\(\int \frac {x (A+B x)}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx\) [15]

Optimal result

Integrand size = 29, antiderivative size = 290 \[ \int \frac {x (A+B x)}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\frac {2 (3 A c e-2 B (c d+b e)) x \sqrt {d+e x}}{3 c e^2 \sqrt {b x+c x^2}}+\frac {2 B \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c e}-\frac {2 \sqrt {b} (3 A c e-2 B (c d+b e)) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{3 c^{3/2} e^2 \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 b^{3/2} B \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{3 c^{3/2} e \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.42 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.91 \[ \int \frac {x (A+B x)}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\frac {2 x \left (B e (b+c x) (d+e x)+\frac {(3 A c e-2 B (c d+b e)) (b+c x) (d+e x)}{c x}+i \sqrt {\frac {b}{c}} e (3 A c e-2 B (c d+b e)) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} \sqrt {x} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \sqrt {\frac {b}{c}} e (B c d+2 b B e-3 A c e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} \sqrt {x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{3 c e^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2184, 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.

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

\(\Big \downarrow \) 2184

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1169

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 1.18 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.31

method	result	size
elliptic	\(\frac {\sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 B \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}-\frac {2 B b \,d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{3 c \,e^{2} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (A -\frac {2 B \left (b e +c d \right )}{3 c e}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(379\)
default	\(-\frac {2 \left (3 A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c d \,e^{2}-3 A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c d \,e^{2}+3 A \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{2} d^{2} e -2 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{2}-B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{2} e +2 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{2}-2 B \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{2} d^{3}-B \,c^{2} e^{3} x^{3}-B b c \,e^{3} x^{2}-B \,c^{2} d \,e^{2} x^{2}-B b c d \,e^{2} x \right ) \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}}{3 e^{3} c^{2} x \left (c e \,x^{2}+b e x +c d x +b d \right )}\)	\(630\)

Input:

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

Output:

((c*x+b)*x*(e*x+d))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/3*B/c/e*(c*e* 
x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)-2/3*B/c/e^2*b*d^2*((x+d/e)/d*e)^(1/2)*((b 
/c+x)/(-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-2/3*B/c/e*( 
b*e+c*d))*d/e*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-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 = 0.10 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.31 \[ \int \frac {x (A+B x)}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {e x + d} B c^{2} e^{2} + {\left (2 \, B c^{2} d^{2} + {\left (B b c - 3 \, A c^{2}\right )} d e + {\left (2 \, B b^{2} - 3 \, A b c\right )} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c - 3 \, A c^{2}\right )} e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )\right )}}{9 \, c^{3} e^{3}} \] Input:

integrate(x*(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 + (2*B*c^2*d^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*(2*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^3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x (A+B x)}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, a -3 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) a b c \,e^{2}-3 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) a \,c^{2} d e +2 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) b^{3} e^{2}+4 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) b^{2} c d e +2 \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{b c \,e^{2} x^{2}+c^{2} d e \,x^{2}+b^{2} e^{2} x +2 b c d e x +c^{2} d^{2} x +b^{2} d e +b c \,d^{2}}d x \right ) b \,c^{2} d^{2}-\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{b c \,e^{2} x^{3}+c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+2 b c d e \,x^{2}+c^{2} d^{2} x^{2}+b^{2} d e x +b c \,d^{2} x}d x \right ) a \,b^{2} d e -\left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}}{b c \,e^{2} x^{3}+c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+2 b c d e \,x^{2}+c^{2} d^{2} x^{2}+b^{2} d e x +b c \,d^{2} x}d x \right ) a b c \,d^{2}}{2 b e +2 c d} \] Input:

int(x*(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 - 3*int((sqrt(x)*sqrt(d + e*x)*sq 
rt(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**2 - 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*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**2 + 4* 
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)*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*c**2*d**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*b**2*d*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*c*d**2)/(2*(b*e + c*d))