\(\int \frac {\sqrt {f+g x} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [75]

Optimal result

Integrand size = 48, antiderivative size = 364 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {5 (c d f-a e g)^3 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 c d g^3 \sqrt {d+e x}}-\frac {5 (c d f-a e g)^2 \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{96 c d g^2 (d+e x)^{3/2}}-\frac {\left (\frac {a e}{c d}-\frac {f}{g}\right ) \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{24 (d+e x)^{5/2}}+\frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{4 c d (d+e x)^{7/2}}-\frac {5 (c d f-a e g)^4 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}\right )}{64 c^{3/2} d^{3/2} g^{7/2}} \] Output:

5/64*(-a*e*g+c*d*f)^3*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2 
)/c/d/g^3/(e*x+d)^(1/2)-5/96*(-a*e*g+c*d*f)^2*(g*x+f)^(1/2)*(a*d*e+(a*e^2+ 
c*d^2)*x+c*d*e*x^2)^(3/2)/c/d/g^2/(e*x+d)^(3/2)-1/24*(a*e/c/d-f/g)*(g*x+f) 
^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5/2)+1/4*(g*x+f)^( 
1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c/d/(e*x+d)^(7/2)-5/64*(-a*e* 
g+c*d*f)^4*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^(3/2)/d^(3/2)/g^(7/2)
 

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {f+g x} \left (15 a^3 e^3 g^3+a^2 c d e^2 g^2 (73 f+118 g x)+a c^2 d^2 e g \left (-55 f^2+36 f g x+136 g^2 x^2\right )+c^3 d^3 \left (15 f^3-10 f^2 g x+8 f g^2 x^2+48 g^3 x^3\right )\right )}{(a e+c d x)^2}-\frac {15 (c d f-a e g)^4 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2}}\right )}{192 c^{3/2} d^{3/2} g^{7/2} (d+e x)^{5/2}} \] Input:

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

Output:

(((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[c]*Sqrt[d]*Sqrt[g]*Sqrt[f + g*x]*( 
15*a^3*e^3*g^3 + a^2*c*d*e^2*g^2*(73*f + 118*g*x) + a*c^2*d^2*e*g*(-55*f^2 
 + 36*f*g*x + 136*g^2*x^2) + c^3*d^3*(15*f^3 - 10*f^2*g*x + 8*f*g^2*x^2 + 
48*g^3*x^3)))/(a*e + c*d*x)^2 - (15*(c*d*f - a*e*g)^4*ArcTanh[(Sqrt[c]*Sqr 
t[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])])/(a*e + c*d*x)^(5/2)))/(1 
92*c^(3/2)*d^(3/2)*g^(7/2)*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1250, 1250, 1250, 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.

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

\(\Big \downarrow \) 1250

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \int \frac {\sqrt {f+g x} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2}}dx}{8 g}\)

\(\Big \downarrow \) 1250

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \left (\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \int \frac {\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{2 g}\right )}{8 g}\)

\(\Big \downarrow \) 1250

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \left (\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 g}\right )}{2 g}\right )}{8 g}\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \left (\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\right )}{8 g}\)

\(\Big \downarrow \) 1268

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \left (\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\right )}{8 g}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \left (\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\right )}{8 g}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 g (d+e x)^{5/2}}-\frac {5 (c d f-a e g) \left (\frac {(f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 g (d+e x)^{3/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 g}\right )}{2 g}\right )}{8 g}\)

Input:

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

Output:

((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(4*g*(d + 
e*x)^(5/2)) - (5*(c*d*f - a*e*g)*(((f + g*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2)^(3/2))/(3*g*(d + e*x)^(3/2)) - ((c*d*f - a*e*g)*(((f + g* 
x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(2*g*Sqrt[d + e*x]) 
- ((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])/(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]*Sqrt[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])))/(4* 
g)))/(2*g)))/(8*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 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. \(731\) vs. \(2(312)=624\).

