\(\int \frac {(A+B x) (d+e x)^{3/2}}{(b x-c x^2)^{3/2}} \, dx\) [156]

Optimal result

Integrand size = 29, antiderivative size = 304 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\frac {2 (b B+2 A c) (c d+b e) x \sqrt {d+e x}}{b^2 c \sqrt {b x-c x^2}}-\frac {2 A (d+e x)^{3/2}}{b \sqrt {b x-c x^2}}-\frac {2 \left (2 A c^2 d+2 b^2 B e+b c (B d+A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{b^{3/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}+\frac {2 (b B+2 A c) d (c d+b e) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{b^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:

2*(2*A*c+B*b)*(b*e+c*d)*x*(e*x+d)^(1/2)/b^2/c/(-c*x^2+b*x)^(1/2)-2*A*(e*x+ 
d)^(3/2)/b/(-c*x^2+b*x)^(1/2)-2*(2*A*c^2*d+2*b^2*B*e+b*c*(A*e+B*d))*x^(1/2 
)*(1-c*x/b)^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2)*x^(1/2)/b^(1/2),(-b*e/c/ 
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)*x^(1/2)*(1-c*x/b)^(1/2)*(1+e*x/d)^(1/2)*EllipticF(c^(1/2)*x^( 
1/2)/b^(1/2),(-b*e/c/d)^(1/2))/b^(3/2)/c^(3/2)/(e*x+d)^(1/2)/(-c*x^2+b*x)^ 
(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.51 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\frac {2 \left (\sqrt {-\frac {b}{c}} \left (2 A c^2 d+2 b^2 B e+b c (B d+A e)\right ) (b-c x) (d+e x)+\sqrt {-\frac {b}{c}} c (d+e x) ((b B+A c) (c d+b e) x+A c d (-b+c x))+i b e \left (2 A c^2 d+2 b^2 B e+b c (B d+A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )-i b (2 b B+A c) e (c d+b e) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )\right )}{b^2 \sqrt {-\frac {b}{c}} c^2 \sqrt {x (b-c x)} \sqrt {d+e x}} \] Input:

Integrate[((A + B*x)*(d + e*x)^(3/2))/(b*x - c*x^2)^(3/2),x]
 

Output:

(2*(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) + Sqrt[-(b/c)]*c*(d + e*x)*((b*B + A*c)*(c*d + b*e)*x + A*c*d*(-b + c 
*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)]*S 
qrt[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^2*Sqrt[-(b/c)]*c^2*Sqrt[x*(b - c*x)]*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, 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

\(\displaystyle -\frac {2 \int \frac {e \left (b (b B-A c) d+\left (2 B e b^2+c (B d+A e) b+2 A c^2 d\right ) x\right )}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {e \int \frac {b (b B-A c) d+\left (2 B e b^2+c (B d+A e) b+2 A c^2 d\right ) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {e \left (\frac {\left (b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}-\frac {d (2 A c+b B) (b e+c d) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 1169

\(\displaystyle -\frac {e \left (\frac {\sqrt {x} \sqrt {b-c x} \left (b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}-\frac {d \sqrt {x} \sqrt {b-c x} (2 A c+b B) (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 122

\(\displaystyle -\frac {e \left (\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b-c x} (2 A c+b B) (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 120

\(\displaystyle -\frac {e \left (\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b-c x} (2 A c+b B) (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 127

\(\displaystyle -\frac {e \left (\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (2 A c+b B) (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

\(\Big \downarrow \) 126

\(\displaystyle -\frac {e \left (\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b} d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (2 A c+b B) (b e+c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}\right )}{b^2 c}-\frac {2 \sqrt {d+e x} \left (A b c d-x \left (b c (A e+B d)+2 A c^2 d+b^2 B e\right )\right )}{b^2 c \sqrt {b x-c x^2}}\)

Input:

Int[((A + B*x)*(d + e*x)^(3/2))/(b*x - c*x^2)^(3/2),x]
 

Output:

(-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)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 120

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]
 

rule 122

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])
 

rule 126

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])
 

rule 127

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])
 

rule 1169

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]
 

rule 1233

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])
 

rule 1269

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(592\) vs. \(2(258)=516\).

