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

Optimal result

Integrand size = 27, antiderivative size = 278 \[ \int \frac {x (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=-\frac {\left (a^2 B d^2-8 b^2 c (B c-A d)+2 a b d (B c-A d)\right ) \sqrt {a x+b x^2}}{8 b^2 d^3}-\frac {(6 b B c-6 A b d-a B d) x \sqrt {a x+b x^2}}{12 b d^2}+\frac {B x^2 \sqrt {a x+b x^2}}{3 d}+\frac {\left (a^3 B d^3-16 b^3 c^2 (B c-A d)+8 a b^2 c d (B c-A d)+2 a^2 b d^2 (B c-A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{8 b^{5/2} d^4}+\frac {2 c^{3/2} \sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^4} \] Output:

-1/8*(a^2*B*d^2-8*b^2*c*(-A*d+B*c)+2*a*b*d*(-A*d+B*c))*(b*x^2+a*x)^(1/2)/b 
^2/d^3-1/12*(-6*A*b*d-B*a*d+6*B*b*c)*x*(b*x^2+a*x)^(1/2)/b/d^2+1/3*B*x^2*( 
b*x^2+a*x)^(1/2)/d+1/8*(a^3*B*d^3-16*b^3*c^2*(-A*d+B*c)+8*a*b^2*c*d*(-A*d+ 
B*c)+2*a^2*b*d^2*(-A*d+B*c))*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(5/2)/ 
d^4+2*c^(3/2)*(-a*d+b*c)^(1/2)*(-A*d+B*c)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/ 
2)/(b*x^2+a*x)^(1/2))/d^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.77 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.95 \[ \int \frac {x (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\sqrt {b} d \sqrt {x} \sqrt {a+b x} \left (-3 a^2 B d^2+2 a b d (-3 B c+3 A d+B d x)+4 b^2 \left (3 A d (-2 c+d x)+B \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )\right )+48 b^{3/2} \sqrt {c} (B c-A d) \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )+48 b^{3/2} \sqrt {c} (B c-A d) \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )-6 \left (-a^3 B d^3+16 b^3 c^2 (B c-A d)+8 a b^2 c d (-B c+A d)+2 a^2 b d^2 (-B c+A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{24 b^{5/2} d^4 \sqrt {x (a+b x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*(Sqrt[b]*d*Sqrt[x]*Sqrt[a + b*x]*(-3*a^2*B*d^2 + 2* 
a*b*d*(-3*B*c + 3*A*d + B*d*x) + 4*b^2*(3*A*d*(-2*c + d*x) + B*(6*c^2 - 3* 
c*d*x + 2*d^2*x^2))) + 48*b^(3/2)*Sqrt[c]*(B*c - A*d)*(b*c - a*d - I*Sqrt[ 
a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sq 
rt[b*c - a*d]]*ArcTan[(Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b* 
c - a*d]]*Sqrt[x])/(Sqrt[c]*(Sqrt[a] - Sqrt[a + b*x]))] + 48*b^(3/2)*Sqrt[ 
c]*(B*c - A*d)*(b*c - a*d + I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) 
 + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[-(b*c) + 2* 
a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(Sqrt[a] - 
Sqrt[a + b*x]))] - 6*(-(a^3*B*d^3) + 16*b^3*c^2*(B*c - A*d) + 8*a*b^2*c*d* 
(-(B*c) + A*d) + 2*a^2*b*d^2*(-(B*c) + A*d))*ArcTanh[(Sqrt[b]*Sqrt[x])/(-S 
qrt[a] + Sqrt[a + b*x])]))/(24*b^(5/2)*d^4*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2153, 2009}

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.

