Integrand size = 26, antiderivative size = 288 \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} B \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {c} e \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2*B*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))*(-a)^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e/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)+2* (-A*e+B*d)*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))*(-a)^(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)/e/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1 /2)
Result contains complex when optimal does not.
Time = 22.33 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.52 \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=-\frac {2 \left (-B e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )+i B \sqrt {c} \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\left (\sqrt {a} B-i A \sqrt {c}\right ) \sqrt {c} e \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {d+e x} \sqrt {a+c x^2}} \]
(-2*(-(B*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2)) + I*B*Sqrt[c]*( Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqr t[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I* ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sq rt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + (Sqrt[a]*B - I*A*Sqrt[c])*Sqrt[c]*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*S qrt[a]*e)]))/(c*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a + c*x^2])
Leaf count is larger than twice the leaf count of optimal. \(614\) vs. \(2(288)=576\).
Time = 0.63 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.13, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {a+c x^2} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 \int \frac {B d-A e-B (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e^2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {2 \left (\frac {B \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\left (A e-B \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{e^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 \left (\frac {B \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (A e-B \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{e^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {2 \left (\frac {B \sqrt {a e^2+c d^2} \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}-\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \left (A e-B \left (d-\frac {\sqrt {a e^2+c d^2}}{\sqrt {c}}\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{e^2}\) |
(-2*((B*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)^(1/4)*(A*e - B*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c]))*(1 + (Sqr t[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^(1/4)*S qrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])))/e^2
3.15.82.3.1 Defintions of rubi rules used
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[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. \(519\) vs. \(2(232)=464\).
Time = 1.03 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.81
| method | result | size |
| default | \(\frac {2 \left (A F\left (\sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}, \sqrt {-\frac {e \sqrt {-a c}-c d}{e \sqrt {-a c}+c d}}\right ) c d e -A \sqrt {-a c}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}, \sqrt {-\frac {e \sqrt {-a c}-c d}{e \sqrt {-a c}+c d}}\right ) e^{2}+B F\left (\sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}, \sqrt {-\frac {e \sqrt {-a c}-c d}{e \sqrt {-a c}+c d}}\right ) a \,e^{2}+B \sqrt {-a c}\, F\left (\sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}, \sqrt {-\frac {e \sqrt {-a c}-c d}{e \sqrt {-a c}+c d}}\right ) d e -B E\left (\sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}, \sqrt {-\frac {e \sqrt {-a c}-c d}{e \sqrt {-a c}+c d}}\right ) a \,e^{2}-B E\left (\sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}, \sqrt {-\frac {e \sqrt {-a c}-c d}{e \sqrt {-a c}+c d}}\right ) c \,d^{2}\right ) \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) e}{e \sqrt {-a c}-c d}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) e}{e \sqrt {-a c}+c d}}\, \sqrt {-\frac {c \left (e x +d \right )}{e \sqrt {-a c}-c d}}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}{c \,e^{2} \left (c e \,x^{3}+c d \,x^{2}+a e x +a d \right )}\) | \(520\) |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 A \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 B \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(556\) |
2*(A*EllipticF((-c*(e*x+d)/(e*(-a*c)^(1/2)-c*d))^(1/2),(-(e*(-a*c)^(1/2)-c *d)/(e*(-a*c)^(1/2)+c*d))^(1/2))*c*d*e-A*(-a*c)^(1/2)*EllipticF((-c*(e*x+d )/(e*(-a*c)^(1/2)-c*d))^(1/2),(-(e*(-a*c)^(1/2)-c*d)/(e*(-a*c)^(1/2)+c*d)) ^(1/2))*e^2+B*EllipticF((-c*(e*x+d)/(e*(-a*c)^(1/2)-c*d))^(1/2),(-(e*(-a*c )^(1/2)-c*d)/(e*(-a*c)^(1/2)+c*d))^(1/2))*a*e^2+B*(-a*c)^(1/2)*EllipticF(( -c*(e*x+d)/(e*(-a*c)^(1/2)-c*d))^(1/2),(-(e*(-a*c)^(1/2)-c*d)/(e*(-a*c)^(1 /2)+c*d))^(1/2))*d*e-B*EllipticE((-c*(e*x+d)/(e*(-a*c)^(1/2)-c*d))^(1/2),( -(e*(-a*c)^(1/2)-c*d)/(e*(-a*c)^(1/2)+c*d))^(1/2))*a*e^2-B*EllipticE((-c*( e*x+d)/(e*(-a*c)^(1/2)-c*d))^(1/2),(-(e*(-a*c)^(1/2)-c*d)/(e*(-a*c)^(1/2)+ c*d))^(1/2))*c*d^2)*((c*x+(-a*c)^(1/2))*e/(e*(-a*c)^(1/2)-c*d))^(1/2)*((-c *x+(-a*c)^(1/2))*e/(e*(-a*c)^(1/2)+c*d))^(1/2)*(-c*(e*x+d)/(e*(-a*c)^(1/2) -c*d))^(1/2)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/c/e^2/(c*e*x^3+c*d*x^2+a*e*x+a* d)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.62 \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c e} B e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + {\left (B d - 3 \, A e\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )}}{3 \, c e^{2}} \]
-2/3*(3*sqrt(c*e)*B*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), 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)) + (B*d - 3*A *e)*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))/(c*e^2)
\[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {a + c x^{2}} \sqrt {d + e x}}\, dx \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + a} \sqrt {e x + d}} \,d x } \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + a} \sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c\,x^2+a}\,\sqrt {d+e\,x}} \,d x \]