\(\int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [99]

Optimal result

Integrand size = 48, antiderivative size = 265 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {5 g^{3/2} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{7/2} d^{7/2}} \] Output:

-2/3*(e*x+d)^(3/2)*(g*x+f)^(5/2)/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/ 
2)-10/3*g*(e*x+d)^(1/2)*(g*x+f)^(3/2)/c^2/d^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e 
*x^2)^(1/2)+5*g^2*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^ 
3/d^3/(e*x+d)^(1/2)+5*g^(3/2)*(-a*e*g+c*d*f)*arctanh(c^(1/2)*d^(1/2)*(e*x+ 
d)^(1/2)*(g*x+f)^(1/2)/g^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^ 
(7/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.71 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{5/2} \left (\sqrt {c} \sqrt {d} (a e+c d x) \sqrt {f+g x} \left (15 a^2 e^2 g^2-10 a c d e g (f-2 g x)+c^2 d^2 \left (-2 f^2-14 f g x+3 g^2 x^2\right )\right )+15 g^{3/2} (c d f-a e g) (a e+c d x)^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{3 c^{7/2} d^{7/2} ((a e+c d x) (d+e x))^{5/2}} \] Input:

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

Output:

((d + e*x)^(5/2)*(Sqrt[c]*Sqrt[d]*(a*e + c*d*x)*Sqrt[f + g*x]*(15*a^2*e^2* 
g^2 - 10*a*c*d*e*g*(f - 2*g*x) + c^2*d^2*(-2*f^2 - 14*f*g*x + 3*g^2*x^2)) 
+ 15*g^(3/2)*(c*d*f - a*e*g)*(a*e + c*d*x)^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]* 
Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])]))/(3*c^(7/2)*d^(7/2)*((a*e + c 
*d*x)*(d + e*x))^(5/2))
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1251, 1251, 1253, 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.

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

\(\Big \downarrow \) 1251

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

\(\Big \downarrow \) 1251

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

\(\Big \downarrow \) 1253

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 1251

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

rule 1253

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
+ (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^(m - 1)*(f + g*x)^n* 
((a + b*x + c*x^2)^(p + 1)/(c*(m - n - 1))), x] - Simp[n*((c*e*f + c*d*g - 
b*e*g)/(c*e*(m - n - 1)))   Int[(d + e*x)^m*(f + g*x)^(n - 1)*(a + b*x + c* 
x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d* 
e + a*e^2, 0] && EqQ[m + p, 0] && GtQ[n, 0] && NeQ[m - n - 1, 0] && (Intege 
rQ[2*p] || IntegerQ[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. \(641\) vs. \(2(225)=450\).

Time = 2.76 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.42

method	result	size
default	\(-\frac {\left (15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,g^{3} x^{2}-15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+30 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{3} e^{3} g^{3}-15 \ln \left (\frac {2 c d g x +a e g +d f c +2 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}-6 c^{2} d^{2} g^{2} x^{2} \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}-40 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, a c d e \,g^{2} x +28 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c^{2} d^{2} f g x -30 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+20 a c d e f g \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}+4 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right ) \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \sqrt {g x +f}}{6 \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}\, \left (c d x +a e \right )^{2} c^{3} d^{3} \sqrt {e x +d}}\)	\(642\)

Input:

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

Output:

-1/6*(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*c^2*d^2*e*g^3*x^2-15*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* 
g^2*x^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^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))*a^3*e^3*g^3-15*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^2-6*c^2*d^2*g^2*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)-40*((c*d*x 
+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)*a*c*d*e*g^2*x+28*((c*d*x+a*e)*(g*x+f))^ 
(1/2)*(c*d*g)^(1/2)*c^2*d^2*f*g*x-30*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^( 
1/2)*a^2*e^2*g^2+20*a*c*d*e*f*g*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)+ 
4*((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)*(g*x+f)^(1/2)/((c*d*x+a*e)*(g*x+f))^(1/2)/(c*d*g)^(1/2)/(c*d*x+ 
a*e)^2/c^3/d^3/(e*x+d)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (225) = 450\).

