\(\int \frac {(f+g x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\) [74]

Optimal result

Integrand size = 48, antiderivative size = 436 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {3 (c d f-a e g)^4 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 g^3 \sqrt {d+e x}}-\frac {(c d f-a e g)^3 \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^2 d^2 g^2 (d+e x)^{3/2}}+\frac {(c d f-a e g)^2 \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{80 c^2 d^2 g (d+e x)^{5/2}}+\frac {3 (c d f-a e g) \sqrt {f+g x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{40 c^2 d^2 (d+e x)^{7/2}}+\frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{5 c d (d+e x)^{7/2}}-\frac {3 (c d f-a e g)^5 \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {f+g x}}\right )}{128 c^{5/2} d^{5/2} g^{7/2}} \] Output:

3/128*(-a*e*g+c*d*f)^4*(g*x+f)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/ 
2)/c^2/d^2/g^3/(e*x+d)^(1/2)-1/64*(-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)^(3/2)/c^2/d^2/g^2/(e*x+d)^(3/2)+1/80*(-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)^(5/2)/c^2/d^2/g/(e*x+ 
d)^(5/2)+3/40*(-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)^(7/2)/c^2/d^2/(e*x+d)^(7/2)+1/5*(g*x+f)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c* 
d*e*x^2)^(7/2)/c/d/(e*x+d)^(7/2)-3/128*(-a*e*g+c*d*f)^5*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^(5/2)/d^(5/2)/g^(7/2)
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.69 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\frac {((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {g} \sqrt {f+g x} \left (-15 a^4 e^4 g^4+10 a^3 c d e^3 g^3 (7 f+g x)+2 a^2 c^2 d^2 e^2 g^2 \left (64 f^2+233 f g x+124 g^2 x^2\right )+2 a c^3 d^3 e g \left (-35 f^3+23 f^2 g x+256 f g^2 x^2+168 g^3 x^3\right )+c^4 d^4 \left (15 f^4-10 f^3 g x+8 f^2 g^2 x^2+176 f g^3 x^3+128 g^4 x^4\right )\right )}{(a e+c d x)^2}-\frac {15 (c d f-a e g)^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{5/2}}\right )}{640 c^{5/2} d^{5/2} g^{7/2} (d+e x)^{5/2}} \] Input:

Integrate[((f + g*x)^(3/2)*(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^4*e^4*g^4 + 10*a^3*c*d*e^3*g^3*(7*f + g*x) + 2*a^2*c^2*d^2*e^2*g^2*( 
64*f^2 + 233*f*g*x + 124*g^2*x^2) + 2*a*c^3*d^3*e*g*(-35*f^3 + 23*f^2*g*x 
+ 256*f*g^2*x^2 + 168*g^3*x^3) + c^4*d^4*(15*f^4 - 10*f^3*g*x + 8*f^2*g^2* 
x^2 + 176*f*g^3*x^3 + 128*g^4*x^4)))/(a*e + c*d*x)^2 - (15*(c*d*f - a*e*g) 
^5*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[f + g*x])/(Sqrt[g]*Sqrt[a*e + c*d*x])])/( 
a*e + c*d*x)^(5/2)))/(640*c^(5/2)*d^(5/2)*g^(7/2)*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1250, 1250, 1250, 1253, 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.

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

\(\Big \downarrow \) 1250

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

\(\Big \downarrow \) 1250

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

\(\Big \downarrow \) 1250

\(\displaystyle \frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \int \frac {\sqrt {d+e x} (f+g x)^{3/2}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 g}\right )}{8 g}\right )}{2 g}\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {3 (c d f-a e g) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{6 g}\right )}{8 g}\right )}{2 g}\)

\(\Big \downarrow \) 1253

\(\displaystyle \frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {3 (c d f-a e g) \left (\frac {(c d f-a e g) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 c d}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{6 g}\right )}{8 g}\right )}{2 g}\)

