)^(3/2)) + (e^2*((Sqrt[e*x]*(A + 3*B*x))/(a*e*Sqrt[a + c*x^2]) + (Sqrt[x]*((3*B*(-((Sqrt[x]*Sqrt[a + c*x^2 ])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(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])))/Sqrt[c] + ((A - (3*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])))/(a*Sqrt[e*x])))/(6*c)
3.5.79.3.1 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*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b *x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[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 = 0.87 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.18
| method | result | size |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (c \,x^{2}+a \right ) e x}\, \left (\frac {\left (-\frac {e B x}{3 c^{3}}-\frac {e A}{3 c^{3}}\right ) \sqrt {c e \,x^{3}+a e x}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 x e c \left (-\frac {e B x}{4 a \,c^{2}}-\frac {A e}{12 a \,c^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) x e c}}+\frac {e^{2} A \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{12 a \,c^{2} \sqrt {c e \,x^{3}+a e x}}-\frac {e^{2} B \sqrt {-a c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, \left (-\frac {2 \sqrt {-a c}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a c}}{c}\right ) c}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{4 a \,c^{2} \sqrt {c e \,x^{3}+a e x}}\right )}{e x \sqrt {c \,x^{2}+a}}\) | \(402\) |
| default | \(\frac {\left (A \sqrt {-a c}\, \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) c \,x^{2}-6 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+3 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a c \,x^{2}+A \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a c}\, a -6 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, E\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+3 B \sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-a c}}{\sqrt {-a c}}}\, \sqrt {-\frac {x c}{\sqrt {-a c}}}\, F\left (\sqrt {\frac {c x +\sqrt {-a c}}{\sqrt {-a c}}}, \frac {\sqrt {2}}{2}\right ) a^{2}+6 B \,c^{2} x^{4}+2 A \,c^{2} x^{3}+2 a B c \,x^{2}-2 a A c x \right ) e \sqrt {e x}}{12 x \,c^{2} a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\) | \(583\) |
1/e/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)*((c*x^2+a)*e*x)^(1/2)*((-1/3*e/c^3*B*x-1 /3*e/c^3*A)*(c*e*x^3+a*e*x)^(1/2)/(x^2+a/c)^2-2*x*e*c*(-1/4*e/a*B/c^2*x-1/ 12*A/a*e/c^2)/((x^2+a/c)*x*e*c)^(1/2)+1/12/a/c^2*e^2*A*(-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/c^2*e^2*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)*(-2*(-a*c)^( 1/2)/c*EllipticE(((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 = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.49 \[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (A c^{2} e x^{4} + 2 \, A a c e x^{2} + A a^{2} e\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right ) + 3 \, {\left (B c^{2} e x^{4} + 2 \, B a c e x^{2} + B a^{2} e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (-\frac {4 \, a}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{c}, 0, x\right )\right ) + {\left (3 \, B c^{2} e x^{3} + A c^{2} e x^{2} + B a c e x - A a c e\right )} \sqrt {c x^{2} + a} \sqrt {e x}}{6 \, {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}} \]
1/6*((A*c^2*e*x^4 + 2*A*a*c*e*x^2 + A*a^2*e)*sqrt(c*e)*weierstrassPInverse (-4*a/c, 0, x) + 3*(B*c^2*e*x^4 + 2*B*a*c*e*x^2 + B*a^2*e)*sqrt(c*e)*weier strassZeta(-4*a/c, 0, weierstrassPInverse(-4*a/c, 0, x)) + (3*B*c^2*e*x^3 + A*c^2*e*x^2 + B*a*c*e*x - A*a*c*e)*sqrt(c*x^2 + a)*sqrt(e*x))/(a*c^4*x^4 + 2*a^2*c^3*x^2 + a^3*c^2)
Result contains complex when optimal does not.
Time = 74.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.28 \[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {A e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right )} + \frac {B e^{\frac {3}{2}} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {5}{2} \\ \frac {11}{4} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{2}} \Gamma \left (\frac {11}{4}\right )} \]
A*e**(3/2)*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)) + B*e**(3/2)*x**(7/2)*gamma(7/4)*hyper((7 /4, 5/2), (11/4,), c*x**2*exp_polar(I*pi)/a)/(2*a**(5/2)*gamma(11/4))
\[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(e x)^{3/2} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (e\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]