\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx\) [67]

Optimal result

Integrand size = 48, antiderivative size = 214 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx=-\frac {3 (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^2 \sqrt {d+e x}}+\frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}+\frac {3 (c d f-a e g)^2 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}\right )}{4 \sqrt {c} \sqrt {d} g^{5/2}} \] Output:

-3/4*(-a*e*g+c*d*f)*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ 
g^2/(e*x+d)^(1/2)+1/2*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2 
)/g/(e*x+d)^(3/2)+3/4*(-a*e*g+c*d*f)^2*arctanh(g^(1/2)*(a*d*e+(a*e^2+c*d^2 
)*x+c*d*e*x^2)^(1/2)/c^(1/2)/d^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(1/2))/c^(1/2)/ 
d^(1/2)/g^(5/2)
 

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {a e+c d x} \sqrt {f+g x} (5 a e g+c d (-3 f+2 g x))+3 (c d f-a e g)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{4 \sqrt {c} \sqrt {d} g^{5/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[a*e + c*d*x 
]*Sqrt[f + g*x]*(5*a*e*g + c*d*(-3*f + 2*g*x)) + 3*(c*d*f - a*e*g)^2*ArcTa 
nh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(4*Sqrt[ 
c]*Sqrt[d]*g^(5/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {1250, 1250, 1268, 66, 221}

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.

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

\(\Big \downarrow \) 1250

\(\displaystyle \frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x} \sqrt {f+g x}}dx}{4 g}\)

\(\Big \downarrow \) 1250

\(\displaystyle \frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 g}\right )}{4 g}\)

\(\Big \downarrow \) 1268

\(\displaystyle \frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 g}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 g}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{\sqrt {c} \sqrt {d} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 g}\)

Input:

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

Output:

(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(2*g*(d + e* 
x)^(3/2)) - (3*(c*d*f - a*e*g)*((Sqrt[f + g*x]*Sqrt[a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2])/(g*Sqrt[d + e*x]) - ((c*d*f - a*e*g)*Sqrt[a*e + c*d*x]*S 
qrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + 
 g*x])])/(Sqrt[c]*Sqrt[d]*g^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x 
^2])))/(4*g)
 

Defintions of rubi rules used

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 221

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

rule 1250

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

rule 1268

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/((d 
 + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f 
 + g*x)^n*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [A] (verified)

Time = 2.64 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \sqrt {g x +f}\, \left (3 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} e^{2} g^{2}-6 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a c d e f g +3 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{2} d^{2} f^{2}+4 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c d g x +10 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, a e g -6 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c d f \right )}{8 \sqrt {e x +d}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, g^{2} \sqrt {c d g}}\)	\(315\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(1/2),x, 
method=_RETURNVERBOSE)
 

Output:

1/8*((e*x+d)*(c*d*x+a*e))^(1/2)/(e*x+d)^(1/2)*(g*x+f)^(1/2)*(3*ln(1/2*(2*c 
*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1 
/2))*a^2*e^2*g^2-6*ln(1/2*(2*c*d*g*x+a*e*g+d*f*c+2*((c*d*x+a*e)*(g*x+f))^( 
1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*a*c*d*e*f*g+3*ln(1/2*(2*c*d*g*x+a*e*g+d 
*f*c+2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2))/(c*d*g)^(1/2))*c^2*d^2*f 
^2+4*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)*c*d*g*x+10*((c*d*x+a*e)*(g* 
x+f))^(1/2)*(c*d*g)^(1/2)*a*e*g-6*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2 
)*c*d*f)/((c*d*x+a*e)*(g*x+f))^(1/2)/g^2/(c*d*g)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 712, normalized size of antiderivative = 3.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx=\left [\frac {4 \, {\left (2 \, c^{2} d^{2} g^{2} x - 3 \, c^{2} d^{2} f g + 5 \, a c d e g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} + 3 \, {\left (c^{2} d^{3} f^{2} - 2 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + {\left (c^{2} d^{2} e f^{2} - 2 \, a c d e^{2} f g + a^{2} e^{3} g^{2}\right )} x\right )} \sqrt {c d g} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {c d g} \sqrt {e x + d} \sqrt {g x + f} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{16 \, {\left (c d e g^{3} x + c d^{2} g^{3}\right )}}, \frac {2 \, {\left (2 \, c^{2} d^{2} g^{2} x - 3 \, c^{2} d^{2} f g + 5 \, a c d e g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} - 3 \, {\left (c^{2} d^{3} f^{2} - 2 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + {\left (c^{2} d^{2} e f^{2} - 2 \, a c d e^{2} f g + a^{2} e^{3} g^{2}\right )} x\right )} \sqrt {-c d g} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + c d f + a e g\right )} \sqrt {-c d g} \sqrt {e x + d} \sqrt {g x + f}}{2 \, {\left (c^{2} d^{2} e g^{2} x^{3} + a c d^{2} e f g + {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (a c d^{2} e g^{2} + {\left (c^{2} d^{3} + a c d e^{2}\right )} f g\right )} x\right )}}\right )}{8 \, {\left (c d e g^{3} x + c d^{2} g^{3}\right )}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(1 
/2),x, algorithm="fricas")
 

