\(\int \frac {x^2 (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx\) [1]

Optimal result

Integrand size = 29, antiderivative size = 387 \[ \int \frac {x^2 (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\frac {\left (5 a^3 B d^3-64 b^3 c^2 (B c-A d)+16 a b^2 c d (B c-A d)+8 a^2 b d^2 (B c-A d)\right ) \sqrt {a x+b x^2}}{64 b^3 d^4}-\frac {\left (5 a^2 B d^2-48 b^2 c (B c-A d)+8 a b d (B c-A d)\right ) x \sqrt {a x+b x^2}}{96 b^2 d^3}-\frac {(8 b B c-8 A b d-a B d) x^2 \sqrt {a x+b x^2}}{24 b d^2}+\frac {B x^3 \sqrt {a x+b x^2}}{4 d}-\frac {\left (5 a^4 B d^4-128 b^4 c^3 (B c-A d)+64 a b^3 c^2 d (B c-A d)+16 a^2 b^2 c d^2 (B c-A d)+8 a^3 b d^3 (B c-A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{64 b^{7/2} d^5}-\frac {2 c^{5/2} \sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^5} \] Output:

1/64*(5*a^3*B*d^3-64*b^3*c^2*(-A*d+B*c)+16*a*b^2*c*d*(-A*d+B*c)+8*a^2*b*d^ 
2*(-A*d+B*c))*(b*x^2+a*x)^(1/2)/b^3/d^4-1/96*(5*a^2*B*d^2-48*b^2*c*(-A*d+B 
*c)+8*a*b*d*(-A*d+B*c))*x*(b*x^2+a*x)^(1/2)/b^2/d^3-1/24*(-8*A*b*d-B*a*d+8 
*B*b*c)*x^2*(b*x^2+a*x)^(1/2)/b/d^2+1/4*B*x^3*(b*x^2+a*x)^(1/2)/d-1/64*(5* 
a^4*B*d^4-128*b^4*c^3*(-A*d+B*c)+64*a*b^3*c^2*d*(-A*d+B*c)+16*a^2*b^2*c*d^ 
2*(-A*d+B*c)+8*a^3*b*d^3*(-A*d+B*c))*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/ 
b^(7/2)/d^5-2*c^(5/2)*(-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^5
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.86 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.63 \[ \int \frac {x^2 (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\sqrt {b} d \sqrt {x} \sqrt {a+b x} \left (15 a^3 B d^3-2 a^2 b d^2 (-12 B c+12 A d+5 B d x)+8 a b^2 d \left (2 A d (-3 c+d x)+B \left (6 c^2-2 c d x+d^2 x^2\right )\right )-16 b^3 \left (-2 A d \left (6 c^2-3 c d x+2 d^2 x^2\right )+B \left (12 c^3-6 c^2 d x+4 c d^2 x^2-3 d^3 x^3\right )\right )\right )+384 b^{5/2} c^{3/2} (B c-A d) \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+384 b^{5/2} c^{3/2} (B c-A d) \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+6 \left (-5 a^4 B d^4+128 b^4 c^3 (B c-A d)-64 a b^3 c^2 d (B c-A d)-16 a^2 b^2 c d^2 (B c-A d)+8 a^3 b d^3 (-B c+A d)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{192 b^{7/2} d^5 \sqrt {x (a+b x)}} \] Input:

Integrate[(x^2*(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]*(15*a^3*B*d^3 - 2* 
a^2*b*d^2*(-12*B*c + 12*A*d + 5*B*d*x) + 8*a*b^2*d*(2*A*d*(-3*c + d*x) + B 
*(6*c^2 - 2*c*d*x + d^2*x^2)) - 16*b^3*(-2*A*d*(6*c^2 - 3*c*d*x + 2*d^2*x^ 
2) + B*(12*c^3 - 6*c^2*d*x + 4*c*d^2*x^2 - 3*d^3*x^3))) + 384*b^(5/2)*c^(3 
/2)*(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]))] + 384*b^(5/2)*c^(3/2)*(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]*Sqr 
t[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*(-5*a^4*B*d^4 
+ 128*b^4*c^3*(B*c - A*d) - 64*a*b^3*c^2*d*(B*c - A*d) - 16*a^2*b^2*c*d^2* 
(B*c - A*d) + 8*a^3*b*d^3*(-(B*c) + A*d))*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt 
[a] + Sqrt[a + b*x])]))/(192*b^(7/2)*d^5*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.

