Integrand size = 28, antiderivative size = 295 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (b B-2 A c) d (c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)*(e*x+d)^(1/2)/b^2/c/(c*x^ 2+b*x)^(1/2)+2*(2*A*c^2*d+2*b^2*B*e-b*c*(A*e+B*d))*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^(3/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2*(-2*A*c+B*b)*d*(-b*e+c*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^(3/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 20.77 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b (d+e x) ((b B-A c) (c d-b e) x-A c d (b+c x))+\sqrt {\frac {b}{c}} \left (\sqrt {\frac {b}{c}} \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) (b+c x) (d+e x)+i b e \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b (-2 b B+A c) e (c d-b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{b^3 c \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(2*(b*(d + e*x)*((b*B - A*c)*(c*d - b*e)*x - A*c*d*(b + c*x)) + Sqrt[b/c]* (Sqrt[b/c]*(2*A*c^2*d + 2*b^2*B*e - b*c*(B*d + A*e))*(b + c*x)*(d + e*x) + I*b*e*(2*A*c^2*d + 2*b^2*B*e - b*c*(B*d + A*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*(-2*b*B + A*c)*e*(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)])))/(b^3*c*Sqrt[x*( b + c*x)]*Sqrt[d + e*x])
Time = 0.53 (sec) , antiderivative size = 310, 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 = {1233, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int \frac {e \left (b (b B+A c) d+\left (2 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {b (b B+A c) d+\left (2 B e b^2-c (B d+A e) b+2 A c^2 d\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {e \left (\frac {\left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}+\frac {d (b B-2 A c) (c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \sqrt {b+c x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}+\frac {d \sqrt {x} \sqrt {b+c x} (b B-2 A c) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {e \left (\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {d \sqrt {x} \sqrt {b+c x} (b B-2 A c) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {e \left (\frac {d \sqrt {x} \sqrt {b+c x} (b B-2 A c) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {e \left (\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (b B-2 A c) (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {e \left (\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (b B-2 A c) (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}\) |
(-2*Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/( b^2*c*Sqrt[b*x + c*x^2]) + (e*((2*Sqrt[-b]*(2*A*c^2*d + 2*b^2*B*e - b*c*(B *d + A*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*B - 2*A*c)*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])))/(b^2*c)
3.13.73.3.1 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))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(578\) vs. \(2(253)=506\).
Time = 1.08 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.96
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d A}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (c e \,x^{2}+c d x \right ) \left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right )}{c^{2} b^{2} \sqrt {\left (x +\frac {b}{c}\right ) \left (c e \,x^{2}+c d x \right )}}+\frac {2 \left (\frac {e \left (A c e -B b e +2 B c d \right )}{c^{2}}-\frac {\left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right ) \left (b e -c d \right )}{c^{2} b^{2}}-\frac {d \left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right )}{c \,b^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (\frac {B \,e^{2}}{c}+\frac {A c d e}{b^{2}}-\frac {\left (A b c e -A \,c^{2} d -b^{2} B e +B b c d \right ) e}{c \,b^{2}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(579\) |
| default | \(\frac {2 \left (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^{2} c^{2} d 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^{3} 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^{3} 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^{2} c^{2} d 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^{3} 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}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} 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^{2} 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^{4} 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^{3} 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^{2} c^{2} d^{2}+A \,x^{2} b \,c^{3} e^{2}-2 A \,x^{2} c^{4} d e -B \,x^{2} b^{2} c^{2} e^{2}+B \,x^{2} b \,c^{3} d e -2 A x \,c^{4} d^{2}-B x \,b^{2} c^{2} d e +B x b \,c^{3} d^{2}-A b \,c^{3} d^{2}\right ) \sqrt {x \left (c x +b \right )}}{x \,c^{3} \left (c x +b \right ) b^{2} \sqrt {e x +d}}\) | \(898\) |
((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)*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*b*c*e-A*c^2*d-B*b^2*e+B*b*c*d)/c^2/b^2/((x+b/c)*(c*e*x^2+c*d*x))^(1/2 )+2*(e*(A*c*e-B*b*e+2*B*c*d)/c^2-(A*b*c*e-A*c^2*d-B*b^2*e+B*b*c*d)/c^2*(b* e-c*d)/b^2-1/c*d*(A*b*c*e-A*c^2*d-B*b^2*e+B*b*c*d)/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*(B*e^ 2/c+A*c*d*e/b^2-(A*b*c*e-A*c^2*d-B*b^2*e+B*b*c*d)/c*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 = 0.14 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.99 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left ({\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{2} + 2 \, {\left (B b^{2} c^{2} + A b c^{3}\right )} d e - {\left (2 \, B b^{3} c - A b^{2} c^{2}\right )} e^{2}\right )} x^{2} + {\left ({\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{2} + 2 \, {\left (B b^{3} c + A b^{2} c^{2}\right )} d e - {\left (2 \, B b^{4} - A b^{3} c\right )} e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d e - {\left (2 \, B b^{2} c^{2} - A b c^{3}\right )} e^{2}\right )} x^{2} + {\left ({\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d e - {\left (2 \, B b^{3} c - A b^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (A b c^{3} d e - {\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d e - {\left (B b^{2} c^{2} - A b c^{3}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} c^{4} e x^{2} + b^{3} c^{3} e x\right )}} \]
2/3*((((B*b*c^3 - 2*A*c^4)*d^2 + 2*(B*b^2*c^2 + A*b*c^3)*d*e - (2*B*b^3*c - A*b^2*c^2)*e^2)*x^2 + ((B*b^2*c^2 - 2*A*b*c^3)*d^2 + 2*(B*b^3*c + A*b^2* c^2)*d*e - (2*B*b^4 - A*b^3*c)*e^2)*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^3 - 2*A*c^4)*d*e - (2*B*b^2*c^2 - A*b*c^3)*e^2)*x^2 + ((B*b^2*c ^2 - 2*A*b*c^3)*d*e - (2*B*b^3*c - A*b^2*c^2)*e^2)*x)*sqrt(c*e)*weierstras sZeta(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^3*d*e - ((B*b*c^3 - 2*A*c^4)*d*e - (B*b^2*c^2 - A*b*c^3)*e^2)*x )*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(b^2*c^4*e*x^2 + b^3*c^3*e*x)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]