Integrand size = 28, antiderivative size = 318 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 \sqrt {d+e x} (4 B c d-b B e-5 A c e-3 B c e x) \sqrt {b x+c x^2}}{15 c e^2}-\frac {2 \sqrt {-b} \left (5 A c e (2 c d-b e)-B \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{3/2} e^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (8 B c d+b B e-10 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 c^{3/2} e^3 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/15*(5*A*c*e*(-b*e+2*c*d)-B*(-2*b^2*e^2-3*b*c*d*e+8*c^2*d^2))*EllipticE( c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^( 1/2)*(e*x+d)^(1/2)/c^(3/2)/e^3/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/15*d*(- b*e+c*d)*(-10*A*c*e+B*b*e+8*B*c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b *e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(3/2)/ e^3/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)-2/15*(-3*B*c*e*x-5*A*c*e-B*b*e+4*B*c*d )*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c/e^2
Result contains complex when optimal does not.
Time = 15.60 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 \left (-b e x (b+c x) (d+e x) (5 A c e+B (-4 c d+b e+3 c e x))-\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (5 A c e (-2 c d+b e)+B \left (8 c^2 d^2-3 b c d e-2 b^2 e^2\right )\right ) (b+c x) (d+e x)-i b e \left (5 A c e (2 c d-b e)+B \left (-8 c^2 d^2+3 b c d e+2 b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e (c d-b e) (5 A c e-2 B (2 c d+b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 b c e^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(-2*(-(b*e*x*(b + c*x)*(d + e*x)*(5*A*c*e + B*(-4*c*d + b*e + 3*c*e*x))) - Sqrt[b/c]*(Sqrt[b/c]*(5*A*c*e*(-2*c*d + b*e) + B*(8*c^2*d^2 - 3*b*c*d*e - 2*b^2*e^2))*(b + c*x)*(d + e*x) - I*b*e*(5*A*c*e*(2*c*d - b*e) + B*(-8*c^ 2*d^2 + 3*b*c*d*e + 2*b^2*e^2))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2 )*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(c*d - b*e) *(5*A*c*e - 2*B*(2*c*d + b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2) *EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b*c*e^3*Sqrt[ x*(b + c*x)]*Sqrt[d + e*x])
Time = 0.54 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1231, 27, 1269, 1169, 122, 120, 127, 126}
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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {2 \int -\frac {b d (4 B c d-b B e-5 A c e)-\left (5 A c e (2 c d-b e)-B \left (8 c^2 d^2-3 b c e d-2 b^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b d (4 B c d-b B e-5 A c e)-\left (5 A c e (2 c d-b e)-B \left (8 c^2 d^2-3 b c e d-2 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-\frac {\left (5 A c e (2 c d-b e)-B \left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (-10 A c e+b B e+8 B c d) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {-\frac {\sqrt {x} \sqrt {b+c x} \left (5 A c e (2 c d-b e)-B \left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (-10 A c e+b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (5 A c e (2 c d-b e)-B \left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (-10 A c e+b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (-10 A c e+b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (5 A c e (2 c d-b e)-B \left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (-10 A c e+b B e+8 B c d) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (5 A c e (2 c d-b e)-B \left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (5 A c e (2 c d-b e)-B \left (-2 b^2 e^2-3 b c d e+8 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (-10 A c e+b B e+8 B c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{15 c e^2}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (-5 A c e-b B e+4 B c d-3 B c e x)}{15 c e^2}\) |
(-2*Sqrt[d + e*x]*(4*B*c*d - b*B*e - 5*A*c*e - 3*B*c*e*x)*Sqrt[b*x + c*x^2 ])/(15*c*e^2) + ((-2*Sqrt[-b]*(5*A*c*e*(2*c*d - b*e) - B*(8*c^2*d^2 - 3*b* c*d*e - 2*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcS in[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d] *Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(8*B*c*d + b*B*e - 10*A*c* e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*S qrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2 ]))/(15*c*e^2)
3.13.56.3.1 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[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.53 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.55
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 B x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e}+\frac {2 \left (A c +B b -\frac {2 B \left (2 b e +2 c d \right )}{5 e}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}-\frac {2 \left (A c +B b -\frac {2 B \left (2 b e +2 c d \right )}{5 e}\right ) b^{2} d \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{3 c^{2} e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (A b -\frac {3 B b d}{5 e}-\frac {2 \left (A c +B b -\frac {2 B \left (2 b e +2 c d \right )}{5 e}\right ) \left (b e +c d \right )}{3 c e}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(494\) |
| default | \(\text {Expression too large to display}\) | \(1129\) |
1/(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*((e*x+d)*x*(c*x+b))^(1/2)/x/(c*x+b)*(2/5 *B/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(A*c+B*b-2/5*B/e*(2*b*e+2 *c*d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)-2/3*(A*c+B*b-2/5*B/e*(2*b *e+2*c*d))/c^2/e*b^2*d*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*(-c* x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+b/c)/b*c)^( 1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(A*b-3/5*B/e*b*d-2/3*(A*c+B*b-2/5*B/e*(2*b *e+2*c*d))/c/e*(b*e+c*d))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/ 2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-b/c+d/e)*Ellipt icE(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b* c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.46 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 \, {\left ({\left (8 \, B c^{3} d^{3} - {\left (7 \, B b c^{2} + 10 \, A c^{3}\right )} d^{2} e - 2 \, {\left (B b^{2} c - 5 \, A b c^{2}\right )} d e^{2} - {\left (2 \, B b^{3} - 5 \, A b^{2} c\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (8 \, B c^{3} d^{2} e - {\left (3 \, B b c^{2} + 10 \, A c^{3}\right )} d e^{2} - {\left (2 \, B b^{2} c - 5 \, A b c^{2}\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (3 \, B c^{3} e^{3} x - 4 \, B c^{3} d e^{2} + {\left (B b c^{2} + 5 \, A c^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, c^{3} e^{4}} \]
-2/45*((8*B*c^3*d^3 - (7*B*b*c^2 + 10*A*c^3)*d^2*e - 2*(B*b^2*c - 5*A*b*c^ 2)*d*e^2 - (2*B*b^3 - 5*A*b^2*c)*e^3)*sqrt(c*e)*weierstrassPInverse(4/3*(c ^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3 *(8*B*c^3*d^2*e - (3*B*b*c^2 + 10*A*c^3)*d*e^2 - (2*B*b^2*c - 5*A*b*c^2)*e ^3)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2* c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e *x + c*d + b*e)/(c*e))) - 3*(3*B*c^3*e^3*x - 4*B*c^3*d*e^2 + (B*b*c^2 + 5* A*c^3)*e^3)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^3*e^4)
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\sqrt {d + e x}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{\sqrt {d+e x}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{\sqrt {d+e\,x}} \,d x \]