Integrand size = 28, antiderivative size = 283 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 (4 B d-3 A e+B e x) \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {-b} (8 B c d-b B e-6 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/3*(-6*A*c*e-B*b*e+8*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)/e^3/c^(1/2)/(1+ e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(B*d*(-5*b*e+8*c*d)-3*A*e*(-b*e+2*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)/e^3/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2 )+2/3*(B*e*x-3*A*e+4*B*d)*(c*x^2+b*x)^(1/2)/e^2/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 13.83 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.95 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \left (b e x (b+c x) (4 B d-3 A e+B e x)+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} (-8 B c d+b B e+6 A c e) (b+c x) (d+e x)-i b e (8 B c d-b B e-6 A c e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e (4 B c d-b B e-3 A c e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b e^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(2*(b*e*x*(b + c*x)*(4*B*d - 3*A*e + B*e*x) + Sqrt[b/c]*(Sqrt[b/c]*(-8*B*c *d + b*B*e + 6*A*c*e)*(b + c*x)*(d + e*x) - I*b*e*(8*B*c*d - b*B*e - 6*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)] + I*b*e*(4*B*c*d - b*B*e - 3*A*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)])))/(3*b*e^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
Time = 0.46 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1230, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {b (4 B d-3 A e)+(8 B c d-b B e-6 A c e) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\int \frac {b (4 B d-3 A e)+(8 B c d-b B e-6 A c e) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\frac {(-6 A c e-b B e+8 B c d) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {(B d (8 c d-5 b e)-3 A e (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}}{3 e^2}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {b+c x} (-6 A c e-b B e+8 B c d) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (-6 A c e-b B e+8 B c d) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {b+c x} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (-6 A c e-b B e+8 B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {b+c x} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}}{3 e^2}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (-6 A c e-b B e+8 B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}-\frac {\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (-6 A c e-b B e+8 B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 e^2}\) |
(2*(4*B*d - 3*A*e + B*e*x)*Sqrt[b*x + c*x^2])/(3*e^2*Sqrt[d + e*x]) - ((2* Sqrt[-b]*(8*B*c*d - b*B*e - 6*A*c*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e* x]*EllipticE[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*d*(8*c*d - 5*b*e) - 3*A*e*(2*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*e^2)
3.13.57.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 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. \(522\) vs. \(2(235)=470\).
Time = 0.85 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.85
| method | result | size |
| elliptic | \(\frac {\sqrt {x \left (c x +b \right )}\, \sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x \right ) \left (A e -B d \right )}{e^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 B \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 e^{2}}+\frac {2 \left (\frac {A b \,e^{2}-A c d e -B b d e +B c \,d^{2}}{e^{3}}-\frac {\left (A e -B d \right ) \left (b e -c d \right )}{e^{3}}+\frac {b \left (A e -B d \right )}{e^{2}}-\frac {B b d}{3 e^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {A c e +B b e -B c d}{e^{2}}+\frac {\left (A e -B d \right ) c}{e^{2}}-\frac {2 B \left (b e +c d \right )}{3 e^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, x \left (c x +b \right )}\) | \(523\) |
| default | \(\frac {2 \sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}\, \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}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c \,e^{2}-6 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 e -6 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}+6 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 -5 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 +8 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}-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}+9 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 -8 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}-3 A \,c^{3} e^{2} x^{2}+B \,e^{2} x^{2} b \,c^{2}+4 B \,c^{3} d e \,x^{2}-3 A x b \,c^{2} e^{2}+4 B x b \,c^{2} d e \right )}{3 x \left (c e \,x^{2}+b e x +c d x +b d \right ) e^{3} c^{2}}\) | \(804\) |
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* (c*e*x^2+b*e*x)*(A*e-B*d)/e^3/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2/3*B/e^2*(c *e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*((A*b*e^2-A*c*d*e-B*b*d*e+B*c*d^2)/e ^3-(A*e-B*d)/e^3*(b*e-c*d)+b/e^2*(A*e-B*d)-1/3*B/e^2*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*(1/ e^2*(A*c*e+B*b*e-B*c*d)+(A*e-B*d)/e^2*c-2/3*B/e^2*(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 = 0.13 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.76 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left ({\left (8 \, B c^{2} d^{3} - {\left (5 \, B b c + 6 \, A c^{2}\right )} d^{2} e - {\left (B b^{2} - 3 \, A b c\right )} d e^{2} + {\left (8 \, B c^{2} d^{2} e - {\left (5 \, B b c + 6 \, A c^{2}\right )} d e^{2} - {\left (B b^{2} - 3 \, A b c\right )} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (8 \, B c^{2} d^{2} e - {\left (B b c + 6 \, A c^{2}\right )} d e^{2} + {\left (8 \, B c^{2} d e^{2} - {\left (B b c + 6 \, A c^{2}\right )} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (B c^{2} e^{3} x + 4 \, B c^{2} d e^{2} - 3 \, A c^{2} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (c^{2} e^{5} x + c^{2} d e^{4}\right )}} \]
2/9*((8*B*c^2*d^3 - (5*B*b*c + 6*A*c^2)*d^2*e - (B*b^2 - 3*A*b*c)*d*e^2 + (8*B*c^2*d^2*e - (5*B*b*c + 6*A*c^2)*d*e^2 - (B*b^2 - 3*A*b*c)*e^3)*x)*sqr t(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^2*d^2*e - (B*b*c + 6*A*c^2)*d*e^2 + (8*B*c^2*d*e^2 - (B*b*c + 6*A*c^2)*e^3)*x)*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*(B*c ^2*e^3*x + 4*B*c^2*d*e^2 - 3*A*c^2*e^3)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/( c^2*e^5*x + c^2*d*e^4)
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]