\(\int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx\) [97]

Optimal result

Integrand size = 26, antiderivative size = 194 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\frac {(2 B d-A e) \sqrt {b x+c x^2}}{d e^2}-\frac {(B d-A e) x \sqrt {b x+c x^2}}{d e (d+e x)}-\frac {(4 B c d-b B e-2 A c e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e^3}+\frac {(B d (4 c d-3 b e)-A e (2 c d-b e)) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{\sqrt {d} e^3 \sqrt {c d-b e}} \] Output:

(-A*e+2*B*d)*(c*x^2+b*x)^(1/2)/d/e^2-(-A*e+B*d)*x*(c*x^2+b*x)^(1/2)/d/e/(e 
*x+d)-(-2*A*c*e-B*b*e+4*B*c*d)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(1/2 
)/e^3+(B*d*(-3*b*e+4*c*d)-A*e*(-b*e+2*c*d))*arctanh((-b*e+c*d)^(1/2)*x/d^( 
1/2)/(c*x^2+b*x)^(1/2))/d^(1/2)/e^3/(-b*e+c*d)^(1/2)
 

Mathematica [A] (verified)

Time = 11.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {(-B d+A e) x^{3/2} (b+c x)}{d+e x}+\frac {e \sqrt {x} (-A e (-c d+b e+c e x)+B d (-2 c d+2 b e+c e x))+\frac {d (-c d+b e) (-4 B c d+b B e+2 A c e) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {b} \sqrt {c} \sqrt {1+\frac {c x}{b}}}-\frac {\sqrt {d} \sqrt {c d-b e} (B d (4 c d-3 b e)+A e (-2 c d+b e)) \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{\sqrt {b+c x}}}{e^3}\right )}{d (-c d+b e) \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(b + c*x)]*(((-(B*d) + A*e)*x^(3/2)*(b + c*x))/(d + e*x) + (e*Sqrt 
[x]*(-(A*e*(-(c*d) + b*e + c*e*x)) + B*d*(-2*c*d + 2*b*e + c*e*x)) + (d*(- 
(c*d) + b*e)*(-4*B*c*d + b*B*e + 2*A*c*e)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b 
]])/(Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]) - (Sqrt[d]*Sqrt[c*d - b*e]*(B*d*(4 
*c*d - 3*b*e) + A*e*(-2*c*d + b*e))*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqr 
t[d]*Sqrt[b + c*x])])/Sqrt[b + c*x])/e^3))/(d*(-(c*d) + b*e)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1230, 1269, 1091, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\sqrt {b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac {\frac {(-2 A c e-b B e+4 B c d) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {(B d (4 c d-3 b e)-A e (2 c d-b e)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\sqrt {b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac {\frac {2 (-2 A c e-b B e+4 B c d) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {(B d (4 c d-3 b e)-A e (2 c d-b e)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt {c} e}-\frac {(B d (4 c d-3 b e)-A e (2 c d-b e)) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\sqrt {b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac {\frac {2 (B d (4 c d-3 b e)-A e (2 c d-b e)) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt {c} e}}{2 e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {b x+c x^2} (-A e+2 B d+B e x)}{e^2 (d+e x)}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (-2 A c e-b B e+4 B c d)}{\sqrt {c} e}-\frac {(B d (4 c d-3 b e)-A e (2 c d-b e)) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e^2}\)

Input:

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

Output:

((2*B*d - A*e + B*e*x)*Sqrt[b*x + c*x^2])/(e^2*(d + e*x)) - ((2*(4*B*c*d - 
 b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e) - (( 
B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/( 
2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(Sqrt[d]*e*Sqrt[c*d - b*e]) 
)/(2*e^2)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (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 [A] (verified)