\(\Big \downarrow \) 1268

\(\displaystyle \frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {3 (c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}}dx}{2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{6 g}\right )}{8 g}\right )}{2 g}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {3 (c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \int \frac {1}{c d-\frac {g (a e+c d x)}{f+g x}}d\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{6 g}\right )}{8 g}\right )}{2 g}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 g (d+e x)^{5/2}}-\frac {(c d f-a e g) \left (\frac {(f+g x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2}}-\frac {3 (c d f-a e g) \left (\frac {(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x}}-\frac {(c d f-a e g) \left (\frac {3 (c d f-a e g) \left (\frac {\sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {\sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}}\right )}{4 c d}+\frac {(f+g x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d \sqrt {d+e x}}\right )}{6 g}\right )}{8 g}\right )}{2 g}\)

Input:

Int[((f + g*x)^(3/2)*(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)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/(5*g*(d + 
e*x)^(5/2)) - ((c*d*f - a*e*g)*(((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)* 
x + c*d*e*x^2)^(3/2))/(4*g*(d + e*x)^(3/2)) - (3*(c*d*f - a*e*g)*(((f + g* 
x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*g*Sqrt[d + e*x]) 
- ((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*c*d*Sqrt[d + e*x]) + (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])/(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])/(Sq 
rt[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*c*d)))/(6*g)))/(8*g)))/(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 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. \(1004\) vs. \(2(376)=752\).

Time = 2.60 (sec) , antiderivative size = 1005, normalized size of antiderivative = 2.31

method	result	size
default	\(\text {Expression too large to display}\)	\(1005\)

Input:

int((g*x+f)^(3/2)*(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:

1/1280*(g*x+f)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(256*c^4*d^4*g^4*x^4*((c* 
d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)+672*a*c^3*d^3*e*g^4*x^3*((c*d*x+a*e) 
*(g*x+f))^(1/2)*(c*d*g)^(1/2)+352*c^4*d^4*f*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^5*e^5*g^5-75*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* 
c*d*e^4*f*g^4+150*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^2*d^2*e^3*f^2*g^3-150*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^3*d^3*e^2*f^3*g^2+75*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^4*d^4*e*f^4*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^5*d^5*f^5+496*a^2*c^2*d^2*e^2*g^4*x^2*((c*d*x+a*e)*(g*x+f))^(1 
/2)*(c*d*g)^(1/2)+1024*a*c^3*d^3*e*f*g^3*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*( 
c*d*g)^(1/2)+16*c^4*d^4*f^2*g^2*x^2*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1 
/2)+20*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)*a^3*c*d*e^3*g^4*x+932*((c 
*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)*a^2*c^2*d^2*e^2*f*g^3*x+92*((c*d*x+ 
a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)*a*c^3*d^3*e*f^2*g^2*x-20*((c*d*x+a*e)*(g 
*x+f))^(1/2)*(c*d*g)^(1/2)*c^4*d^4*f^3*g*x-30*((c*d*x+a*e)*(g*x+f))^(1/2)* 
(c*d*g)^(1/2)*a^4*e^4*g^4+140*((c*d*x+a*e)*(g*x+f))^(1/2)*(c*d*g)^(1/2)...
 

Fricas [A] (verification not implemented)

Time = 3.22 (sec) , antiderivative size = 1392, normalized size of antiderivative = 3.19 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^(3/2)*(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:

