Integrand size = 26, antiderivative size = 345 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d+e x} (a (B d+A e)-(A c d-a B e) x)}{a c \sqrt {a+c x^2}}-\frac {(A c d-3 a B e) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {A \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]
-(a*(A*e+B*d)-(A*c*d-B*a*e)*x)*(e*x+d)^(1/2)/a/c/(c*x^2+a)^(1/2)-(A*c*d-3* B*a*e)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+ d*(-a)^(1/2)*c^(1/2)))^(1/2))*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/c^(3/2)/(-a) ^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+A* (a*e^2+c*d^2)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e /(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(1+c*x^2/a)^(1/2)*((e*x+d)*c^(1/2)/(e *(-a)^(1/2)+d*c^(1/2)))^(1/2)/c^(3/2)/(-a)^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^( 1/2)
Result contains complex when optimal does not.
Time = 24.46 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.35 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {e (-A c d+3 a B e) \left (a+c x^2\right )}{d+e x}+c (A c d x-a (A e+B (d+e x)))-\frac {i c (A c d-3 a B e) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}-\sqrt {a} \left (3 i \sqrt {a} B+A \sqrt {c}\right ) c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{a c^2 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((e*(-(A*c*d) + 3*a*B*e)*(a + c*x^2))/(d + e*x) + c*(A*c*d* x - a*(A*e + B*(d + e*x))) - (I*c*(A*c*d - 3*a*B*e)*Sqrt[-d - (I*Sqrt[a]*e )/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a ]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c ]*d + I*Sqrt[a]*e)])/e - Sqrt[a]*((3*I)*Sqrt[a]*B + A*Sqrt[c])*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt [-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticF[I*Arc Sinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[ a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(a*c^2*Sqrt[a + c*x^2])
Time = 0.81 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {684, 27, 599, 25, 1511, 1416, 1509}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {\int \frac {e (a (3 B d+A e)-(A c d-3 a B e) x)}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a c}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {a (3 B d+A e)-(A c d-3 a B e) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{2 a c}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {\int -\frac {A \left (c d^2+a e^2\right )-(A c d-3 a B e) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a c e}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {A \left (c d^2+a e^2\right )-(A c d-3 a B e) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a c e}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {\frac {\sqrt {a e^2+c d^2} \left (-A \sqrt {c} \sqrt {a e^2+c d^2}-3 a B e+A c d\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} (A c d-3 a B e) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}}{a c e}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-A \sqrt {c} \sqrt {a e^2+c d^2}-3 a B e+A c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} (A c d-3 a B e) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}}{a c e}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-A \sqrt {c} \sqrt {a e^2+c d^2}-3 a B e+A c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} (A c d-3 a B e) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}}{a c e}-\frac {\sqrt {d+e x} (a (A e+B d)-x (A c d-a B e))}{a c \sqrt {a+c x^2}}\) |
-((Sqrt[d + e*x]*(a*(B*d + A*e) - (A*c*d - a*B*e)*x))/(a*c*Sqrt[a + c*x^2] )) - (-(((A*c*d - 3*a*B*e)*Sqrt[c*d^2 + a*e^2]*(-((Sqrt[d + e*x]*Sqrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])/((a + (c*d^2)/ e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1 /4)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])*Sqrt[(a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2)/((a + (c*d^2)/e^2)*(1 + (Sqrt [c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])^2)]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)], (1 + (Sqrt[c]*d)/Sqrt[c*d^2 + a*e^2])/2]) /(c^(1/4)*Sqrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e ^2])))/Sqrt[c]) + ((c*d^2 + a*e^2)^(3/4)*(A*c*d - 3*a*B*e - A*Sqrt[c]*Sqrt [c*d^2 + a*e^2])*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])*Sqrt[(a + ( c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2)/((a + (c*d^2)/e^ 2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])^2)]*EllipticF[2*ArcTan[(c ^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)], (1 + (Sqrt[c]*d)/Sqrt[c*d^2 + a*e^2])/2])/(2*c^(3/4)*Sqrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c *(d + e*x)^2)/e^2]))/(a*c*e)
3.15.84.3.1 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[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Leaf count of result is larger than twice the leaf count of optimal. \(717\) vs. \(2(285)=570\).
Time = 2.06 (sec) , antiderivative size = 718, normalized size of antiderivative = 2.08
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (x c e +c d \right ) \left (-\frac {\left (A c d -B a e \right ) x}{2 a \,c^{2}}+\frac {A e +B d}{2 c^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (x c e +c d \right )}}+\frac {2 \left (\frac {e \left (A e +2 B d \right )}{c}-\frac {A a \,e^{2}-A c \,d^{2}+2 B a d e}{a c}+\frac {e \left (A e +B d \right )}{2 c}-\frac {d \left (A c d -B a e \right )}{c a}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {B \,e^{2}}{c}-\frac {e \left (A c d -B a e \right )}{2 a c}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(718\) |
| default | \(\text {Expression too large to display}\) | \(1319\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2*(c*e*x+c*d)*(- 1/2*(A*c*d-B*a*e)/a/c^2*x+1/2*(A*e+B*d)/c^2)/((x^2+a/c)*(c*e*x+c*d))^(1/2) +2*(e*(A*e+2*B*d)/c-(A*a*e^2-A*c*d^2+2*B*a*d*e)/a/c+1/2/c*e*(A*e+B*d)-1/c* d*(A*c*d-B*a*e)/a)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/ 2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(- d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x +d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2 )/c))^(1/2))+2*(B*e^2/c-1/2*e*(A*c*d-B*a*e)/a/c)*(d/e-(-a*c)^(1/2)/c)*((x+ d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c) )^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+ a*e*x+a*d)^(1/2)*((-d/e-(-a*c)^(1/2)/c)*EllipticE(((x+d/e)/(d/e-(-a*c)^(1/ 2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^( 1/2)/c*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/ c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (A a c d^{2} + 6 \, B a^{2} d e + 3 \, A a^{2} e^{2} + {\left (A c^{2} d^{2} + 6 \, B a c d e + 3 \, A a c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (A a c d e - 3 \, B a^{2} e^{2} + {\left (A c^{2} d e - 3 \, B a c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (B a c d e + A a c e^{2} - {\left (A c^{2} d e - B a c e^{2}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a c^{3} e x^{2} + a^{2} c^{2} e\right )}} \]
1/3*((A*a*c*d^2 + 6*B*a^2*d*e + 3*A*a^2*e^2 + (A*c^2*d^2 + 6*B*a*c*d*e + 3 *A*a*c*e^2)*x^2)*sqrt(c*e)*weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^ 2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 3*(A*a*c*d*e - 3*B*a^2*e^2 + (A*c^2*d*e - 3*B*a*c*e^2)*x^2)*sqrt(c*e)*weierstrassZeta(4/ 3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), weierstras sPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3) , 1/3*(3*e*x + d)/e)) - 3*(B*a*c*d*e + A*a*c*e^2 - (A*c^2*d*e - B*a*c*e^2) *x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(a*c^3*e*x^2 + a^2*c^2*e)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]