\(\int \frac {(d+e x)^2}{(f+g x) (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [286]

Optimal result

Integrand size = 44, antiderivative size = 139 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)}{(c d f-a e g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {c d f-a e g} (d+e x)}{\sqrt {e f-d g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{(c d f-a e g)^{3/2}} \] Output:

(-2*e*x-2*d)/(-a*e*g+c*d*f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+2*(-d* 
g+e*f)^(1/2)*arctanh((-a*e*g+c*d*f)^(1/2)*(e*x+d)/(-d*g+e*f)^(1/2)/(a*d*e+ 
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(-a*e*g+c*d*f)^(3/2)
 

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {c d f-a e g} (d+e x)+2 \sqrt {-e f+d g} \sqrt {a e+c d x} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {-e f+d g} \sqrt {a e+c d x}}\right )}{(c d f-a e g)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)^2/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^( 
3/2)),x]
 

Output:

(-2*Sqrt[c*d*f - a*e*g]*(d + e*x) + 2*Sqrt[-(e*f) + d*g]*Sqrt[a*e + c*d*x] 
*Sqrt[d + e*x]*ArcTan[(Sqrt[c*d*f - a*e*g]*Sqrt[d + e*x])/(Sqrt[-(e*f) + d 
*g]*Sqrt[a*e + c*d*x])])/((c*d*f - a*e*g)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e* 
x)])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.27, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1264, 27, 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.

\(\displaystyle \int \frac {(d+e x)^2}{(f+g x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1264

\(\displaystyle -\frac {2 \int -\frac {\left (c d^2-a e^2\right )^2 (e f-d g)}{2 (c d f-a e g) (f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{\left (c d^2-a e^2\right )^2}-\frac {2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(e f-d g) \int \frac {1}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{c d f-a e g}-\frac {2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {2 (e f-d g) \int \frac {1}{4 (e f-d g) (c d f-a e g)-\frac {\left (c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\left (-\frac {c f d^2+a e (e f-2 d g)-\left (a e^2 g-c d (2 e f-d g)\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}\right )}{c d f-a e g}-\frac {2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {e f-d g} \text {arctanh}\left (\frac {-x \left (a e^2 g-c d (2 e f-d g)\right )+a e (e f-2 d g)+c d^2 f}{2 \sqrt {e f-d g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{3/2}}-\frac {2 (d+e x)}{\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}\)

Input:

Int[(d + e*x)^2/((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)), 
x]
 

Output:

(-2*(d + e*x))/((c*d*f - a*e*g)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
]) + (Sqrt[e*f - d*g]*ArcTanh[(c*d^2*f + a*e*(e*f - 2*d*g) - (a*e^2*g - c* 
d*(2*e*f - d*g))*x)/(2*Sqrt[e*f - d*g]*Sqrt[c*d*f - a*e*g]*Sqrt[a*d*e + (c 
*d^2 + a*e^2)*x + c*d*e*x^2])])/(c*d*f - a*e*g)^(3/2)
 

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 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 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 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(952\) vs. \(2(129)=258\).

Time = 2.07 (sec) , antiderivative size = 953, normalized size of antiderivative = 6.86

method	result	size
default	\(\frac {e \left (e g \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )+\frac {4 d g \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {2 e f \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{g^{2}}+\frac {\left (d^{2} g^{2}-2 d e f g +e^{2} f^{2}\right ) \left (\frac {g^{2}}{\left (a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}\right ) \sqrt {c d \left (x +\frac {f}{g}\right )^{2} e +\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}}-\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) g \left (2 d e c \left (x +\frac {f}{g}\right )+\frac {a \,e^{2} g +c \,d^{2} g -2 c d e f}{g}\right )}{\left (a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}\right ) \left (\frac {4 d e c \left (a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}\right )}{g^{2}}-\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right )^{2}}{g^{2}}\right ) \sqrt {c d \left (x +\frac {f}{g}\right )^{2} e +\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}}-\frac {g^{2} \ln \left (\frac {\frac {2 a d e \,g^{2}-2 a \,e^{2} f g -2 c \,d^{2} f g +2 c d e \,f^{2}}{g^{2}}+\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}\, \sqrt {c d \left (x +\frac {f}{g}\right )^{2} e +\frac {\left (a \,e^{2} g +c \,d^{2} g -2 c d e f \right ) \left (x +\frac {f}{g}\right )}{g}+\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}}{x +\frac {f}{g}}\right )}{\left (a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}\right ) \sqrt {\frac {a d e \,g^{2}-a \,e^{2} f g -c \,d^{2} f g +c d e \,f^{2}}{g^{2}}}}\right )}{g^{3}}\)	\(953\)