Time = 1.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {c}\, \left (e x +d \right ) \left (-2 B c \,d^{2}+e \left (A c +\frac {3 B b}{2}\right ) d -\frac {A b \,e^{2}}{2}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )-\left (-2 \left (e x +d \right ) \left (-2 B c d +e \left (A c +\frac {B b}{2}\right )\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+\sqrt {c}\, e \left (-2 B d +e \left (-B x +A \right )\right ) \sqrt {x \left (c x +b \right )}\right ) \sqrt {d \left (b e -c d \right )}}{\sqrt {c}\, e^{3} \left (e x +d \right ) \sqrt {d \left (b e -c d \right )}}\)	\(173\)
risch	\(\frac {B x \left (c x +b \right )}{e^{2} \sqrt {x \left (c x +b \right )}}+\frac {\frac {\left (2 A c e +B b e -4 B c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}-\frac {2 \left (A b \,e^{2}-2 A c d e -2 B b d e +3 B c \,d^{2}\right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {2 d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right ) \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{3}}}{2 e^{2}}\)	\(486\)
default	\(\frac {B \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}+\frac {\left (A e -B d \right ) \left (\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{d \left (b e -c d \right )}\right )}{e^{3}}\)	\(882\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (174) = 348\).

Time = 0.28 (sec) , antiderivative size = 1509, normalized size of antiderivative = 7.78 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*((4*B*c^2*d^4 - (5*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^ 
2 + (4*B*c^2*d^3*e - (5*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 2*A*b*c)*d*e^3 
)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (4*B*c^2*d^3 + 
 A*b*c*d*e^2 - (3*B*b*c + 2*A*c^2)*d^2*e + (4*B*c^2*d^2*e + A*b*c*e^3 - (3 
*B*b*c + 2*A*c^2)*d*e^2)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x 
 + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(2*B*c^2*d^3*e 
+ A*b*c*d*e^3 - (2*B*b*c + A*c^2)*d^2*e^2 + (B*c^2*d^2*e^2 - B*b*c*d*e^3)* 
x)*sqrt(c*x^2 + b*x))/(c^2*d^3*e^3 - b*c*d^2*e^4 + (c^2*d^2*e^4 - b*c*d*e^ 
5)*x), -1/2*(2*(4*B*c^2*d^3 + A*b*c*d*e^2 - (3*B*b*c + 2*A*c^2)*d^2*e + (4 
*B*c^2*d^2*e + A*b*c*e^3 - (3*B*b*c + 2*A*c^2)*d*e^2)*x)*sqrt(-c*d^2 + b*d 
*e)*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) + (4*B*c^ 
2*d^4 - (5*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^2 + (4*B*c^2*d 
^3*e - (5*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 2*A*b*c)*d*e^3)*x)*sqrt(c)*l 
og(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(2*B*c^2*d^3*e + A*b*c*d*e 
^3 - (2*B*b*c + A*c^2)*d^2*e^2 + (B*c^2*d^2*e^2 - B*b*c*d*e^3)*x)*sqrt(c*x 
^2 + b*x))/(c^2*d^3*e^3 - b*c*d^2*e^4 + (c^2*d^2*e^4 - b*c*d*e^5)*x), 1/2* 
(2*(4*B*c^2*d^4 - (5*B*b*c + 2*A*c^2)*d^3*e + (B*b^2 + 2*A*b*c)*d^2*e^2 + 
(4*B*c^2*d^3*e - (5*B*b*c + 2*A*c^2)*d^2*e^2 + (B*b^2 + 2*A*b*c)*d*e^3)*x) 
*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) + (4*B*c^2*d^3 + A* 
b*c*d*e^2 - (3*B*b*c + 2*A*c^2)*d^2*e + (4*B*c^2*d^2*e + A*b*c*e^3 - (3...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 1383, normalized size of antiderivative = 7.13 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) 
- sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*a*b*c*d*e**2 - sqrt(d)*sqrt( 
b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e) 
*sqrt(c))/(sqrt(d)*sqrt(c)))*a*b*c*e**3*x + 2*sqrt(d)*sqrt(b*e - c*d)*atan 
((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt 
(d)*sqrt(c)))*a*c**2*d**2*e + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c 
*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))* 
a*c**2*d*e**2*x + 3*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e 
)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c*d**2* 
e + 3*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x 
) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c*d*e**2*x - 4*sqrt(d 
)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)* 
sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**2*d**3 - 4*sqrt(d)*sqrt(b*e - c*d 
)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c)) 
/(sqrt(d)*sqrt(c)))*b*c**2*d**2*e*x - sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b 
*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt 
(c)))*a*b*c*d*e**2 - sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt( 
e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*a*b*c*e**3* 
x + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x 
) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*a*c**2*d**2*e + 2*sqrt(...