Integrand size = 28, antiderivative size = 253 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (A b-(b B-2 A c) x) \sqrt {d+e x}}{b^2 \sqrt {b x+c x^2}}-\frac {2 (b B-2 A c) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (b B d-2 A c d+A b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} \sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2*(A*b-(-2*A*c+B*b)*x)*(e*x+d)^(1/2)/b^2/(c*x^2+b*x)^(1/2)-2*(-2*A*c+B*b) *EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*x^(1/2)*(1+c*x/b)^( 1/2)*(e*x+d)^(1/2)/(-b)^(3/2)/c^(1/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2* (A*b*e-2*A*c*d+B*b*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2) )*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(3/2)/c^(1/2)/(e*x+d)^(1/2) /(c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 11.99 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.83 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {-2 i \sqrt {\frac {b}{c}} c (b B-2 A c) e \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-2 (b B-A c) \left (b (d+e x)-i \sqrt {\frac {b}{c}} c e \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{b^2 c \sqrt {x (b+c x)} \sqrt {d+e x}} \]
((-2*I)*Sqrt[b/c]*c*(b*B - 2*A*c)*e*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)] - 2*(b*B - A*c) *(b*(d + e*x) - I*Sqrt[b/c]*c*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)]))/(b^2*c*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
Time = 0.42 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1234, 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.
\(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1234 |
\(\displaystyle -\frac {2 \int -\frac {e (A b-(b B-2 A c) x)}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \int \frac {A b-(b B-2 A c) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {e \left (\frac {(A b e-2 A c d+b B d) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}-\frac {(b B-2 A c) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}\right )}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 1169 |
\(\displaystyle \frac {e \left (\frac {\sqrt {x} \sqrt {b+c x} (A b e-2 A c d+b B d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x} (b B-2 A c) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {e \left (\frac {\sqrt {x} \sqrt {b+c x} (A b e-2 A c d+b B d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (b B-2 A c) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {e \left (\frac {\sqrt {x} \sqrt {b+c x} (A b e-2 A c d+b B d) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (b B-2 A c) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {e \left (\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (A b e-2 A c d+b B d) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (b B-2 A c) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {e \left (\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (A b e-2 A c d+b B d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (b B-2 A c) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2}-\frac {2 \sqrt {d+e x} (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}}\) |
(-2*(A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(b^2*Sqrt[b*x + c*x^2]) + (e*(( -2*Sqrt[-b]*(b*B - 2*A*c)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Elliptic E[ArcSin[(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]*(b*B*d - 2*A*c*d + A*b*e)*Sqrt[x]* Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqr t[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/b^2
3.13.74.3.1 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*(a + b*x + c*x^2)^(p + 1)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 0] && (IntegerQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(477\) vs. \(2(211)=422\).
Time = 0.93 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.89
method | result | size |
elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) A}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}-\frac {2 \left (c e \,x^{2}+c d x \right ) \left (A c -B b \right )}{b^{2} c \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (\frac {B e}{c}+\frac {\left (A c -B b \right ) \left (b e -c d \right )}{c \,b^{2}}+\frac {d \left (A c -B b \right )}{b^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {A c e}{b^{2}}+\frac {\left (A c -B b \right ) e}{b^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(478\) |
default | \(\frac {2 \left (A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c e -2 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d -2 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c e +2 A \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d +B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e -B \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d -2 A \,x^{2} c^{3} e +B \,x^{2} b \,c^{2} e -A x b \,c^{2} e -2 A x \,c^{3} d +B x b \,c^{2} d -A d b \,c^{2}\right ) \sqrt {x \left (c x +b \right )}}{x \left (c x +b \right ) b^{2} c^{2} \sqrt {e x +d}}\) | \(610\) |
((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2*(c*e*x^2+b*e *x+c*d*x+b*d)*A/b^2/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2*(c*e*x^2+c*d*x)* (A*c-B*b)/b^2/c/((x+b/c)*(c*e*x^2+c*d*x))^(1/2)+2*(B*e/c+(A*c-B*b)/c*(b*e- c*d)/b^2+d*(A*c-B*b)/b^2)*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*c*e/b^2+(A*c-B*b)*e/b^2)*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 = 0.10 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.79 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d + {\left (B b^{2} c + A b c^{2}\right )} e\right )} x^{2} + {\left ({\left (B b^{2} c - 2 \, A b c^{2}\right )} d + {\left (B b^{3} + A b^{2} c\right )} e\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} e x^{2} + {\left (B b^{2} c - 2 \, A b c^{2}\right )} e x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (A b c^{2} e - {\left (B b c^{2} - 2 \, A c^{3}\right )} e x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} c^{3} e x^{2} + b^{3} c^{2} e x\right )}} \]
2/3*((((B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + ((B*b^2*c - 2* A*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*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*b*c^2 - 2*A*c^3)*e*x^2 + (B*b^2*c - 2*A*b*c^2)*e*x)*sqrt(c*e)*weierstr assZeta(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*(A*b*c^2*e - (B*b*c^2 - 2*A*c^3)*e*x)*sqrt(c*x^2 + b*x)*sqrt(e*x + d ))/(b^2*c^3*e*x^2 + b^3*c^2*e*x)
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]