\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx\) [68]

Optimal result

Integrand size = 48, antiderivative size = 198 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx=\frac {3 c d \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g^2 \sqrt {d+e x}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{g (d+e x)^{3/2} \sqrt {f+g x}}-\frac {3 \sqrt {c} \sqrt {d} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}\right )}{g^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} (-2 a e g+c d (3 f+g x))-3 \sqrt {c} \sqrt {d} (c d f-a e g) \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{g^{5/2} \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \] Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(Sqrt[g]*Sqrt[a*e + c*d*x]*(-2*a*e*g + c* 
d*(3*f + g*x)) - 3*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)*Sqrt[f + g*x]*ArcTanh[( 
Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(g^(5/2)*Sqr 
t[(a*e + c*d*x)*(d + e*x)]*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {1249, 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.

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1250

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(g*(d + e*x)^(3/2)*Sqrt 
[f + g*x]) + (3*c*d*((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]*Sqrt[d + e*x 
]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])])/(S 
qrt[c]*Sqrt[d]*g^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])))/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 1249

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*(n + 1))), x] + Simp[c*(m/(e*g*(n + 1)))   Int[(d + e*x) 
^(m + 1)*(f + g*x)^(n + 1)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b 
, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0] && G 
tQ[p, 0] && LtQ[n, -1] &&  !(IntegerQ[n + p] && LeQ[n + p + 2, 0])
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(168)=336\).

Time = 2.59 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.88

method	result	size
default	\(\frac {\left (3 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a c d e \,g^{2} x -3 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{2} d^{2} f g x +3 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a c d e f g -3 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{2} d^{2} f^{2}+2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c d g x -4 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, a e g +6 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c d f \right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, g^{2} \sqrt {g x +f}\, \sqrt {e x +d}}\)	\(373\)

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)^(3/2),x, 
method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.35 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx=\left [\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt {e x + d} \sqrt {g x + f} - 3 \, {\left (c d^{2} f^{2} - a d e f g + {\left (c d e f g - a e^{2} g^{2}\right )} x^{2} + {\left (c d e f^{2} - a d e g^{2} + {\left (c d^{2} - a e^{2}\right )} f g\right )} x\right )} \sqrt {\frac {c d}{g}} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 4 \, {\left (2 \, c d g^{2} x + c d f g + a e g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} \sqrt {\frac {c d}{g}} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{4 \, {\left (e g^{3} x^{2} + d f g^{2} + {\left (e f g^{2} + d g^{3}\right )} x\right )}}, \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d g x + 3 \, c d f - 2 \, a e g\right )} \sqrt {e x + d} \sqrt {g x + f} + 3 \, {\left (c d^{2} f^{2} - a d e f g + {\left (c d e f g - a e^{2} g^{2}\right )} x^{2} + {\left (c d e f^{2} - a d e g^{2} + {\left (c d^{2} - a e^{2}\right )} f g\right )} x\right )} \sqrt {-\frac {c d}{g}} \arctan \left (\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f} \sqrt {-\frac {c d}{g}} g}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{2 \, {\left (e g^{3} x^{2} + d f g^{2} + {\left (e f g^{2} + d g^{3}\right )} x\right )}}\right ] \] 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)^(3 
/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

Output:

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

Maxima [F]

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx=\frac {\sqrt {c d x + a e} {\left (\frac {{\left (c d x + a e\right )} {\left | c \right |} {\left | d \right |}}{g} + \frac {3 \, {\left (c d f g {\left | c \right |} {\left | d \right |} - a e g^{2} {\left | c \right |} {\left | d \right |}\right )}}{g^{3}}\right )}}{\sqrt {c^{2} d^{2} f - a c d e g + {\left (c d x + a e\right )} c d g}} + \frac {3 \, {\left (c d f {\left | c \right |} {\left | d \right |} - a e g {\left | c \right |} {\left | d \right |}\right )} \log \left ({\left | -\sqrt {c d g} \sqrt {c d x + a e} + \sqrt {c^{2} d^{2} f - a c d e g + {\left (c d x + a e\right )} c d g} \right |}\right )}{\sqrt {c d g} g^{2}} \] 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)^(3 
/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.71 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^{3/2}} \, dx=\frac {-8 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a e \,g^{2}+12 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c d f g +4 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c d \,g^{2} x +12 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a e f g +12 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a e \,g^{2} x -12 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) c d \,f^{2}-12 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) c d f g x -9 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, a e f g -9 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, a e \,g^{2} x +9 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c d \,f^{2}+9 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c d f g x}{4 g^{3} \left (g x +f \right )} \] 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)^(3/2),x)
 

Output:

( - 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*e*g**2 + 12*sqrt(f + g*x)*sqrt(a*e 
 + c*d*x)*c*d*f*g + 4*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c*d*g**2*x + 12*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*e*f*g + 12*sqrt(g)*sqrt(d)*sqrt(c)*log((s 
qrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d 
*f))*a*e*g**2*x - 12*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*d*f**2 - 12*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*d*f*g*x - 9*sqrt(g)*sqrt(d)*sqrt(c)*a*e*f 
*g - 9*sqrt(g)*sqrt(d)*sqrt(c)*a*e*g**2*x + 9*sqrt(g)*sqrt(d)*sqrt(c)*c*d* 
f**2 + 9*sqrt(g)*sqrt(d)*sqrt(c)*c*d*f*g*x)/(4*g**3*(f + g*x))