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{8 b^{5/2} d}+\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{4 b^{3/2} d^2}+\frac {c^{3/2} \sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{d^4}-\frac {c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) (B c-A d)}{\sqrt {b} d^4}+\frac {c \sqrt {a x+b x^2} (B c-A d)}{d^3}-\frac {(a+2 b x) \sqrt {a x+b x^2} (B c-A d)}{4 b d^2}-\frac {a B (a+2 b x) \sqrt {a x+b x^2}}{8 b^2 d}+\frac {B \left (a x+b x^2\right )^{3/2}}{3 b d}\)

Input:

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

Output:

(c*(B*c - A*d)*Sqrt[a*x + b*x^2])/d^3 - (a*B*(a + 2*b*x)*Sqrt[a*x + b*x^2] 
)/(8*b^2*d) - ((B*c - A*d)*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b*d^2) + (B*( 
a*x + b*x^2)^(3/2))/(3*b*d) + (a^3*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2] 
])/(8*b^(5/2)*d) + (a^2*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]] 
)/(4*b^(3/2)*d^2) - (c*(2*b*c - a*d)*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/Sqrt[ 
a*x + b*x^2]])/(Sqrt[b]*d^4) + (c^(3/2)*Sqrt[b*c - a*d]*(B*c - A*d)*ArcTan 
h[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/ 
d^4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2153

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e* 
x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m, n, p}, x] && PolyQ[Px, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ 
[m] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] && IGtQ[n, 0])
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(-\frac {-\frac {d \sqrt {x \left (b x +a \right )}\, \left (8 B \,b^{2} d^{2} x^{2}+12 A \,b^{2} d^{2} x +2 B a b \,d^{2} x -12 B \,b^{2} c d x +6 A a b \,d^{2}-24 A \,b^{2} c d -3 a^{2} B \,d^{2}-6 a b B c d +24 B \,b^{2} c^{2}\right )}{24 b^{2}}+\frac {\left (2 A \,a^{2} b \,d^{3}+8 A a \,b^{2} c \,d^{2}-16 A \,b^{3} c^{2} d -a^{3} B \,d^{3}-2 B \,a^{2} b c \,d^{2}-8 B a \,b^{2} c^{2} d +16 B \,b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{8 b^{\frac {5}{2}}}+\frac {2 \left (a d -b c \right ) \left (A d -B c \right ) c^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{d^{4}}\)	\(258\)
risch	\(\frac {\left (8 B \,b^{2} d^{2} x^{2}+12 A \,b^{2} d^{2} x +2 B a b \,d^{2} x -12 B \,b^{2} c d x +6 A a b \,d^{2}-24 A \,b^{2} c d -3 a^{2} B \,d^{2}-6 a b B c d +24 B \,b^{2} c^{2}\right ) x \left (b x +a \right )}{24 b^{2} d^{3} \sqrt {x \left (b x +a \right )}}-\frac {\frac {\left (2 A \,a^{2} b \,d^{3}+8 A a \,b^{2} c \,d^{2}-16 A \,b^{3} c^{2} d -a^{3} B \,d^{3}-2 B \,a^{2} b c \,d^{2}-8 B a \,b^{2} c^{2} d +16 B \,b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {16 c^{2} \left (A a \,d^{2}-A b c d -B a c d +B b \,c^{2}\right ) b^{2} \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{16 d^{3} b^{2}}\)	\(381\)
default	\(\frac {A d \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )+B d \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )-B c \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{d^{2}}-\frac {c \left (A d -B c \right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}\)	\(498\)

Input:

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

Output:

-1/d^4*(-1/24*d*(x*(b*x+a))^(1/2)*(8*B*b^2*d^2*x^2+12*A*b^2*d^2*x+2*B*a*b* 
d^2*x-12*B*b^2*c*d*x+6*A*a*b*d^2-24*A*b^2*c*d-3*B*a^2*d^2-6*B*a*b*c*d+24*B 
*b^2*c^2)/b^2+1/8*(2*A*a^2*b*d^3+8*A*a*b^2*c*d^2-16*A*b^3*c^2*d-B*a^3*d^3- 
2*B*a^2*b*c*d^2-8*B*a*b^2*c^2*d+16*B*b^3*c^3)/b^(5/2)*arctanh((x*(b*x+a))^ 
(1/2)/x/b^(1/2))+2*(a*d-b*c)*(A*d-B*c)*c^2/(c*(a*d-b*c))^(1/2)*arctan((x*( 
b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 2.50 (sec) , antiderivative size = 1148, normalized size of antiderivative = 4.13 \[ \int \frac {x (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/48*(3*(16*B*b^3*c^3 - 8*(B*a*b^2 + 2*A*b^3)*c^2*d - 2*(B*a^2*b - 4*A*a* 
b^2)*c*d^2 - (B*a^3 - 2*A*a^2*b)*d^3)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 
 + a*x)*sqrt(b)) - 48*(B*b^3*c^2 - A*b^3*c*d)*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*(8*B*b^3*d^3*x^2 + 24*B*b^3*c^2*d - 6*(B*a*b^2 + 4*A*b^3)*c*d^2 - 3*(B* 
a^2*b - 2*A*a*b^2)*d^3 - 2*(6*B*b^3*c*d^2 - (B*a*b^2 + 6*A*b^3)*d^3)*x)*sq 
rt(b*x^2 + a*x))/(b^3*d^4), -1/48*(96*(B*b^3*c^2 - A*b^3*c*d)*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)) - 3* 
(16*B*b^3*c^3 - 8*(B*a*b^2 + 2*A*b^3)*c^2*d - 2*(B*a^2*b - 4*A*a*b^2)*c*d^ 
2 - (B*a^3 - 2*A*a^2*b)*d^3)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*s 
qrt(b)) - 2*(8*B*b^3*d^3*x^2 + 24*B*b^3*c^2*d - 6*(B*a*b^2 + 4*A*b^3)*c*d^ 
2 - 3*(B*a^2*b - 2*A*a*b^2)*d^3 - 2*(6*B*b^3*c*d^2 - (B*a*b^2 + 6*A*b^3)*d 
^3)*x)*sqrt(b*x^2 + a*x))/(b^3*d^4), 1/24*(3*(16*B*b^3*c^3 - 8*(B*a*b^2 + 
2*A*b^3)*c^2*d - 2*(B*a^2*b - 4*A*a*b^2)*c*d^2 - (B*a^3 - 2*A*a^2*b)*d^3)* 
sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - 24*(B*b^3*c^2 - A* 
b^3*c*d)*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)) + (8*B*b^3*d^3*x^2 + 24*B*b^3*c^2*d - 
6*(B*a*b^2 + 4*A*b^3)*c*d^2 - 3*(B*a^2*b - 2*A*a*b^2)*d^3 - 2*(6*B*b^3*c*d 
^2 - (B*a*b^2 + 6*A*b^3)*d^3)*x)*sqrt(b*x^2 + a*x))/(b^3*d^4), -1/24*(48*( 
B*b^3*c^2 - A*b^3*c*d)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d)...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.64 \[ \int \frac {x (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\frac {-48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c d +48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}-48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c d +48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}+3 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,d^{3}-30 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c \,d^{2}+14 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} d^{3} x +24 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{2} d -12 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c \,d^{2} x +8 \sqrt {x}\, \sqrt {b x +a}\, b^{3} d^{3} x^{2}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{3}-18 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b c \,d^{2}+72 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,b^{2} c^{2} d -48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{3} c^{3}}{24 b^{2} d^{4}} \] Input:

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

Output:

( - 48*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*b**2*c*d + 48*sqrt(c)*s 
qrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqr 
t(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**3*c**2 - 48*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sq 
rt(c)*sqrt(b)))*a*b**2*c*d + 48*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**3*c**2 + 3*sqrt(x)*sqrt(a + b*x)*a**2*b*d**3 - 30*sqrt(x)*sqrt(a + b*x) 
*a*b**2*c*d**2 + 14*sqrt(x)*sqrt(a + b*x)*a*b**2*d**3*x + 24*sqrt(x)*sqrt( 
a + b*x)*b**3*c**2*d - 12*sqrt(x)*sqrt(a + b*x)*b**3*c*d**2*x + 8*sqrt(x)* 
sqrt(a + b*x)*b**3*d**3*x**2 - 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt 
(b))/sqrt(a))*a**3*d**3 - 18*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*a**2*b*c*d**2 + 72*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*a*b**2*c**2*d - 48*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*b**3*c**3)/(24*b**2*d**4)