\(\int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x-c x^2}} \, dx\) [149]

Optimal result

Integrand size = 29, antiderivative size = 252 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x-c x^2}} \, dx=-\frac {2 B \sqrt {d+e x} \sqrt {b x-c x^2}}{3 c}+\frac {2 \sqrt {b} (B c d+2 b B e+3 A c e) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{3 c^{3/2} e \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}-\frac {2 \sqrt {b} B d (c d+b e) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{3 c^{3/2} e \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.00 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x-c x^2}} \, dx=\frac {2 \sqrt {x (b-c x)} \left (-B c (d+e x)-\frac {\sqrt {-\frac {b}{c}} (B c d+2 b B e+3 A c e) (b-c x) (d+e x)+i b e (B c d+2 b B e+3 A c e) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )-i (2 b B+3 A c) e (c d+b e) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )}{\sqrt {-\frac {b}{c}} e x (b-c x)}\right )}{3 c^2 \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[x*(b - c*x)]*(-(B*c*(d + e*x)) - (Sqrt[-(b/c)]*(B*c*d + 2*b*B*e + 
3*A*c*e)*(b - c*x)*(d + e*x) + I*b*e*(B*c*d + 2*b*B*e + 3*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*(2*b*B + 3*A*c)*e*(c*d + b*e)*Sqrt[1 - b/(c*x)]*Sq 
rt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], -((c*d) 
/(b*e))])/(Sqrt[-(b/c)]*e*x*(b - c*x))))/(3*c^2*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, 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

\(\displaystyle -\frac {2 \int -\frac {(b B+3 A c) d+(B c d+2 b B e+3 A c e) x}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(b B+3 A c) d+(B c d+2 b B e+3 A c e) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {(3 A c e+2 b B e+B c d) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}-\frac {B d (b e+c d) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {\frac {\sqrt {x} \sqrt {b-c x} (3 A c e+2 b B e+B c d) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}-\frac {B d \sqrt {x} \sqrt {b-c x} (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} (3 A c e+2 b B e+B c d) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {B d \sqrt {x} \sqrt {b-c x} (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} (3 A c e+2 b B e+B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {B d \sqrt {x} \sqrt {b-c x} (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} (3 A c e+2 b B e+B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {B d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} (3 A c e+2 b B e+B c d) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b} B d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}}{3 c}-\frac {2 B \sqrt {b x-c x^2} \sqrt {d+e x}}{3 c}\)

Input:

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

Output:

(-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[(Sqr 
t[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)
 

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 1236

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])
 

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 [A] (verified)

Time = 1.39 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.58

method	result	size
elliptic	\(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 B \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 c}+\frac {2 \left (A d +\frac {B b d}{3 c}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (A e +B d +\frac {2 B \left (b e -c d \right )}{3 c}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}}\)	\(399\)
default	\(\frac {2 \sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}\, \left (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 {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b 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 {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) c^{2} d^{2} e -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 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 ) c^{2} 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 {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b^{2} 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 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^{2} 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 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 ) c^{2} d^{3}+B \,c^{2} e^{3} x^{3}-B b c \,e^{3} x^{2}+B \,c^{2} d \,e^{2} x^{2}-B b c d \,e^{2} x \right )}{3 x \left (-c e \,x^{2}+b e x -c d x +b d \right ) c^{2} e^{2}}\)	\(766\)

Input:

int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+b*x)^(1/2),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)*(-2/3*B/c*(- 
c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)+2*(A*d+1/3*B/c*b*d)*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*(A* 
e+B*d+2/3*B/c*(b*e-c*d))*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)*Ellipt 
icE(((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 [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.52 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {b x-c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {-c x^{2} + b x} \sqrt {e x + d} B c^{2} e^{2} - {\left (B c^{2} d^{2} - 2 \, {\left (B b c + 3 \, A c^{2}\right )} d e - {\left (2 \, B b^{2} + 3 \, A b c\right )} e^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right ) - 3 \, {\left (B c^{2} d e + {\left (2 \, B b c + 3 \, A c^{2}\right )} e^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right )\right )\right )}}{9 \, c^{3} e^{2}} \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+b*x)^(1/2),x, algorithm="fricas")
 

Output:

-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)
 

Sympy [F]

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

integrate((B*x+A)*(e*x+d)**(1/2)/(-c*x**2+b*x)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+b*x)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+b*x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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