Time = 2.63 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.95

method	result	size
elliptic	\(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 \left (-c e \,x^{2}-c d x \right ) \left (A c e b +A \,c^{2} d +b^{2} B e +B b c d \right )}{b^{2} c^{2} \sqrt {\left (x -\frac {b}{c}\right ) \left (-c e \,x^{2}-c d x \right )}}-\frac {2 \left (-c e \,x^{2}+b e x -c d x +b d \right ) d A}{b^{2} \sqrt {x \left (-c e \,x^{2}+b e x -c d x +b d \right )}}+\frac {2 \left (-\frac {e \left (A c e +B b e +2 B c d \right )}{c^{2}}+\frac {\left (A c e b +A \,c^{2} d +b^{2} B e +B b c d \right ) \left (b e +c d \right )}{c^{2} b^{2}}-\frac {d \left (A c e b +A \,c^{2} d +b^{2} B e +B b c d \right )}{c \,b^{2}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {B \,e^{2}}{c}-\frac {\left (A c e b +A \,c^{2} d +b^{2} B e +B b c d \right ) e}{c \,b^{2}}-\frac {A c d e}{b^{2}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}}\)	\(593\)
default	\(-\frac {2 \left (A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{2} c d \,e^{2}+A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b \,c^{2} d^{2} e -A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{2} c d \,e^{2}-3 A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b \,c^{2} d^{2} e -2 A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) c^{3} d^{3}+2 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{3} d \,e^{2}+2 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{2} c \,d^{2} e -2 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{3} d \,e^{2}-3 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{2} c \,d^{2} e -B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b \,c^{2} d^{3}-A b \,c^{2} e^{3} x^{2}-2 A \,c^{3} d \,e^{2} x^{2}-B \,b^{2} c \,e^{3} x^{2}-B b \,c^{2} d \,e^{2} x^{2}-2 A \,c^{3} d^{2} e x -B \,b^{2} c d \,e^{2} x -B b \,c^{2} d^{2} e x +A b \,d^{2} e \,c^{2}\right ) \sqrt {x \left (-c x +b \right )}}{x \left (-c x +b \right ) c^{2} e \,b^{2} \sqrt {e x +d}}\)	\(892\)

Input:

int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/(e*x+d)^(1/2)*((-c*x+b)*x*(e*x+d))^(1/2)/(x*(-c*x+b))^(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)/b^2/c^2/((x-b/c)*(-c*e*x^2-c*d* 
x))^(1/2)-2*(-c*e*x^2+b*e*x-c*d*x+b*d)*d/b^2*A/(x*(-c*e*x^2+b*e*x-c*d*x+b* 
d))^(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)*d/e*((x+d 
/e)/d*e)^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c*e*x^3+b*e*x^2 
-c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1/2 
))+2*(-B*e^2/c-(A*b*c*e+A*c^2*d+B*b^2*e+B*b*c*d)/c*e/b^2-A*c*d*e/b^2)*d/e* 
((x+d/e)/d*e)^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c*e*x^3+b* 
e*x^2-c*d*x^2+b*d*x)^(1/2)*((-d/e-b/c)*EllipticE(((x+d/e)/d*e)^(1/2),(-d/e 
/(-d/e-b/c))^(1/2))+b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1 
/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (258) = 516\).

Time = 0.13 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.93 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left ({\left (B b c^{3} + 2 \, A c^{4}\right )} d^{2} - 2 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} d e - {\left (2 \, B b^{3} c + A b^{2} c^{2}\right )} e^{2}\right )} x^{2} - {\left ({\left (B b^{2} c^{2} + 2 \, A b c^{3}\right )} d^{2} - 2 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} d e - {\left (2 \, B b^{4} + A b^{3} c\right )} e^{2}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right ) + 3 \, {\left ({\left ({\left (B b c^{3} + 2 \, A c^{4}\right )} d e + {\left (2 \, B b^{2} c^{2} + A b c^{3}\right )} e^{2}\right )} x^{2} - {\left ({\left (B b^{2} c^{2} + 2 \, A b c^{3}\right )} d e + {\left (2 \, B b^{3} c + A b^{2} c^{2}\right )} e^{2}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right )\right ) - 3 \, {\left (A b c^{3} d e - {\left ({\left (B b c^{3} + 2 \, A c^{4}\right )} d e + {\left (B b^{2} c^{2} + A b c^{3}\right )} e^{2}\right )} x\right )} \sqrt {-c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} c^{4} e x^{2} - b^{3} c^{3} e x\right )}} \] Input:

integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+b*x)^(3/2),x, algorithm="fricas")
 