Input:

int((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RE 
TURNVERBOSE)
 

Output:

e/g^2*(e*g*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2) 
/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e 
^2+c*d^2)*x+c*d*x^2*e)^(1/2))+4*d*g*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2 
-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-2*e*f*(2*c*d*e*x 
+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x 
^2*e)^(1/2))+(d^2*g^2-2*d*e*f*g+e^2*f^2)/g^3*(1/(a*d*e*g^2-a*e^2*f*g-c*d^2 
*f*g+c*d*e*f^2)*g^2/(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g) 
+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2)-(a*e^2*g+c*d^2*g-2*c 
*d*e*f)*g/(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)*(2*d*e*c*(x+f/g)+(a*e^ 
2*g+c*d^2*g-2*c*d*e*f)/g)/(4*d*e*c*(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^ 
2)/g^2-(a*e^2*g+c*d^2*g-2*c*d*e*f)^2/g^2)/(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2* 
g-2*c*d*e*f)/g*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2 
)-1/(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)*g^2/((a*d*e*g^2-a*e^2*f*g-c* 
d^2*f*g+c*d*e*f^2)/g^2)^(1/2)*ln((2*(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f 
^2)/g^2+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g*(x+f/g)+2*((a*d*e*g^2-a*e^2*f*g-c*d^ 
2*f*g+c*d*e*f^2)/g^2)^(1/2)*(c*d*(x+f/g)^2*e+(a*e^2*g+c*d^2*g-2*c*d*e*f)/g 
*(x+f/g)+(a*d*e*g^2-a*e^2*f*g-c*d^2*f*g+c*d*e*f^2)/g^2)^(1/2))/(x+f/g)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (127) = 254\).

Time = 1.34 (sec) , antiderivative size = 758, normalized size of antiderivative = 5.45 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {{\left (c d x + a e\right )} \sqrt {\frac {e f - d g}{c d f - a e g}} \log \left (\frac {8 \, a^{2} d^{2} e^{2} g^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} f^{2} - 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} f g + {\left (8 \, c^{2} d^{2} e^{2} f^{2} - 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} f g + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} g^{2}\right )} x^{2} + 2 \, {\left (4 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} f^{2} - {\left (3 \, c^{2} d^{4} + 10 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} f g + 4 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} g^{2}\right )} x - 4 \, {\left (2 \, a^{2} d e^{2} g^{2} + {\left (c^{2} d^{3} + a c d e^{2}\right )} f^{2} - {\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} f g + {\left (2 \, c^{2} d^{2} e f^{2} - {\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e f - d g}{c d f - a e g}}}{g^{2} x^{2} + 2 \, f g x + f^{2}}\right ) + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (a c d e f - a^{2} e^{2} g + {\left (c^{2} d^{2} f - a c d e g\right )} x\right )}}, -\frac {{\left (c d x + a e\right )} \sqrt {-\frac {e f - d g}{c d f - a e g}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e g - {\left (c d^{2} + a e^{2}\right )} f - {\left (2 \, c d e f - {\left (c d^{2} + a e^{2}\right )} g\right )} x\right )} \sqrt {-\frac {e f - d g}{c d f - a e g}}}{2 \, {\left (a d e^{2} f - a d^{2} e g + {\left (c d e^{2} f - c d^{2} e g\right )} x^{2} + {\left ({\left (c d^{2} e + a e^{3}\right )} f - {\left (c d^{3} + a d e^{2}\right )} g\right )} x\right )}}\right ) + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{a c d e f - a^{2} e^{2} g + {\left (c^{2} d^{2} f - a c d e g\right )} x}\right ] \] Input:

integrate((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, alg 
orithm="fricas")
 

Output:

[-1/2*((c*d*x + a*e)*sqrt((e*f - d*g)/(c*d*f - a*e*g))*log((8*a^2*d^2*e^2* 
g^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*f^2 - 8*(a*c*d^3*e + a^2*d*e^3)* 
f*g + (8*c^2*d^2*e^2*f^2 - 8*(c^2*d^3*e + a*c*d*e^3)*f*g + (c^2*d^4 + 6*a* 
c*d^2*e^2 + a^2*e^4)*g^2)*x^2 + 2*(4*(c^2*d^3*e + a*c*d*e^3)*f^2 - (3*c^2* 
d^4 + 10*a*c*d^2*e^2 + 3*a^2*e^4)*f*g + 4*(a*c*d^3*e + a^2*d*e^3)*g^2)*x - 
 4*(2*a^2*d*e^2*g^2 + (c^2*d^3 + a*c*d*e^2)*f^2 - (3*a*c*d^2*e + a^2*e^3)* 
f*g + (2*c^2*d^2*e*f^2 - (c^2*d^3 + 3*a*c*d*e^2)*f*g + (a*c*d^2*e + a^2*e^ 
3)*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt((e*f - d*g)/(c 
*d*f - a*e*g)))/(g^2*x^2 + 2*f*g*x + f^2)) + 4*sqrt(c*d*e*x^2 + a*d*e + (c 
*d^2 + a*e^2)*x))/(a*c*d*e*f - a^2*e^2*g + (c^2*d^2*f - a*c*d*e*g)*x), -(( 
c*d*x + a*e)*sqrt(-(e*f - d*g)/(c*d*f - a*e*g))*arctan(-1/2*sqrt(c*d*e*x^2 
 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e*g - (c*d^2 + a*e^2)*f - (2*c*d*e*f 
- (c*d^2 + a*e^2)*g)*x)*sqrt(-(e*f - d*g)/(c*d*f - a*e*g))/(a*d*e^2*f - a* 
d^2*e*g + (c*d*e^2*f - c*d^2*e*g)*x^2 + ((c*d^2*e + a*e^3)*f - (c*d^3 + a* 
d*e^2)*g)*x)) + 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a*c*d*e*f 
- a^2*e^2*g + (c^2*d^2*f - a*c*d*e*g)*x)]
 

Sympy [F]

\[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \] Input:

integrate((e*x+d)**2/(g*x+f)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x 
)
 

Output:

Integral((d + e*x)**2/(((d + e*x)*(a*e + c*d*x))**(3/2)*(f + g*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, alg 
orithm="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*e^2)/g>0)', see `assume?` fo 
r more det
 

Giac [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, alg 
orithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{-1,[1,1,5]%%%},[2,1,3,0]%%%}+%%%{%%%{2,[2,3,3]%%%},[2, 
1,2,0]%%%
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\left (f+g\,x\right )\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int((d + e*x)^2/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), 
x)
 

Output:

int((d + e*x)^2/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), 
 x)
 

Reduce [B] (verification not implemented)

Time = 2.39 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.15 \[ \int \frac {(d+e x)^2}{(f+g x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {\sqrt {d g -e f}\, \sqrt {c d x +a e}\, \sqrt {a e g -c d f}\, \mathrm {log}\left (\sqrt {g}\, \sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {d g -e f}\, \sqrt {a e g -c d f}+a \,e^{2} g +c \,d^{2} g -2 c d e f}+\sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c d +\sqrt {d g -e f}\, \sqrt {c d x +a e}\, \sqrt {a e g -c d f}\, \mathrm {log}\left (\sqrt {g}\, \sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {d g -e f}\, \sqrt {a e g -c d f}+a \,e^{2} g +c \,d^{2} g -2 c d e f}+\sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c d -\sqrt {d g -e f}\, \sqrt {c d x +a e}\, \sqrt {a e g -c d f}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {d g -e f}\, \sqrt {a e g -c d f}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}\, g +2 c d e f +2 c d e g x \right ) c d +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a e g -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c d f +2 \sqrt {e x +d}\, a c d e g -2 \sqrt {e x +d}\, c^{2} d^{2} f}{\sqrt {c d x +a e}\, c d \left (a^{2} e^{2} g^{2}-2 a c d e f g +c^{2} d^{2} f^{2}\right )} \] Input:

int((e*x+d)^2/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

(sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f)*log(sqrt(g)*sqrt(e) 
*sqrt(a*e + c*d*x) - sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt(a 
*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sqrt(g)*sqrt(d)*sqrt(c) 
*sqrt(d + e*x))*c*d + sqrt(d*g - e*f)*sqrt(a*e + c*d*x)*sqrt(a*e*g - c*d*f 
)*log(sqrt(g)*sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqrt(e)*sqrt(d)*sqrt(c)*s 
qrt(d*g - e*f)*sqrt(a*e*g - c*d*f) + a*e**2*g + c*d**2*g - 2*c*d*e*f) + sq 
rt(g)*sqrt(d)*sqrt(c)*sqrt(d + e*x))*c*d - sqrt(d*g - e*f)*sqrt(a*e + c*d* 
x)*sqrt(a*e*g - c*d*f)*log(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d*g - e*f)*sqrt( 
a*e*g - c*d*f) + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*g + 2*c*d*e*f + 2*c*d*e*g*x)*c*d + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c 
*d*x)*a*e*g - 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c*d*f + 2*sqrt(d 
 + e*x)*a*c*d*e*g - 2*sqrt(d + e*x)*c**2*d**2*f)/(sqrt(a*e + c*d*x)*c*d*(a 
**2*e**2*g**2 - 2*a*c*d*e*f*g + c**2*d**2*f**2))