Output:

[1/16*(4*(2*c^2*d^2*g^2*x - 3*c^2*d^2*f*g + 5*a*c*d*e*g^2)*sqrt(c*d*e*x^2 
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) + 3*(c^2*d^3*f^2 
- 2*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + (c^2*d^2*e*f^2 - 2*a*c*d*e^2*f*g + a^2 
*e^3*g^2)*x)*sqrt(c*d*g)*log(-(8*c^2*d^2*e*g^2*x^3 + c^2*d^3*f^2 + 6*a*c*d 
^2*e*f*g + a^2*d*e^2*g^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*( 
2*c*d*g*x + c*d*f + a*e*g)*sqrt(c*d*g)*sqrt(e*x + d)*sqrt(g*x + f) + 8*(c^ 
2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (c^2*d^2*e*f^2 + 2*(4*c^2*d 
^3 + 3*a*c*d*e^2)*f*g + (8*a*c*d^2*e + a^2*e^3)*g^2)*x)/(e*x + d)))/(c*d*e 
*g^3*x + c*d^2*g^3), 1/8*(2*(2*c^2*d^2*g^2*x - 3*c^2*d^2*f*g + 5*a*c*d*e*g 
^2)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f 
) - 3*(c^2*d^3*f^2 - 2*a*c*d^2*e*f*g + a^2*d*e^2*g^2 + (c^2*d^2*e*f^2 - 2* 
a*c*d*e^2*f*g + a^2*e^3*g^2)*x)*sqrt(-c*d*g)*arctan(1/2*sqrt(c*d*e*x^2 + a 
*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*g*x + c*d*f + a*e*g)*sqrt(-c*d*g)*sqrt(e* 
x + d)*sqrt(g*x + f)/(c^2*d^2*e*g^2*x^3 + a*c*d^2*e*f*g + (c^2*d^2*e*f*g + 
 (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 + (a*c*d^2*e*g^2 + (c^2*d^3 + a*c*d*e^2)*f 
*g)*x)))/(c*d*e*g^3*x + c*d^2*g^3)]
 

Sympy [F]

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

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+ 
f)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(1 
/2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx=\frac {{\left (\sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} {\left (\frac {2 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} {\left | e \right |}}{c d e^{2} g} - \frac {3 \, {\left (c d e^{2} f g {\left | e \right |} - a e^{3} g^{2} {\left | e \right |}\right )}}{c d e^{2} g^{3}}\right )} - \frac {3 \, {\left (c^{2} d^{2} e^{2} f^{2} {\left | e \right |} - 2 \, a c d e^{3} f g {\left | e \right |} + a^{2} e^{4} g^{2} {\left | e \right |}\right )} \log \left ({\left | -\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} \sqrt {c d g} + \sqrt {c^{2} d^{2} e^{2} f - a c d e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d g} \right |}\right )}{\sqrt {c d g} g^{2}}\right )} {\left | c \right |} {\left | d \right |}}{4 \, c d e^{2} {\left | e \right |}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(1 
/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.18 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}} \, dx=\frac {5 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a c d e \,g^{2}-3 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g +2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x +3 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a^{2} e^{2} g^{2}-6 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a c d e f g +3 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) c^{2} d^{2} f^{2}}{4 c d \,g^{3}} \] Input:

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

Output:

(5*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c*d*e*g**2 - 3*sqrt(f + g*x)*sqrt(a*e 
 + c*d*x)*c**2*d**2*f*g + 2*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**2*d**2*g**2 
*x + 3*sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sq 
rt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a**2*e**2*g**2 - 6*sqrt(g)*sqrt( 
d)*sqrt(c)*log((sqrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x)) 
/sqrt(a*e*g - c*d*f))*a*c*d*e*f*g + 3*sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g) 
*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*c 
**2*d**2*f**2)/(4*c*d*g**3)