Time = 2.65 (sec) , antiderivative size = 732, normalized size of antiderivative = 2.01

method	result	size
default	\(-\frac {\sqrt {g x +f}\, \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (-96 c^{3} d^{3} g^{3} x^{3} \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d 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 ) a^{4} e^{4} g^{4}-60 \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} c d \,e^{3} f \,g^{3}+90 \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^{2} d^{2} e^{2} f^{2} g^{2}-60 \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^{3} d^{3} e \,f^{3} 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^{4} d^{4} f^{4}-272 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}-16 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, \sqrt {c d g}-236 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, a^{2} c d \,e^{2} g^{3} x -72 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, a \,c^{2} d^{2} e f \,g^{2} x +20 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, c^{3} d^{3} f^{2} g x -30 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, a^{3} e^{3} g^{3}-146 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, a^{2} c d \,e^{2} f \,g^{2}+110 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, a \,c^{2} d^{2} e \,f^{2} g -30 \sqrt {c d g}\, \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, c^{3} d^{3} f^{3}\right )}{384 \sqrt {e x +d}\, c d \sqrt {\left (c d x +a e \right ) \left (g x +f \right )}\, g^{3} \sqrt {c d g}}\)	\(732\)

Input:

int((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(5/2),x, 
method=_RETURNVERBOSE)
 

Output:

-1/384*(g*x+f)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(-96*c^3*d^3*g^3*x^3*((c* 
d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)+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))*a^4*e^4*g^4-60*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*c*d*e^3*f*g^3+90*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^2*d^2*e^2*f^2*g^2 
-60*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^3*d^3*e*f^3*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^4*d^4*f^4-272*a 
*c^2*d^2*e*g^3*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)-16*c^3*d^3*f* 
g^2*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)-236*(c*d*g)^(1/2)*((c*d* 
x+a*e)*(g*x+f))^(1/2)*a^2*c*d*e^2*g^3*x-72*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x 
+f))^(1/2)*a*c^2*d^2*e*f*g^2*x+20*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/2 
)*c^3*d^3*f^2*g*x-30*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/2)*a^3*e^3*g^3 
-146*(c*d*g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/2)*a^2*c*d*e^2*f*g^2+110*(c*d* 
g)^(1/2)*((c*d*x+a*e)*(g*x+f))^(1/2)*a*c^2*d^2*e*f^2*g-30*(c*d*g)^(1/2)*(( 
c*d*x+a*e)*(g*x+f))^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/c/d/((c*d*x+a*e)*(g*x 
+f))^(1/2)/g^3/(c*d*g)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 1126, normalized size of antiderivative = 3.09 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5 
/2),x, algorithm="fricas")
 

Output:

[1/768*(4*(48*c^4*d^4*g^4*x^3 + 15*c^4*d^4*f^3*g - 55*a*c^3*d^3*e*f^2*g^2 
+ 73*a^2*c^2*d^2*e^2*f*g^3 + 15*a^3*c*d*e^3*g^4 + 8*(c^4*d^4*f*g^3 + 17*a* 
c^3*d^3*e*g^4)*x^2 - 2*(5*c^4*d^4*f^2*g^2 - 18*a*c^3*d^3*e*f*g^3 - 59*a^2* 
c^2*d^2*e^2*g^4)*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^4*d^5*f^4 - 4*a*c^3*d^4*e*f^3*g + 6*a^2*c^2*d^3* 
e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3 + a^4*d*e^4*g^4 + (c^4*d^4*e*f^4 - 4*a 
*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 4*a^3*c*d*e^4*f*g^3 + a^4 
*e^5*g^4)*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^2*d 
^2*e*g^4*x + c^2*d^3*g^4), 1/384*(2*(48*c^4*d^4*g^4*x^3 + 15*c^4*d^4*f^3*g 
 - 55*a*c^3*d^3*e*f^2*g^2 + 73*a^2*c^2*d^2*e^2*f*g^3 + 15*a^3*c*d*e^3*g^4 
+ 8*(c^4*d^4*f*g^3 + 17*a*c^3*d^3*e*g^4)*x^2 - 2*(5*c^4*d^4*f^2*g^2 - 18*a 
*c^3*d^3*e*f*g^3 - 59*a^2*c^2*d^2*e^2*g^4)*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^4*d^5*f^4 - 4*a*c^3*d^ 
4*e*f^3*g + 6*a^2*c^2*d^3*e^2*f^2*g^2 - 4*a^3*c*d^2*e^3*f*g^3 + a^4*d*e^4* 
g^4 + (c^4*d^4*e*f^4 - 4*a*c^3*d^3*e^2*f^3*g + 6*a^2*c^2*d^2*e^3*f^2*g^2 - 
 4*a^3*c*d*e^4*f*g^3 + a^4*e^5*g^4)*x)*sqrt(-c*d*g)*arctan(1/2*sqrt(c*d...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

integrate((g*x+f)**(1/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+ 
d)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5 
/2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4837 vs. \(2 (312) = 624\).