Time = 0.69 (sec) , antiderivative size = 1055, normalized size of antiderivative = 3.98 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/12*(4*(3*c^2*d^2*g^2*x^2 - 2*c^2*d^2*f^2 - 10*a*c*d*e*f*g + 15*a^2*e^2* 
g^2 - 2*(7*c^2*d^2*f*g - 10*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*(a^2*c*d^2*e^2*f*g - a^3*d* 
e^3*g^2 + (c^3*d^3*e*f*g - a*c^2*d^2*e^2*g^2)*x^3 + ((c^3*d^4 + 2*a*c^2*d^ 
2*e^2)*f*g - (a*c^2*d^3*e + 2*a^2*c*d*e^3)*g^2)*x^2 + ((2*a*c^2*d^3*e + a^ 
2*c*d*e^3)*f*g - (2*a^2*c*d^2*e^2 + a^3*e^4)*g^2)*x)*sqrt(g/(c*d))*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 + 8*(c^ 
2*d^2*e*f*g + (c^2*d^3 + a*c*d*e^2)*g^2)*x^2 - 4*(2*c^2*d^2*g*x + c^2*d^2* 
f + a*c*d*e*g)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*s 
qrt(g*x + f)*sqrt(g/(c*d)) + (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^5*d^5*e*x^3 + a^2*c^3 
*d^4*e^2 + (c^5*d^6 + 2*a*c^4*d^4*e^2)*x^2 + (2*a*c^4*d^5*e + a^2*c^3*d^3* 
e^3)*x), 1/6*(2*(3*c^2*d^2*g^2*x^2 - 2*c^2*d^2*f^2 - 10*a*c*d*e*f*g + 15*a 
^2*e^2*g^2 - 2*(7*c^2*d^2*f*g - 10*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*(a^2*c*d^2*e^2*f*g - 
 a^3*d*e^3*g^2 + (c^3*d^3*e*f*g - a*c^2*d^2*e^2*g^2)*x^3 + ((c^3*d^4 + 2*a 
*c^2*d^2*e^2)*f*g - (a*c^2*d^3*e + 2*a^2*c*d*e^3)*g^2)*x^2 + ((2*a*c^2*d^3 
*e + a^2*c*d*e^3)*f*g - (2*a^2*c*d^2*e^2 + a^3*e^4)*g^2)*x)*sqrt(-g/(c*d)) 
*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)*c*d*sqrt(-g/(c*d))/(2*c*d*e*g*x^2 + c*d^2*f + a*d*e*g + (c*d*e*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (225) = 450\).

Time = 0.49 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.95 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left ({\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left (\frac {3 \, {\left (c^{5} d^{5} e^{2} f g^{5} - a c^{4} d^{4} e^{3} g^{6}\right )} {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )}}{c^{6} d^{6} e^{4} f g {\left | g \right |} - a c^{5} d^{5} e^{5} g^{2} {\left | g \right |}} - \frac {20 \, {\left (c^{5} d^{5} e^{4} f^{2} g^{5} - 2 \, a c^{4} d^{4} e^{5} f g^{6} + a^{2} c^{3} d^{3} e^{6} g^{7}\right )}}{c^{6} d^{6} e^{4} f g {\left | g \right |} - a c^{5} d^{5} e^{5} g^{2} {\left | g \right |}}\right )} + \frac {15 \, {\left (c^{5} d^{5} e^{6} f^{3} g^{5} - 3 \, a c^{4} d^{4} e^{7} f^{2} g^{6} + 3 \, a^{2} c^{3} d^{3} e^{8} f g^{7} - a^{3} c^{2} d^{2} e^{9} g^{8}\right )}}{c^{6} d^{6} e^{4} f g {\left | g \right |} - a c^{5} d^{5} e^{5} g^{2} {\left | g \right |}}\right )}}{3 \, {\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )}^{2}} - \frac {5 \, {\left (c d f g^{3} - a e g^{4}\right )} \log \left ({\left | -\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} + \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \right |}\right )}{\sqrt {c d g} c^{3} d^{3} {\left | g \right |}} \] Input:

