\(\int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x-c x^2}} \, dx\) [148]

Optimal result

Integrand size = 29, antiderivative size = 337 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x-c x^2}} \, dx=-\frac {2 (3 B c d+4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x-c x^2}}{15 c^2}-\frac {2 B (d+e x)^{3/2} \sqrt {b x-c x^2}}{5 c}+\frac {2 \sqrt {b} \left (10 A c e (2 c d+b e)+B \left (3 c^2 d^2+13 b c d e+8 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}-\frac {2 \sqrt {b} d (c d+b e) (3 B c d+4 b B e+5 A c e) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:

-2/15*(5*A*c*e+4*B*b*e+3*B*c*d)*(e*x+d)^(1/2)*(-c*x^2+b*x)^(1/2)/c^2-2/5*B 
*(e*x+d)^(3/2)*(-c*x^2+b*x)^(1/2)/c+2/15*b^(1/2)*(10*A*c*e*(b*e+2*c*d)+B*( 
8*b^2*e^2+13*b*c*d*e+3*c^2*d^2))*x^(1/2)*(1-c*x/b)^(1/2)*(e*x+d)^(1/2)*Ell 
ipticE(c^(1/2)*x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/c^(5/2)/e/(1+e*x/d)^(1/2) 
/(-c*x^2+b*x)^(1/2)-2/15*b^(1/2)*d*(b*e+c*d)*(5*A*c*e+4*B*b*e+3*B*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^(5/2)/e/(e*x+d)^(1/2)/(-c*x^2+b*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.45 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x-c x^2}} \, dx=\frac {2 \sqrt {x (b-c x)} \left (-\frac {\left (10 A c e (2 c d+b e)+B \left (3 c^2 d^2+13 b c d e+8 b^2 e^2\right )\right ) (d+e x)}{e \sqrt {x}}-c \sqrt {x} (d+e x) (5 A c e+B (6 c d+4 b e+3 c e x))+\frac {i \sqrt {-\frac {b}{c}} c \left (10 A c e (2 c d+b e)+B \left (3 c^2 d^2+13 b c d e+8 b^2 e^2\right )\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )}{b-c x}+\frac {i (c d+b e) \left (15 A c^2 d+8 b^2 B e+b c (9 B d+10 A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )}{\sqrt {-\frac {b}{c}} (b-c x)}\right )}{15 c^3 \sqrt {x} \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[x*(b - c*x)]*(-(((10*A*c*e*(2*c*d + b*e) + B*(3*c^2*d^2 + 13*b*c*d 
*e + 8*b^2*e^2))*(d + e*x))/(e*Sqrt[x])) - c*Sqrt[x]*(d + e*x)*(5*A*c*e + 
B*(6*c*d + 4*b*e + 3*c*e*x)) + (I*Sqrt[-(b/c)]*c*(10*A*c*e*(2*c*d + b*e) + 
 B*(3*c^2*d^2 + 13*b*c*d*e + 8*b^2*e^2))*Sqrt[1 - b/(c*x)]*Sqrt[1 + d/(e*x 
)]*x*EllipticE[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], -((c*d)/(b*e))])/(b - c*x) 
 + (I*(c*d + b*e)*(15*A*c^2*d + 8*b^2*B*e + b*c*(9*B*d + 10*A*e))*Sqrt[1 - 
 b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], - 
((c*d)/(b*e))])/(Sqrt[-(b/c)]*(b - c*x))))/(15*c^3*Sqrt[x]*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {1236, 27, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle -\frac {2 \int -\frac {\sqrt {d+e x} ((b B+5 A c) d+(3 B c d+4 b B e+5 A c e) x)}{2 \sqrt {b x-c x^2}}dx}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {d+e x} ((b B+5 A c) d+(3 B c d+4 b B e+5 A c e) x)}{\sqrt {b x-c x^2}}dx}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {-\frac {2 \int -\frac {d \left (4 B e b^2+c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d+b e)+B \left (3 c^2 d^2+13 b c e d+8 b^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {d \left (4 B e b^2+c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d+b e)+B \left (3 c^2 d^2+13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {\left (10 A c e (b e+2 c d)+B \left (8 b^2 e^2+13 b c d e+3 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}-\frac {d (b e+c d) (5 A c e+4 b B e+3 B c d) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {\frac {\frac {\sqrt {x} \sqrt {b-c x} \left (10 A c e (b e+2 c d)+B \left (8 b^2 e^2+13 b c d e+3 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}-\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) (5 A c e+4 b B e+3 B c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {\frac {\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (10 A c e (b e+2 c d)+B \left (8 b^2 e^2+13 b c d e+3 c^2 d^2\right )\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) (5 A c e+4 b B e+3 B c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {\frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (10 A c e (b e+2 c d)+B \left (8 b^2 e^2+13 b c d e+3 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) (5 A c e+4 b B e+3 B c d) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (10 A c e (b e+2 c d)+B \left (8 b^2 e^2+13 b c d e+3 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) (5 A c e+4 b B e+3 B c d) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (10 A c e (b e+2 c d)+B \left (8 b^2 e^2+13 b c d e+3 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b} d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) (5 A c e+4 b B e+3 B c d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}}{3 c}-\frac {2 \sqrt {b x-c x^2} \sqrt {d+e x} (5 A c e+4 b B e+3 B c d)}{3 c}}{5 c}-\frac {2 B \sqrt {b x-c x^2} (d+e x)^{3/2}}{5 c}\)

