Integrand size = 26, antiderivative size = 356 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {d+e x} (a (B d-A e)-(A c d+a B e) x)}{a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}}-\frac {(A c d+a B e) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {A \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} \sqrt {c} \sqrt {d+e x} \sqrt {a+c x^2}} \]
-(a*(-A*e+B*d)-(A*c*d+B*a*e)*x)*(e*x+d)^(1/2)/a/(a*e^2+c*d^2)/(c*x^2+a)^(1 /2)-(A*c*d+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)/( a*e^2+c*d^2)/(-a)^(1/2)/c^(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*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)/(-a)^(1/2)/c^(1/2)/(e*x+d)^(1/2)/ (c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.43 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.47 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (2 (A c d x+a (-B d+A e+B e x))-\frac {2 \left (e^2 (A c d+a B e) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )+\sqrt {c} \left (-i \sqrt {c} d+\sqrt {a} e\right ) (A c d+a B e) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} \left (i \sqrt {a} B-A \sqrt {c}\right ) \sqrt {c} e \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{2 a \left (c d^2+a e^2\right ) \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*(2*(A*c*d*x + a*(-(B*d) + A*e + B*e*x)) - (2*(e^2*(A*c*d + a*B*e)*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + Sqrt[c]*((-I)*Sqrt[c ]*d + Sqrt[a]*e)*(A*c*d + a*B*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e *x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*Elli pticE[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)] + Sqrt[a]*(I*Sqrt[a]*B - A*Sqr t[c])*Sqrt[c]*e*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x ))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^( 3/2)*EllipticF[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)]))/(c*e*Sqrt[-d - (I*S qrt[a]*e)/Sqrt[c]]*(d + e*x))))/(2*a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])
Time = 0.78 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {686, 27, 599, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a+c x^2\right )^{3/2} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle -\frac {\int \frac {c e (a (B d-A e)+(A c d+a B e) x)}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a c \left (a e^2+c d^2\right )}-\frac {\sqrt {d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {a (B d-A e)+(A c d+a B e) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{2 a \left (a e^2+c d^2\right )}-\frac {\sqrt {d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\int \frac {A \left (c d^2+a e^2\right )-(A c d+a B e) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e \left (a e^2+c d^2\right )}-\frac {\sqrt {d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {\sqrt {a e^2+c d^2} (a B e+A c d) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (-A \sqrt {c} \sqrt {a e^2+c d^2}+a B e+A c d\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}}{a e \left (a e^2+c d^2\right )}-\frac {\sqrt {d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {\sqrt {a e^2+c d^2} (a B e+A c d) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-A \sqrt {c} \sqrt {a e^2+c d^2}+a B e+A c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}}{a e \left (a e^2+c d^2\right )}-\frac {\sqrt {d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {\sqrt {a e^2+c d^2} (a B e+A c d) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (-A \sqrt {c} \sqrt {a e^2+c d^2}+a B e+A c d\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}}{a e \left (a e^2+c d^2\right )}-\frac {\sqrt {d+e x} (a (B d-A e)-x (a B e+A c d))}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}\) |
-((Sqrt[d + e*x]*(a*(B*d - A*e) - (A*c*d + a*B*e)*x))/(a*(c*d^2 + a*e^2)*S qrt[a + c*x^2])) + (((A*c*d + 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 + 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*Ar cTan[(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*e*(c*d^2 + a*e^2))
3.15.86.3.1 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))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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. \(725\) vs. \(2(296)=592\).
Time = 2.54 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.04
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (x c e +c d \right ) \left (-\frac {\left (A c d +B a e \right ) x}{2 \left (e^{2} a +c \,d^{2}\right ) a c}-\frac {A e -B d}{2 \left (e^{2} a +c \,d^{2}\right ) c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (x c e +c d \right )}}+\frac {2 \left (\frac {A}{a}-\frac {e \left (A e -B d \right )}{2 \left (e^{2} a +c \,d^{2}\right )}-\frac {d \left (A c d +B a e \right )}{\left (e^{2} a +c \,d^{2}\right ) a}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}-\frac {e \left (A c d +B a e \right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{a \left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(726\) |
| default | \(\text {Expression too large to display}\) | \(1317\) |
((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*e^2+c*d^2)/a/c*x-1/2*(A*e-B*d)/(a*e^2+c*d^2)/c)/((x^2 +a/c)*(c*e*x+c*d))^(1/2)+2*(A/a-1/2*e*(A*e-B*d)/(a*e^2+c*d^2)-d*(A*c*d+B*a *e)/(a*e^2+c*d^2)/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))-e*(A*c*d+B*a*e)/a/(a*e^2+c*d^2)*(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 = 0.20 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (A a c d^{2} - 2 \, B a^{2} d e + 3 \, A a^{2} e^{2} + {\left (A c^{2} d^{2} - 2 \, B a c d e + 3 \, A a c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (A a c d e + B a^{2} e^{2} + {\left (A c^{2} d e + B a c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (B a c d e - A a c e^{2} - {\left (A c^{2} d e + B a c e^{2}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a^{2} c^{2} d^{2} e + a^{3} c e^{3} + {\left (a c^{3} d^{2} e + a^{2} c^{2} e^{3}\right )} x^{2}\right )}} \]
1/3*((A*a*c*d^2 - 2*B*a^2*d*e + 3*A*a^2*e^2 + (A*c^2*d^2 - 2*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 + B*a^2*e^2 + (A*c^2*d*e + 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), weierstrassPIn verse(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^2*c^2*d^2*e + a^3*c*e^3 + (a*c^3*d^2*e + a^2*c^2*e^3)*x^2)
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (a + c x^{2}\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d}} \,d x } \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} \sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (c\,x^2+a\right )}^{3/2}\,\sqrt {d+e\,x}} \,d x \]