\(\int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x-c x^2}} \, dx\) [151]

Optimal result

Integrand size = 29, antiderivative size = 259 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x-c x^2}} \, dx=-\frac {2 (B d-A e) \sqrt {b x-c x^2}}{d (c d+b e) \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {c} (B d-A e) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{d e (c d+b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}+\frac {2 \sqrt {b} B \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 19.59 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x-c x^2}} \, dx=\frac {2 \sqrt {-\frac {b}{c}} d (B d-A e) (b-c x)-2 i b e (-B d+A e) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )+2 i A e (c d+b e) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )}{\sqrt {-\frac {b}{c}} d e (c d+b e) \sqrt {x (b-c x)} \sqrt {d+e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1237, 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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1169

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(-2*(B*d - A*e)*Sqrt[b*x - c*x^2])/(d*(c*d + b*e)*Sqrt[d + e*x]) + ((-2*Sq 
rt[b]*Sqrt[c]*(B*d - A*e)*Sqrt[x]*Sqrt[1 - (c*x)/b]*Sqrt[d + e*x]*Elliptic 
E[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[b]], -((b*e)/(c*d))])/(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]))/(d*(c*d + b*e))
 

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 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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. \(452\) vs. \(2(217)=434\).

Time = 2.72 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.75

method	result	size
elliptic	\(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 \left (-c e \,x^{2}+b e x \right ) \left (A e -B d \right )}{\left (b e +c d \right ) e d \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {B}{e}+\frac {A e -B d}{e d}-\frac {b \left (A e -B d \right )}{\left (b e +c d \right ) d}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (A e -B d \right ) c \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{\left (b e +c d \right ) e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}}\)	\(453\)
default	\(\frac {2 \left (A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b d \,e^{2}+A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) c \,d^{2} e -A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b d \,e^{2}-A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) c \,d^{2} e +B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) b \,d^{2} e +B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {\left (-c x +b \right ) e}{b e +c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {\frac {d c}{b e +c d}}\right ) c \,d^{3}-A c \,e^{3} x^{2}+B c d \,e^{2} x^{2}+A x b \,e^{3}-B x b d \,e^{2}\right ) \sqrt {x \left (-c x +b \right )}\, \sqrt {e x +d}}{d \,e^{2} \left (b e +c d \right ) x \left (-c e \,x^{2}+b e x -c d x +b d \right )}\)	\(530\)

Input:

int((B*x+A)/(e*x+d)^(3/2)/(-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*(-c*e*x^2 
+b*e*x)/(b*e+c*d)/e/d*(A*e-B*d)/((x+d/e)*(-c*e*x^2+b*e*x))^(1/2)+2*(B/e+1/ 
e*(A*e-B*d)/d-b/(b*e+c*d)/d*(A*e-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)*Elli 
pticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1/2))+2*(A*e-B*d)*c/(b*e+c*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. 449 vs. \(2 (217) = 434\).

Time = 0.09 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.73 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {b x-c x^2}} \, dx=-\frac {2 \, {\left ({\left (B c d^{3} + A b d e^{2} + 2 \, {\left (B b + A c\right )} d^{2} e + {\left (B c d^{2} e + A b e^{3} + 2 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right ) + 3 \, {\left (B c d^{2} e - A c d e^{2} + {\left (B c d e^{2} - A c e^{3}\right )} x\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right )\right ) + 3 \, {\left (B c d e^{2} - A c e^{3}\right )} \sqrt {-c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (c^{2} d^{3} e^{2} + b c d^{2} e^{3} + {\left (c^{2} d^{2} e^{3} + b c d e^{4}\right )} x\right )}} \] Input:

integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(1/2),x, algorithm="fricas")
 

Output:

-2/3*((B*c*d^3 + A*b*d*e^2 + 2*(B*b + A*c)*d^2*e + (B*c*d^2*e + A*b*e^3 + 
2*(B*b + A*c)*d*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*c*d^2*e - 
 A*c*d*e^2 + (B*c*d*e^2 - A*c*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*d*e^2 
- A*c*e^3)*sqrt(-c*x^2 + b*x)*sqrt(e*x + d))/(c^2*d^3*e^2 + b*c*d^2*e^3 + 
(c^2*d^2*e^3 + b*c*d*e^4)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(d + e*x)*sqrt(b - c*x))/(sqrt(x)*b*d**2 + 2*sqrt(x)*b*d*e*x + sq 
rt(x)*b*e**2*x**2 - sqrt(x)*c*d**2*x - 2*sqrt(x)*c*d*e*x**2 - sqrt(x)*c*e* 
*2*x**3),x)*a + int((sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x))/(b*d**2 + 2*b*d* 
e*x + b*e**2*x**2 - c*d**2*x - 2*c*d*e*x**2 - c*e**2*x**3),x)*b