\(\int \frac {(A+B x) \sqrt {a x+b x^2}}{(c+d x)^2} \, dx\) [9]

Optimal result

Integrand size = 26, antiderivative size = 195 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{(c+d x)^2} \, dx=\frac {(2 B c-A d) \sqrt {a x+b x^2}}{c d^2}-\frac {(B c-A d) x \sqrt {a x+b x^2}}{c d (c+d x)}-\frac {(4 b B c-2 A b d-a B d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b} d^3}+\frac {(2 b c (2 B c-A d)-a d (3 B c-A d)) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{\sqrt {c} d^3 \sqrt {b c-a d}} \] Output:

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

Mathematica [A] (verified)

Time = 10.93 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{(c+d x)^2} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {(-B c+A d) x^{3/2} (a+b x)}{c+d x}+\frac {d \sqrt {x} (a d (2 B c-A d)+A b d (c-d x)+b B c (-2 c+d x))+\frac {c (-b c+a d) (-4 b B c+2 A b d+a B d) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {1+\frac {b x}{a}}}-\frac {\sqrt {c} \sqrt {b c-a d} (2 b c (2 B c-A d)+a d (-3 B c+A d)) \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {a+b x}}}{d^3}\right )}{c (-b c+a d) \sqrt {x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 192, 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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

((2*B*c - A*d + B*d*x)*Sqrt[a*x + b*x^2])/(d^2*(c + d*x)) - ((2*(4*b*B*c - 
 2*A*b*d - a*B*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*d) - (( 
2*b*c*(2*B*c - A*d) - a*d*(3*B*c - A*d))*ArcTanh[(a*c + (2*b*c - a*d)*x)/( 
2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/(Sqrt[c]*d*Sqrt[b*c - a*d]) 
)/(2*d^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 = 0.74 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.89

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 372 vs. \(2 (175) = 350\).

Time = 0.30 (sec) , antiderivative size = 1509, normalized size of antiderivative = 7.74 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((B*x+A)*(b*x^2+a*x)^(1/2)/(d*x+c)^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(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.08 \[ \int \frac {(A+B x) \sqrt {a x+b x^2}}{(c+d x)^2} \, dx=\frac {-\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a c d -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,d^{2} x +4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b \,c^{2}+4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c d x -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a c d -\sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,d^{2} x +4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b \,c^{2}+4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b c d x -\sqrt {x}\, \sqrt {b x +a}\, a c \,d^{2}+2 \sqrt {x}\, \sqrt {b x +a}\, b \,c^{2} d +\sqrt {x}\, \sqrt {b x +a}\, b c \,d^{2} x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,c^{2} d +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a c \,d^{2} x -4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b \,c^{3}-4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b \,c^{2} d x}{c \,d^{3} \left (d x +c \right )} \] Input:

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

Output:

( - sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) 
- sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*c*d - sqrt(c)*sqrt(a*d - b 
*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b 
))/(sqrt(c)*sqrt(b)))*a*d**2*x + 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d 
- b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b) 
))*b*c**2 + 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt 
(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b*c*d*x - sqrt(c)* 
sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sq 
rt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*c*d - sqrt(c)*sqrt(a*d - b*c)*atan((sq 
rt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)* 
sqrt(b)))*a*d**2*x + 4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqr 
t(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b*c**2 + 
4*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + 
sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b*c*d*x - sqrt(x)*sqrt(a + b*x 
)*a*c*d**2 + 2*sqrt(x)*sqrt(a + b*x)*b*c**2*d + sqrt(x)*sqrt(a + b*x)*b*c* 
d**2*x + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*c**2*d 
 + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*c*d**2*x - 4 
*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b*c**3 - 4*sqrt(b) 
*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b*c**2*d*x)/(c*d**3*(c + d 
*x))