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

\(\Big \downarrow \) 2153

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 a^4 B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{64 b^{7/2} d}-\frac {a^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{8 b^{5/2} d^2}-\frac {a^2 c \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (B c-A d)}{4 b^{3/2} d^3}+\frac {5 a^2 B (a+2 b x) \sqrt {a x+b x^2}}{64 b^3 d}-\frac {c^{5/2} \sqrt {b c-a d} (B c-A d) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{d^5}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) (B c-A d)}{\sqrt {b} d^5}+\frac {a (a+2 b x) \sqrt {a x+b x^2} (B c-A d)}{8 b^2 d^2}-\frac {c^2 \sqrt {a x+b x^2} (B c-A d)}{d^4}+\frac {c (a+2 b x) \sqrt {a x+b x^2} (B c-A d)}{4 b d^3}-\frac {\left (a x+b x^2\right )^{3/2} (B c-A d)}{3 b d^2}-\frac {5 a B \left (a x+b x^2\right )^{3/2}}{24 b^2 d}+\frac {B x \left (a x+b x^2\right )^{3/2}}{4 b d}\)

Input:

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

Output:

-((c^2*(B*c - A*d)*Sqrt[a*x + b*x^2])/d^4) + (5*a^2*B*(a + 2*b*x)*Sqrt[a*x 
 + b*x^2])/(64*b^3*d) + (c*(B*c - A*d)*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b 
*d^3) + (a*(B*c - A*d)*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(8*b^2*d^2) - (5*a*B 
*(a*x + b*x^2)^(3/2))/(24*b^2*d) - ((B*c - A*d)*(a*x + b*x^2)^(3/2))/(3*b* 
d^2) + (B*x*(a*x + b*x^2)^(3/2))/(4*b*d) - (5*a^4*B*ArcTanh[(Sqrt[b]*x)/Sq 
rt[a*x + b*x^2]])/(64*b^(7/2)*d) - (a^2*c*(B*c - A*d)*ArcTanh[(Sqrt[b]*x)/ 
Sqrt[a*x + b*x^2]])/(4*b^(3/2)*d^3) - (a^3*(B*c - A*d)*ArcTanh[(Sqrt[b]*x) 
/Sqrt[a*x + b*x^2]])/(8*b^(5/2)*d^2) + (c^2*(2*b*c - a*d)*(B*c - A*d)*ArcT 
anh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*d^5) - (c^(5/2)*Sqrt[b*c - a* 
d]*(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])])/d^5
 

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 = 0.75 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(-\frac {-16 c^{3} \left (b^{\frac {7}{2}} a d -b^{\frac {9}{2}} c \right ) \left (A d -B c \right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+\left (\left (\left (\frac {5}{8} a^{4} B -A \,a^{3} b \right ) d^{4}+\left (-2 a^{2} A \,b^{2}+B \,a^{3} b \right ) c \,d^{3}+\left (-8 A a \,b^{3}+2 B \,a^{2} b^{2}\right ) c^{2} d^{2}+\left (16 A \,b^{4}+8 B a \,b^{3}\right ) c^{3} d -16 B \,b^{4} c^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )+\left (\left (\left (-2 B \,x^{3}-\frac {8}{3} A \,x^{2}\right ) d^{3}+4 c x \left (\frac {2 B x}{3}+A \right ) d^{2}-8 c^{2} \left (\frac {B x}{2}+A \right ) d +8 B \,c^{3}\right ) b^{\frac {7}{2}}+d \left (\left (-\frac {2 x \left (\frac {B x}{2}+A \right ) d^{2}}{3}+2 \left (\frac {B x}{3}+A \right ) c d -2 B \,c^{2}\right ) b^{\frac {5}{2}}+d \left (\left (\left (\frac {5 B x}{12}+A \right ) d -B c \right ) b^{\frac {3}{2}}-\frac {5 B a d \sqrt {b}}{8}\right ) a \right ) a \right ) d \sqrt {x \left (b x +a \right )}\right ) \sqrt {c \left (a d -b c \right )}}{8 b^{\frac {7}{2}} \sqrt {c \left (a d -b c \right )}\, d^{5}}\)	\(330\)
risch	\(-\frac {\left (-48 B \,b^{3} d^{3} x^{3}-64 A \,b^{3} d^{3} x^{2}-8 B a \,b^{2} d^{3} x^{2}+64 B \,b^{3} c \,d^{2} x^{2}-16 A a \,b^{2} d^{3} x +96 A \,b^{3} c \,d^{2} x +10 B \,a^{2} b \,d^{3} x +16 B a \,b^{2} c \,d^{2} x -96 B \,b^{3} c^{2} d x +24 A \,a^{2} b \,d^{3}+48 A a \,b^{2} c \,d^{2}-192 A \,b^{3} c^{2} d -15 a^{3} B \,d^{3}-24 B \,a^{2} b c \,d^{2}-48 B a \,b^{2} c^{2} d +192 B \,b^{3} c^{3}\right ) x \left (b x +a \right )}{192 b^{3} d^{4} \sqrt {x \left (b x +a \right )}}+\frac {\frac {\left (8 A \,a^{3} b \,d^{4}+16 A \,a^{2} b^{2} c \,d^{3}+64 A a \,b^{3} c^{2} d^{2}-128 A \,b^{4} c^{3} d -5 a^{4} B \,d^{4}-8 B \,a^{3} b c \,d^{3}-16 B \,a^{2} b^{2} c^{2} d^{2}-64 B a \,b^{3} c^{3} d +128 B \,b^{4} c^{4}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {128 c^{3} \left (A a \,d^{2}-A b c d -B a c d +B b \,c^{2}\right ) b^{3} \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{128 b^{3} d^{4}}\)	\(507\)
default	\(\frac {c^{2} \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^{4}}-\frac {-d \left (A d -B c \right ) \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 )+A c 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 \,c^{2} \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^{2} \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \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 )}{8 b}\right )}{d^{3}}\)	\(619\)