Input:

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

Output:

(-2*B*(d + e*x)^(3/2)*Sqrt[b*x - c*x^2])/(5*c) + ((-2*(3*B*c*d + 4*b*B*e + 
 5*A*c*e)*Sqrt[d + e*x]*Sqrt[b*x - c*x^2])/(3*c) + ((2*Sqrt[b]*(10*A*c*e*( 
2*c*d + b*e) + B*(3*c^2*d^2 + 13*b*c*d*e + 8*b^2*e^2))*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]*d*(c 
*d + b*e)*(3*B*c*d + 4*b*B*e + 5*A*c*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))])/(S 
qrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x - c*x^2]))/(3*c))/(5*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 = 1.60 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.59

method	result	size
elliptic	\(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 B e x \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{5 c}-\frac {2 \left (A \,e^{2}+2 B d e +\frac {2 B e \left (2 b e -2 c d \right )}{5 c}\right ) \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (A \,d^{2}+\frac {\left (A \,e^{2}+2 B d e +\frac {2 B e \left (2 b e -2 c d \right )}{5 c}\right ) b d}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (2 A d e +B \,d^{2}+\frac {3 B e b d}{5 c}+\frac {2 \left (A \,e^{2}+2 B d e +\frac {2 B e \left (2 b e -2 c d \right )}{5 c}\right ) \left (b e -c d \right )}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}}\)	\(536\)
default	\(\text {Expression too large to display}\)	\(1197\)

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)*(-2/5*B*e/c* 
x*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)-2/3*(A*e^2+2*B*d*e+2/5*B*e/c*(2*b 
*e-2*c*d))/c/e*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)+2*(A*d^2+1/3*(A*e^2+ 
2*B*d*e+2/5*B*e/c*(2*b*e-2*c*d))/c/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)*E 
llipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1/2))+2*(2*A*d*e+B*d^2+3/5 
*B*e/c*b*d+2/3*(A*e^2+2*B*d*e+2/5*B*e/c*(2*b*e-2*c*d))/c/e*(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)*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 [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.39 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x-c x^2}} \, dx=\frac {2 \, {\left ({\left (3 \, B c^{3} d^{3} - {\left (8 \, B b c^{2} + 25 \, A c^{3}\right )} d^{2} e - {\left (17 \, B b^{2} c + 25 \, A b c^{2}\right )} d e^{2} - 2 \, {\left (4 \, B b^{3} + 5 \, A b^{2} c\right )} e^{3}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right ) + 3 \, {\left (3 \, B c^{3} d^{2} e + {\left (13 \, B b c^{2} + 20 \, A c^{3}\right )} d e^{2} + 2 \, {\left (4 \, B b^{2} c + 5 \, A b c^{2}\right )} e^{3}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} + 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d - b e}{3 \, c e}\right )\right ) - 3 \, {\left (3 \, B c^{3} e^{3} x + 6 \, B c^{3} d e^{2} + {\left (4 \, B b c^{2} + 5 \, A c^{3}\right )} e^{3}\right )} \sqrt {-c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, c^{4} e^{2}} \] Input:

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

Output:

2/45*((3*B*c^3*d^3 - (8*B*b*c^2 + 25*A*c^3)*d^2*e - (17*B*b^2*c + 25*A*b*c 
^2)*d*e^2 - 2*(4*B*b^3 + 5*A*b^2*c)*e^3)*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*(3*B*c^3*d^2*e + (13*B*b*c^2 + 20*A*c^3)*d*e^2 + 2*(4*B*b^2*c + 5*A*b 
*c^2)*e^3)*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*(3*B*c^3*e^3*x + 6*B*c^3*d*e^2 + (4*B* 
b*c^2 + 5*A*c^3)*e^3)*sqrt(-c*x^2 + b*x)*sqrt(e*x + d))/(c^4*e^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(20*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*a*c*d*e + 6*sqrt(x)*sqrt(d + e*x)* 
sqrt(b - c*x)*b**2*d*e - 4*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b**2*e**2*x 
 + 10*sqrt(x)*sqrt(d + e*x)*sqrt(b - c*x)*b*c*d**2 + 4*sqrt(x)*sqrt(d + e* 
x)*sqrt(b - c*x)*b*c*d*e*x + 10*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**2*c*e**4 + 10*int((sqrt(x)*sqrt(d + e*x)*s 
qrt(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**2*d*e**3 - 20*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**3*d**2*e**2 
+ 8*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 
**4*e**4 + 5*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*c*d*e**3 - 10*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**2*d**2*e**2 - 3*int((sqrt(x)*sqrt(d + e*x)*s 
qrt(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**3*d**3*e - 10*int((sqrt(x)*s 
qrt(d + e*x)*sqrt(b - c*x))/(b**2*d*e*x + b**2*e**2*x**2 - b*c*d**2*x -...