Output:

-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)*weierst 
rassZeta(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)
 

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\left (- x \left (- b + c x\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((B*x+A)*(e*x+d)**(3/2)/(-c*x**2+b*x)**(3/2),x)
 

Output:

Integral((A + B*x)*(d + e*x)**(3/2)/(-x*(-b + c*x))**(3/2), x)
 

Maxima [F]

\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (-c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+b*x)^(3/2),x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(e*x + d)^(3/2)/(-c*x^2 + b*x)^(3/2), x)
 

Giac [F]

\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (-c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+b*x)^(3/2),x, algorithm="giac")
 

Output:

integrate((B*x + A)*(e*x + d)^(3/2)/(-c*x^2 + b*x)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (b\,x-c\,x^2\right )}^{3/2}} \,d x \] Input:

int(((A + B*x)*(d + e*x)^(3/2))/(b*x - c*x^2)^(3/2),x)
 

Output:

int(((A + B*x)*(d + e*x)^(3/2))/(b*x - c*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x-c x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:

int((B*x+A)*(e*x+d)^(3/2)/(-c*x^2+b*x)^(3/2),x)
 

Output:

( - 4*sqrt(d + e*x)*sqrt(b - c*x)*a*c*d - 4*sqrt(d + e*x)*sqrt(b - c*x)*b* 
*2*e*x + 2*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d + s 
qrt(x)*b**2*e*x - 2*sqrt(x)*b*c*d*x - 2*sqrt(x)*b*c*e*x**2 + sqrt(x)*c**2* 
d*x**2 + sqrt(x)*c**2*e*x**3),x)*a*b**2*c*e**2 + 2*sqrt(x)*int((sqrt(d + e 
*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d + sqrt(x)*b**2*e*x - 2*sqrt(x)*b*c*d* 
x - 2*sqrt(x)*b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + sqrt(x)*c**2*e*x**3),x)*a 
*b*c**2*d*e - 2*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2* 
d + sqrt(x)*b**2*e*x - 2*sqrt(x)*b*c*d*x - 2*sqrt(x)*b*c*e*x**2 + sqrt(x)* 
c**2*d*x**2 + sqrt(x)*c**2*e*x**3),x)*a*b*c**2*e**2*x - 2*sqrt(x)*int((sqr 
t(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d + sqrt(x)*b**2*e*x - 2*sqrt(x) 
*b*c*d*x - 2*sqrt(x)*b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + sqrt(x)*c**2*e*x** 
3),x)*a*c**3*d*e*x + sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)* 
b**2*d + sqrt(x)*b**2*e*x - 2*sqrt(x)*b*c*d*x - 2*sqrt(x)*b*c*e*x**2 + sqr 
t(x)*c**2*d*x**2 + sqrt(x)*c**2*e*x**3),x)*b**4*e**2 + sqrt(x)*int((sqrt(d 
 + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d + sqrt(x)*b**2*e*x - 2*sqrt(x)*b* 
c*d*x - 2*sqrt(x)*b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + sqrt(x)*c**2*e*x**3), 
x)*b**3*c*d*e - sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2* 
d + sqrt(x)*b**2*e*x - 2*sqrt(x)*b*c*d*x - 2*sqrt(x)*b*c*e*x**2 + sqrt(x)* 
c**2*d*x**2 + sqrt(x)*c**2*e*x**3),x)*b**3*c*e**2*x - sqrt(x)*int((sqrt(d 
+ e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d + sqrt(x)*b**2*e*x - 2*sqrt(x)*...