Input:

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

Output:

-1/8/b^(7/2)*(-16*c^3*(b^(7/2)*a*d-b^(9/2)*c)*(A*d-B*c)*arctan((x*(b*x+a)) 
^(1/2)/x*c/(c*(a*d-b*c))^(1/2))+(((5/8*a^4*B-A*a^3*b)*d^4+(-2*A*a^2*b^2+B* 
a^3*b)*c*d^3+(-8*A*a*b^3+2*B*a^2*b^2)*c^2*d^2+(16*A*b^4+8*B*a*b^3)*c^3*d-1 
6*B*b^4*c^4)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))+(((-2*B*x^3-8/3*A*x^2)*d 
^3+4*c*x*(2/3*B*x+A)*d^2-8*c^2*(1/2*B*x+A)*d+8*B*c^3)*b^(7/2)+d*((-2/3*x*( 
1/2*B*x+A)*d^2+2*(1/3*B*x+A)*c*d-2*B*c^2)*b^(5/2)+d*(((5/12*B*x+A)*d-B*c)* 
b^(3/2)-5/8*B*a*d*b^(1/2))*a)*a)*d*(x*(b*x+a))^(1/2))*(c*(a*d-b*c))^(1/2)) 
/(c*(a*d-b*c))^(1/2)/d^5
 

Fricas [A] (verification not implemented)

