\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx\) [77]

Optimal result

Integrand size = 48, antiderivative size = 270 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=-\frac {15 c d (c d f-a e g) \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g^3 \sqrt {d+e x}}+\frac {5 c d \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{2 g^2 (d+e x)^{3/2}}-\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}+\frac {15 \sqrt {c} \sqrt {d} (c d f-a e g)^2 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}\right )}{4 g^{7/2}} \] Output:

-15/4*c*d*(-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^3/(e*x+d)^(1/2)+5/2*c*d*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* 
x^2)^(3/2)/g^2/(e*x+d)^(3/2)-2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/g/( 
e*x+d)^(5/2)/(g*x+f)^(1/2)+15/4*c^(1/2)*d^(1/2)*(-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))/g^(7/2)
 

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\sqrt {g} \sqrt {a e+c d x} \left (-8 a^2 e^2 g^2+a c d e g (25 f+9 g x)+c^2 d^2 \left (-15 f^2-5 f g x+2 g^2 x^2\right )\right )+15 \sqrt {c} \sqrt {d} (c d f-a e g)^2 \sqrt {f+g x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{4 g^{7/2} \sqrt {(a e+c d x) (d+e x)} \sqrt {f+g x}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*( 
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]*(-8*a^2*e^2*g^ 
2 + a*c*d*e*g*(25*f + 9*g*x) + c^2*d^2*(-15*f^2 - 5*f*g*x + 2*g^2*x^2)) + 
15*Sqrt[c]*Sqrt[d]*(c*d*f - a*e*g)^2*Sqrt[f + g*x]*ArcTanh[(Sqrt[c]*Sqrt[d 
]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(4*g^(7/2)*Sqrt[(a*e + c*d 
*x)*(d + e*x)]*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1249, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx\)

\(\Big \downarrow \) 1249

\(\displaystyle \frac {5 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2} \sqrt {f+g x}}dx}{g}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}\)

\(\Big \downarrow \) 1250

\(\displaystyle \frac {5 c d \left (\frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x} \sqrt {f+g x}}dx}{4 g}\right )}{g}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}\)

\(\Big \downarrow \) 1250

\(\displaystyle \frac {5 c d \left (\frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 g}\right )}{4 g}\right )}{g}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}\)

\(\Big \downarrow \) 1268

\(\displaystyle \frac {5 c d \left (\frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 g}\right )}{g}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {5 c d \left (\frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 g}\right )}{g}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {5 c d \left (\frac {\sqrt {f+g x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x}}-\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{\sqrt {c} \sqrt {d} g^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{4 g}\right )}{g}-\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{g (d+e x)^{5/2} \sqrt {f+g x}}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/((d + e*x)^(5/2)*(f + g* 
x)^(3/2)),x]
 

Output:

(-2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(g*(d + e*x)^(5/2)*Sqrt 
[f + g*x]) + (5*c*d*((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]*Sqrt[d + e*x]*ArcTanh[(Sqrt[g]*Sqrt[a*e + c*d*x])/(Sqr 
t[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)))/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. \(624\) vs. \(2(228)=456\).

Time = 2.78 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.31

method	result	size
default	\(\frac {\left (15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{2} g x +15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}-30 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,f^{2} g +15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f^{3}+4 c^{2} d^{2} g^{2} x^{2} \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}+18 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, a c d e \,g^{2} x -10 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c^{2} d^{2} f g x -16 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+50 a c d e f g \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}-30 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}}{8 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, g^{3} \sqrt {g x +f}\, \sqrt {e x +d}}\)	\(625\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3/2),x, 
method=_RETURNVERBOSE)
 

Output:

1/8*(15*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*c*d*e^2*g^3*x-30*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^2*d^2*e*f 
*g^2*x+15*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^3*d^3*f^2*g*x+15*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*c*d*e^2*f 
*g^2-30*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^2*d^2*e*f^2*g+15*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^3*d^3*f^3+4 
*c^2*d^2*g^2*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)+18*((c*d*x+a*e) 
*(g*x+f))^(1/2)*(c*d*g)^(1/2)*a*c*d*e*g^2*x-10*((c*d*x+a*e)*(g*x+f))^(1/2) 
*(c*d*g)^(1/2)*c^2*d^2*f*g*x-16*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)* 
a^2*e^2*g^2+50*a*c*d*e*f*g*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)-30*(( 
c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)*c^2*d^2*f^2)*((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^3/(g*x+f)^(1/2)/(e*x+d) 
^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 915, normalized size of antiderivative = 3.39 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3 
/2),x, algorithm="fricas")
 

Output:

[1/16*(4*(2*c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 25*a*c*d*e*f*g - 8*a^2*e^2* 
g^2 - (5*c^2*d^2*f*g - 9*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + 
 a*e^2)*x)*sqrt(e*x + d)*sqrt(g*x + f) + 15*(c^2*d^3*f^3 - 2*a*c*d^2*e*f^2 
*g + a^2*d*e^2*f*g^2 + (c^2*d^2*e*f^2*g - 2*a*c*d*e^2*f*g^2 + a^2*e^3*g^3) 
*x^2 + (c^2*d^2*e*f^3 + a^2*d*e^2*g^3 + (c^2*d^3 - 2*a*c*d*e^2)*f^2*g - (2 
*a*c*d^2*e - a^2*e^3)*f*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*(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^4*x^2 + d*f*g^3 + (e*f*g^3 + d*g^4)*x), 1/8*(2*(2* 
c^2*d^2*g^2*x^2 - 15*c^2*d^2*f^2 + 25*a*c*d*e*f*g - 8*a^2*e^2*g^2 - (5*c^2 
*d^2*f*g - 9*a*c*d*e*g^2)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*s 
qrt(e*x + d)*sqrt(g*x + f) - 15*(c^2*d^3*f^3 - 2*a*c*d^2*e*f^2*g + a^2*d*e 
^2*f*g^2 + (c^2*d^2*e*f^2*g - 2*a*c*d*e^2*f*g^2 + a^2*e^3*g^3)*x^2 + (c^2* 
d^2*e*f^3 + a^2*d*e^2*g^3 + (c^2*d^3 - 2*a*c*d*e^2)*f^2*g - (2*a*c*d^2*e - 
 a^2*e^3)*f*g^2)*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^4*x^2 + d*f*g^ 
3 + (e*f*g^3 + d*g^4)*x)]
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(5/2)/(g*x+ 
f)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\frac {\sqrt {c d x + a e} {\left ({\left (c d x + a e\right )} {\left (\frac {2 \, {\left (c d x + a e\right )} {\left | c \right |} {\left | d \right |}}{g} - \frac {5 \, {\left (c d f g^{3} {\left | c \right |} {\left | d \right |} - a e g^{4} {\left | c \right |} {\left | d \right |}\right )}}{g^{5}}\right )} - \frac {15 \, {\left (c^{2} d^{2} f^{2} g^{2} {\left | c \right |} {\left | d \right |} - 2 \, a c d e f g^{3} {\left | c \right |} {\left | d \right |} + a^{2} e^{2} g^{4} {\left | c \right |} {\left | d \right |}\right )}}{g^{5}}\right )}}{4 \, \sqrt {c^{2} d^{2} f - a c d e g + {\left (c d x + a e\right )} c d g}} - \frac {15 \, {\left (c^{2} d^{2} f^{2} {\left | c \right |} {\left | d \right |} - 2 \, a c d e f g {\left | c \right |} {\left | d \right |} + a^{2} e^{2} g^{2} {\left | c \right |} {\left | d \right |}\right )} \log \left ({\left | -\sqrt {c d g} \sqrt {c d x + a e} + \sqrt {c^{2} d^{2} f - a c d e g + {\left (c d x + a e\right )} c d g} \right |}\right )}{4 \, \sqrt {c d g} g^{3}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)/(g*x+f)^(3 
/2),x, algorithm="giac")
 

Output:

1/4*sqrt(c*d*x + a*e)*((c*d*x + a*e)*(2*(c*d*x + a*e)*abs(c)*abs(d)/g - 5* 
(c*d*f*g^3*abs(c)*abs(d) - a*e*g^4*abs(c)*abs(d))/g^5) - 15*(c^2*d^2*f^2*g 
^2*abs(c)*abs(d) - 2*a*c*d*e*f*g^3*abs(c)*abs(d) + a^2*e^2*g^4*abs(c)*abs( 
d))/g^5)/sqrt(c^2*d^2*f - a*c*d*e*g + (c*d*x + a*e)*c*d*g) - 15/4*(c^2*d^2 
*f^2*abs(c)*abs(d) - 2*a*c*d*e*f*g*abs(c)*abs(d) + a^2*e^2*g^2*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^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^{3/2}} \, dx=\frac {-8 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{3}+25 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a c d e f \,g^{2}+9 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a c d e \,g^{3} x -15 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2} g -5 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} f \,g^{2} x +2 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{3} x^{2}+15 \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} f \,g^{2}+15 \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^{3} x -30 \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^{2} g -30 \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^{2} x +15 \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^{3}+15 \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} g x -10 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, a^{2} e^{2} f \,g^{2}-10 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, a^{2} e^{2} g^{3} x +20 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, a c d e \,f^{2} g +20 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, a c d e f \,g^{2} x -10 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f^{3}-10 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, c^{2} d^{2} f^{2} g x}{4 g^{4} \left (g x +f \right )} \] Input:

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

Output:

( - 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*e**2*g**3 + 25*sqrt(f + g*x)*sq 
rt(a*e + c*d*x)*a*c*d*e*f*g**2 + 9*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c*d*e 
*g**3*x - 15*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**2*d**2*f**2*g - 5*sqrt(f + 
 g*x)*sqrt(a*e + c*d*x)*c**2*d**2*f*g**2*x + 2*sqrt(f + g*x)*sqrt(a*e + c* 
d*x)*c**2*d**2*g**3*x**2 + 15*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**2*e**2 
*f*g**2 + 15*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**2*e**2*g**3*x - 30*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**2*g - 30*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**2*x + 15*sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g)*sqr 
t(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**3 + 15*sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g)*sqrt(a*e + c*d*x) + sq 
rt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*c**2*d**2*f**2*g*x - 10* 
sqrt(g)*sqrt(d)*sqrt(c)*a**2*e**2*f*g**2 - 10*sqrt(g)*sqrt(d)*sqrt(c)*a**2 
*e**2*g**3*x + 20*sqrt(g)*sqrt(d)*sqrt(c)*a*c*d*e*f**2*g + 20*sqrt(g)*sqrt 
(d)*sqrt(c)*a*c*d*e*f*g**2*x - 10*sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*f**3 - 
 10*sqrt(g)*sqrt(d)*sqrt(c)*c**2*d**2*f**2*g*x)/(4*g**4*(f + g*x))