[1/2560*(4*(128*c^5*d^5*g^5*x^4 + 15*c^5*d^5*f^4*g - 70*a*c^4*d^4*e*f^3*g^ 
2 + 128*a^2*c^3*d^3*e^2*f^2*g^3 + 70*a^3*c^2*d^2*e^3*f*g^4 - 15*a^4*c*d*e^ 
4*g^5 + 16*(11*c^5*d^5*f*g^4 + 21*a*c^4*d^4*e*g^5)*x^3 + 8*(c^5*d^5*f^2*g^ 
3 + 64*a*c^4*d^4*e*f*g^4 + 31*a^2*c^3*d^3*e^2*g^5)*x^2 - 2*(5*c^5*d^5*f^3* 
g^2 - 23*a*c^4*d^4*e*f^2*g^3 - 233*a^2*c^3*d^3*e^2*f*g^4 - 5*a^3*c^2*d^2*e 
^3*g^5)*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^5*d^6*f^5 - 5*a*c^4*d^5*e*f^4*g + 10*a^2*c^3*d^4*e^2*f^3* 
g^2 - 10*a^3*c^2*d^3*e^3*f^2*g^3 + 5*a^4*c*d^2*e^4*f*g^4 - a^5*d*e^5*g^5 + 
 (c^5*d^5*e*f^5 - 5*a*c^4*d^4*e^2*f^4*g + 10*a^2*c^3*d^3*e^3*f^3*g^2 - 10* 
a^3*c^2*d^2*e^4*f^2*g^3 + 5*a^4*c*d*e^5*f*g^4 - a^5*e^6*g^5)*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^3*d^3*e*g^4*x + c^3*d^4*g^ 
4), 1/1280*(2*(128*c^5*d^5*g^5*x^4 + 15*c^5*d^5*f^4*g - 70*a*c^4*d^4*e*f^3 
*g^2 + 128*a^2*c^3*d^3*e^2*f^2*g^3 + 70*a^3*c^2*d^2*e^3*f*g^4 - 15*a^4*c*d 
*e^4*g^5 + 16*(11*c^5*d^5*f*g^4 + 21*a*c^4*d^4*e*g^5)*x^3 + 8*(c^5*d^5*f^2 
*g^3 + 64*a*c^4*d^4*e*f*g^4 + 31*a^2*c^3*d^3*e^2*g^5)*x^2 - 2*(5*c^5*d^5*f 
^3*g^2 - 23*a*c^4*d^4*e*f^2*g^3 - 233*a^2*c^3*d^3*e^2*f*g^4 - 5*a^3*c^2...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

integrate((g*x+f)**(3/2)*(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]

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

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 13705 vs. \(2 (376) = 752\).

Time = 1.84 (sec) , antiderivative size = 13705, normalized size of antiderivative = 31.43 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^(3/2)*(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:

1/1920*(480*(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)*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)*c*d))*abs(g)/(e*g^2))*a^2*f*abs(e)^2/(e^2*g) - 80* 
(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 
)*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*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...
 

Mupad [F(-1)]

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

int(((f + g*x)^(3/2)*(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)^(3/2)*(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 = 0.84 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.94 \[ \int \frac {(f+g x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int((g*x+f)^(3/2)*(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**4*c*d*e**4*g**5 + 70*sqrt(f + g* 
x)*sqrt(a*e + c*d*x)*a**3*c**2*d**2*e**3*f*g**4 + 10*sqrt(f + g*x)*sqrt(a* 
e + c*d*x)*a**3*c**2*d**2*e**3*g**5*x + 128*sqrt(f + g*x)*sqrt(a*e + c*d*x 
)*a**2*c**3*d**3*e**2*f**2*g**3 + 466*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2 
*c**3*d**3*e**2*f*g**4*x + 248*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a**2*c**3*d 
**3*e**2*g**5*x**2 - 70*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e*f**3 
*g**2 + 46*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e*f**2*g**3*x + 512 
*sqrt(f + g*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e*f*g**4*x**2 + 336*sqrt(f + 
g*x)*sqrt(a*e + c*d*x)*a*c**4*d**4*e*g**5*x**3 + 15*sqrt(f + g*x)*sqrt(a*e 
 + c*d*x)*c**5*d**5*f**4*g - 10*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**5*d**5* 
f**3*g**2*x + 8*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**5*d**5*f**2*g**3*x**2 + 
 176*sqrt(f + g*x)*sqrt(a*e + c*d*x)*c**5*d**5*f*g**4*x**3 + 128*sqrt(f + 
g*x)*sqrt(a*e + c*d*x)*c**5*d**5*g**5*x**4 + 15*sqrt(g)*sqrt(d)*sqrt(c)*lo 
g((sqrt(g)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - 
 c*d*f))*a**5*e**5*g**5 - 75*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**4*c*d*e 
**4*f*g**4 + 150*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**3*c**2*d**2*e**3*f* 
*2*g**3 - 150*sqrt(g)*sqrt(d)*sqrt(c)*log((sqrt(g)*sqrt(a*e + c*d*x) + sqr 
t(d)*sqrt(c)*sqrt(f + g*x))/sqrt(a*e*g - c*d*f))*a**2*c**3*d**3*e**2*f*...