Time = 8.57 (sec) , antiderivative size = 1608, normalized size of antiderivative = 4.16 \[ \int \frac {x^2 (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/384*(3*(128*B*b^4*c^4 - 64*(B*a*b^3 + 2*A*b^4)*c^3*d - 16*(B*a^2*b^2 - 
4*A*a*b^3)*c^2*d^2 - 8*(B*a^3*b - 2*A*a^2*b^2)*c*d^3 - (5*B*a^4 - 8*A*a^3* 
b)*d^4)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 384*(B*b^4* 
c^3 - A*b^4*c^2*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*(48*B*b^4*d^4*x^3 - 192* 
B*b^4*c^3*d + 48*(B*a*b^3 + 4*A*b^4)*c^2*d^2 + 24*(B*a^2*b^2 - 2*A*a*b^3)* 
c*d^3 + 3*(5*B*a^3*b - 8*A*a^2*b^2)*d^4 - 8*(8*B*b^4*c*d^3 - (B*a*b^3 + 8* 
A*b^4)*d^4)*x^2 + 2*(48*B*b^4*c^2*d^2 - 8*(B*a*b^3 + 6*A*b^4)*c*d^3 - (5*B 
*a^2*b^2 - 8*A*a*b^3)*d^4)*x)*sqrt(b*x^2 + a*x))/(b^4*d^5), 1/384*(768*(B* 
b^4*c^3 - A*b^4*c^2*d)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d)*sq 
rt(b*x^2 + a*x)/(b*c*x + a*c)) + 3*(128*B*b^4*c^4 - 64*(B*a*b^3 + 2*A*b^4) 
*c^3*d - 16*(B*a^2*b^2 - 4*A*a*b^3)*c^2*d^2 - 8*(B*a^3*b - 2*A*a^2*b^2)*c* 
d^3 - (5*B*a^4 - 8*A*a^3*b)*d^4)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a* 
x)*sqrt(b)) + 2*(48*B*b^4*d^4*x^3 - 192*B*b^4*c^3*d + 48*(B*a*b^3 + 4*A*b^ 
4)*c^2*d^2 + 24*(B*a^2*b^2 - 2*A*a*b^3)*c*d^3 + 3*(5*B*a^3*b - 8*A*a^2*b^2 
)*d^4 - 8*(8*B*b^4*c*d^3 - (B*a*b^3 + 8*A*b^4)*d^4)*x^2 + 2*(48*B*b^4*c^2* 
d^2 - 8*(B*a*b^3 + 6*A*b^4)*c*d^3 - (5*B*a^2*b^2 - 8*A*a*b^3)*d^4)*x)*sqrt 
(b*x^2 + a*x))/(b^4*d^5), -1/192*(3*(128*B*b^4*c^4 - 64*(B*a*b^3 + 2*A*b^4 
)*c^3*d - 16*(B*a^2*b^2 - 4*A*a*b^3)*c^2*d^2 - 8*(B*a^3*b - 2*A*a^2*b^2)*c 
*d^3 - (5*B*a^4 - 8*A*a^3*b)*d^4)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqr...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 10.39 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.50 \[ \int \frac {x^2 (A+B x) \sqrt {a x+b x^2}}{c+d x} \, dx=\frac {384 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{3} c^{2} d -384 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{4} c^{3}+384 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{3} c^{2} d -384 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{4} c^{3}-9 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b \,d^{4}-24 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} c \,d^{3}+6 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{2} d^{4} x +240 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} c^{2} d^{2}-112 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} c \,d^{3} x +72 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{3} d^{4} x^{2}-192 \sqrt {x}\, \sqrt {b x +a}\, b^{4} c^{3} d +96 \sqrt {x}\, \sqrt {b x +a}\, b^{4} c^{2} d^{2} x -64 \sqrt {x}\, \sqrt {b x +a}\, b^{4} c \,d^{3} x^{2}+48 \sqrt {x}\, \sqrt {b x +a}\, b^{4} d^{4} x^{3}+9 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{4} d^{4}+24 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} b c \,d^{3}+144 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b^{2} c^{2} d^{2}-576 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,b^{3} c^{3} d +384 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{4} c^{4}}{192 b^{3} d^{5}} \] Input:

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

Output:

(384*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**3*c**2*d - 384*sqrt(c) 
*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*s 
qrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**4*c**3 + 384*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**3*c**2*d - 384*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)*sqr 
t(b)))*b**4*c**3 - 9*sqrt(x)*sqrt(a + b*x)*a**3*b*d**4 - 24*sqrt(x)*sqrt(a 
 + b*x)*a**2*b**2*c*d**3 + 6*sqrt(x)*sqrt(a + b*x)*a**2*b**2*d**4*x + 240* 
sqrt(x)*sqrt(a + b*x)*a*b**3*c**2*d**2 - 112*sqrt(x)*sqrt(a + b*x)*a*b**3* 
c*d**3*x + 72*sqrt(x)*sqrt(a + b*x)*a*b**3*d**4*x**2 - 192*sqrt(x)*sqrt(a 
+ b*x)*b**4*c**3*d + 96*sqrt(x)*sqrt(a + b*x)*b**4*c**2*d**2*x - 64*sqrt(x 
)*sqrt(a + b*x)*b**4*c*d**3*x**2 + 48*sqrt(x)*sqrt(a + b*x)*b**4*d**4*x**3 
 + 9*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**4*d**4 + 24 
*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**3*b*c*d**3 + 14 
4*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*b**2*c**2*d* 
*2 - 576*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b**3*c** 
3*d + 384*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b**4*c**4 
)/(192*b**3*d**5)