Time = 0.75 (sec) , antiderivative size = 4837, normalized size of antiderivative = 13.29 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(5 
/2),x, algorithm="giac")
 

Output:

1/192*(48*(4*((c*d*e^2*f*g - a*e^3*g^2)*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) + 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* 
f*abs(g)/g^2 - 4*((c*d*e^2*f*g - a*e^3*g^2)*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) + 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) 
)*d*abs(g)/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)*(2*e^2*f + 2*(e*x + d)*e*g - 2*d*e*g - (5*c^2*d^2*e^2*f - 4*c 
^2*d^3*e*g - a*c*d*e^3*g)/(c^2*d^2))*sqrt(e^2*f + (e*x + d)*e*g - d*e*g) - 
 (3*c^2*d^2*e^4*f^2*g - 4*c^2*d^3*e^3*f*g^2 - 2*a*c*d*e^5*f*g^2 + 4*a*c*d^ 
2*e^4*g^3 - a^2*e^6*g^3)*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*d))*abs(g)/(e*g^2))*a^2*abs(e)^2/(e^2*g) - 16*(24* 
((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g - d*e*g)*sq 
rt(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) + 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))*d*e*f*abs(g)/g^ 
2 - 24*((c*d*e^2*f*g - a*e^3*g^2)*log(abs(-sqrt(e^2*f + (e*x + d)*e*g -...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.67 \[ \int \frac {\sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {15 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a^{3} c d \,e^{3} g^{4}+73 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} f \,g^{3}+118 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{2} g^{4} x -55 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,f^{2} g^{2}+36 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e f \,g^{3} x +136 \sqrt {g x +f}\, \sqrt {c d x +a e}\, a \,c^{3} d^{3} e \,g^{4} x^{2}+15 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{4} d^{4} f^{3} g -10 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{4} d^{4} f^{2} g^{2} x +8 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{4} d^{4} f \,g^{3} x^{2}+48 \sqrt {g x +f}\, \sqrt {c d x +a e}\, c^{4} d^{4} g^{4} x^{3}-15 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a^{4} e^{4} g^{4}+60 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a^{3} c d \,e^{3} f \,g^{3}-90 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}+60 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) a \,c^{3} d^{3} e \,f^{3} g -15 \sqrt {g}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {g}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {g x +f}}{\sqrt {a e g -c d f}}\right ) c^{4} d^{4} f^{4}}{192 c^{2} d^{2} g^{4}} \] Input:

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

Output:

(15*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**3*c*d*e**3*g**4 + 73*sqrt(f + g*x)* 
sqrt(a*e + c*d*x)*a**2*c**2*d**2*e**2*f*g**3 + 118*sqrt(f + g*x)*sqrt(a*e 
+ c*d*x)*a**2*c**2*d**2*e**2*g**4*x - 55*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a 
*c**3*d**3*e*f**2*g**2 + 36*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c**3*d**3*e* 
f*g**3*x + 136*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c**3*d**3*e*g**4*x**2 + 1 
5*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**4*d**4*f**3*g - 10*sqrt(f + g*x)*sqrt 
(a*e + c*d*x)*c**4*d**4*f**2*g**2*x + 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c* 
*4*d**4*f*g**3*x**2 + 48*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**4*d**4*g**4*x* 
*3 - 15*sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*s 
qrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a**4*e**4*g**4 + 60*sqrt(g)*sqr 
t(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**3*c*d*e**3*f*g**3 - 90*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*c**2*d**2*e**2*f**2*g**2 + 60*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**3*d**3*e*f**3*g - 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))*c**4*d 
**4*f**4)/(192*c**2*d**2*g**4)