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

Output:

1/3*sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2*f + (e*x + d)*e*g - d*e*g)*c*d*g) 
*sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*((e^2*f + (e*x + d)*e*g - d*e*g)*(3*( 
c^5*d^5*e^2*f*g^5 - a*c^4*d^4*e^3*g^6)*(e^2*f + (e*x + d)*e*g - d*e*g)/(c^ 
6*d^6*e^4*f*g*abs(g) - a*c^5*d^5*e^5*g^2*abs(g)) - 20*(c^5*d^5*e^4*f^2*g^5 
 - 2*a*c^4*d^4*e^5*f*g^6 + a^2*c^3*d^3*e^6*g^7)/(c^6*d^6*e^4*f*g*abs(g) - 
a*c^5*d^5*e^5*g^2*abs(g))) + 15*(c^5*d^5*e^6*f^3*g^5 - 3*a*c^4*d^4*e^7*f^2 
*g^6 + 3*a^2*c^3*d^3*e^8*f*g^7 - a^3*c^2*d^2*e^9*g^8)/(c^6*d^6*e^4*f*g*abs 
(g) - a*c^5*d^5*e^5*g^2*abs(g)))/(c*d*e^2*f*g - a*e^3*g^2 - (e^2*f + (e*x 
+ d)*e*g - d*e*g)*c*d*g)^2 - 5*(c*d*f*g^3 - a*e*g^4)*log(abs(-sqrt(e^2*f + 
 (e*x + d)*e*g - d*e*g)*sqrt(c*d*g) + sqrt(-c*d*e^2*f*g + a*e^3*g^2 + (e^2 
*f + (e*x + d)*e*g - d*e*g)*c*d*g)))/(sqrt(c*d*g)*c^3*d^3*abs(g))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {-30 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a^{2} e^{2} g^{2}+30 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a c d e f g -30 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a c d e \,g^{2} x +30 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) c^{2} d^{2} f g x -5 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+5 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d e f g -5 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x +5 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x +30 \sqrt {g x +f}\, a^{2} c d \,e^{2} g^{2}-20 \sqrt {g x +f}\, a \,c^{2} d^{2} e f g +40 \sqrt {g x +f}\, a \,c^{2} d^{2} e \,g^{2} x -4 \sqrt {g x +f}\, c^{3} d^{3} f^{2}-28 \sqrt {g x +f}\, c^{3} d^{3} f g x +6 \sqrt {g x +f}\, c^{3} d^{3} g^{2} x^{2}}{6 \sqrt {c d x +a e}\, c^{4} d^{4} \left (c d x +a e \right )} \] Input:

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

Output:

( - 30*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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**2 
 + 30*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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 
0*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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*g**2*x + 30 
*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*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*g*x - 5* 
sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**2*e**2*g**2 + 5*sqrt(g)*sqrt( 
d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c*d*e*f*g - 5*sqrt(g)*sqrt(d)*sqrt(c)*sqrt( 
a*e + c*d*x)*a*c*d*e*g**2*x + 5*sqrt(g)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)* 
c**2*d**2*f*g*x + 30*sqrt(f + g*x)*a**2*c*d*e**2*g**2 - 20*sqrt(f + g*x)*a 
*c**2*d**2*e*f*g + 40*sqrt(f + g*x)*a*c**2*d**2*e*g**2*x - 4*sqrt(f + g*x) 
*c**3*d**3*f**2 - 28*sqrt(f + g*x)*c**3*d**3*f*g*x + 6*sqrt(f + g*x)*c**3* 
d**3*g**2*x**2)/(6*sqrt(a*e + c*d*x)*c**4*d**4*(a*e + c*d*x))