Integrand size = 28, antiderivative size = 254 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 B \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}+\frac {2 \sqrt {-b} (B c d-2 b B e+3 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} e \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} B d (c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 c^{3/2} e \sqrt {d+e x} \sqrt {b x+c x^2}} \]
2/3*(3*A*c*e-2*B*b*e+B*c*d)*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/(1+e*x/ d)^(1/2)/(c*x^2+b*x)^(1/2)-2/3*B*d*(-b*e+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/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*B*(e*x+d)^(1/2)*(c*x^2+b* x)^(1/2)/c
Result contains complex when optimal does not.
Time = 11.02 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 x \left (B (b+c x) (d+e x)+\frac {(B c d-2 b B e+3 A c e) (b+c x) (d+e x)}{c e x}+i \sqrt {\frac {b}{c}} (B c d-2 b B e+3 A c e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} \sqrt {x} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\frac {i \sqrt {\frac {b}{c}} (2 b B-3 A c) (-c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b}\right )}{3 c \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(2*x*(B*(b + c*x)*(d + e*x) + ((B*c*d - 2*b*B*e + 3*A*c*e)*(b + c*x)*(d + e*x))/(c*e*x) + I*Sqrt[b/c]*(B*c*d - 2*b*B*e + 3*A*c*e)*Sqrt[1 + b/(c*x)]* Sqrt[1 + d/(e*x)]*Sqrt[x]*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b *e)] + (I*Sqrt[b/c]*(2*b*B - 3*A*c)*(-(c*d) + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[ 1 + d/(e*x)]*Sqrt[x]*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]) /b))/(3*c*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
Time = 0.43 (sec) , antiderivative size = 258, 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 = {1236, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 \int -\frac {(b B-3 A c) d-(B c d-2 b B e+3 A c e) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\int \frac {(b B-3 A c) d-(B c d-2 b B e+3 A c e) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\frac {B d (c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}-\frac {(3 A c e-2 b B e+B c d) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}}{3 c}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\frac {B d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x} (3 A c e-2 b B e+B c d) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}}{3 c}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\frac {B d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (3 A c e-2 b B e+B c d) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}}{3 c}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\frac {B d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (3 A c e-2 b B e+B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}}{3 c}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\frac {B d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (3 A c e-2 b B e+B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}}{3 c}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}-\frac {\frac {2 \sqrt {-b} B d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (3 A c e-2 b B e+B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}}{3 c}\) |
(2*B*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) - ((-2*Sqrt[-b]*(B*c*d - 2*b*B *e + 3*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sq rt[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]*B*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt [1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/ (Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2]))/(3*c)
3.13.66.3.1 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[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
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 = 0.50 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.51
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 B \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c}+\frac {2 \left (d A -\frac {B b d}{3 c}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (A e +B d -\frac {2 B \left (b e +c d \right )}{3 c}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(383\) |
| default | \(-\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (3 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c \,e^{2}-3 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d e -B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}-2 B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e^{2}+3 B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e -B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}-B \,c^{3} e^{2} x^{3}-B \,e^{2} x^{2} b \,c^{2}-B \,c^{3} d e \,x^{2}-B x b \,c^{2} d e \right )}{3 x \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{3} e}\) | \(629\) |
((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/3*B/c*(c*e*x^ 3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(d*A-1/3*B/c*b*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)*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(A*e+B*d-2 /3*B/c*(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)*EllipticE(((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 = 0.12 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {e x + d} B c^{2} e^{2} - {\left (B c^{2} d^{2} + 2 \, {\left (B b c - 3 \, A c^{2}\right )} d e - {\left (2 \, B b^{2} - 3 \, A b c\right )} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 3 \, {\left (B c^{2} d e - {\left (2 \, B b c - 3 \, A c^{2}\right )} e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )\right )}}{9 \, c^{3} e^{2}} \]
2/9*(3*sqrt(c*x^2 + b*x)*sqrt(e*x + d)*B*c^2*e^2 - (B*c^2*d^2 + 2*(B*b*c - 3*A*c^2)*d*e - (2*B*b^2 - 3*A*b*c)*e^2)*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*(B*c^2*d*e - (2*B*b*c - 3*A*c^2)*e^2)*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))))/(c^3*e^2)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\sqrt {x \left (b + c x\right )}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{\sqrt {c\,x^2+b\,x}} \,d x \]