\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx\) [20]

Optimal result

Integrand size = 46, antiderivative size = 335 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=-\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g^2 \sqrt {d+e x} (f+g x)^3}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^2 (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g^2 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 g (d+e x)^{3/2} (f+g x)^4}+\frac {3 c^4 d^4 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{64 g^{5/2} (c d f-a e g)^{5/2}} \] Output:

-1/8*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(e*x+d)^(1/2)/(g*x+f) 
^3+1/32*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^2/(-a*e*g+c*d*f) 
/(e*x+d)^(1/2)/(g*x+f)^2+3/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1 
/2)/g^2/(-a*e*g+c*d*f)^2/(e*x+d)^(1/2)/(g*x+f)-1/4*(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)^4+3/64*c^4*d^4*arctan(g^(1/2)*(a* 
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e*g+c*d*f)^(1/2)/(e*x+d)^(1/2))/g 
^(5/2)/(-a*e*g+c*d*f)^(5/2)
 

Mathematica [A] (verified)

Time = 1.82 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.72 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\frac {c^4 d^4 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {g} \left (-16 a^3 e^3 g^3+24 a^2 c d e^2 g^2 (f-g x)-2 a c^2 d^2 e g \left (f^2-22 f g x+g^2 x^2\right )+c^3 d^3 \left (-3 f^3-11 f^2 g x+11 f g^2 x^2+3 g^3 x^3\right )\right )}{c^4 d^4 (c d f-a e g)^2 (a e+c d x) (f+g x)^4}+\frac {3 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{5/2} (a e+c d x)^{3/2}}\right )}{64 g^{5/2} (d+e x)^{3/2}} \] 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)^5),x]
 

Output:

(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[g]*(-16*a^3*e^3*g^3 + 24*a 
^2*c*d*e^2*g^2*(f - g*x) - 2*a*c^2*d^2*e*g*(f^2 - 22*f*g*x + g^2*x^2) + c^ 
3*d^3*(-3*f^3 - 11*f^2*g*x + 11*f*g^2*x^2 + 3*g^3*x^3)))/(c^4*d^4*(c*d*f - 
 a*e*g)^2*(a*e + c*d*x)*(f + g*x)^4) + (3*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x 
])/Sqrt[c*d*f - a*e*g]])/((c*d*f - a*e*g)^(5/2)*(a*e + c*d*x)^(3/2))))/(64 
*g^(5/2)*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1249, 1249, 1254, 1254, 1255, 218}

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.

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1249

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1254

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

\(\Big \downarrow \) 1255

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

\(\Big \downarrow \) 218

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

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)^5),x]
 

Output:

-1/4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(g*(d + e*x)^(3/2)*(f + 
 g*x)^4) + (3*c*d*(-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(g*Sqr 
t[d + e*x]*(f + g*x)^3) + (c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2 
]/(2*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^2) + (3*c*d*(Sqrt[a*d*e + (c* 
d^2 + a*e^2)*x + c*d*e*x^2]/((c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)) + (c 
*d*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d* 
f - a*e*g]*Sqrt[d + e*x])])/(Sqrt[g]*(c*d*f - a*e*g)^(3/2))))/(4*(c*d*f - 
a*e*g))))/(6*g)))/(8*g)
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[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 1254

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

rule 1255

Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + 
 (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2   Subst[Int[1/(c*(e*f + d*g) - b*e 
*g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ 
a, b, c, d, e, f, g}, 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. \(654\) vs. \(2(297)=594\).

Time = 2.75 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.96

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} g^{4} x^{4}+12 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f \,g^{3} x^{3}+18 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f^{2} g^{2} x^{2}+12 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f^{3} g x -3 c^{3} d^{3} g^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+3 \,\operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right ) c^{4} d^{4} f^{4}+2 a \,c^{2} d^{2} e \,g^{3} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-11 c^{3} d^{3} f \,g^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+24 a^{2} c d \,e^{2} g^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}-44 a \,c^{2} d^{2} e f \,g^{2} x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+11 c^{3} d^{3} f^{2} g x \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}+16 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a^{3} e^{3} g^{3}-24 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a^{2} c d \,e^{2} f \,g^{2}+2 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, a \,c^{2} d^{2} e \,f^{2} g +3 \sqrt {\left (a e g -d f c \right ) g}\, \sqrt {c d x +a e}\, c^{3} d^{3} f^{3}\right )}{64 \sqrt {e x +d}\, \sqrt {\left (a e g -d f c \right ) g}\, \left (g x +f \right )^{4} g^{2} \left (a e g -d f c \right )^{2} \sqrt {c d x +a e}}\)	\(655\)

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)^5,x,meth 
od=_RETURNVERBOSE)
 

Output:

-1/64*((e*x+d)*(c*d*x+a*e))^(1/2)*(3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*c^4*d^4*g^4*x^4+12*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d 
*f)*g)^(1/2))*c^4*d^4*f*g^3*x^3+18*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d 
*f)*g)^(1/2))*c^4*d^4*f^2*g^2*x^2+12*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c 
*d*f)*g)^(1/2))*c^4*d^4*f^3*g*x-3*c^3*d^3*g^3*x^3*(c*d*x+a*e)^(1/2)*((a*e* 
g-c*d*f)*g)^(1/2)+3*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c 
^4*d^4*f^4+2*a*c^2*d^2*e*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2) 
-11*c^3*d^3*f*g^2*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+24*a^2*c*d 
*e^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-44*a*c^2*d^2*e*f*g^2* 
x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+11*c^3*d^3*f^2*g*x*(c*d*x+a*e) 
^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+16*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2 
)*a^3*e^3*g^3-24*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d*e^2*f*g 
^2+2*((a*e*g-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*a*c^2*d^2*e*f^2*g+3*((a*e*g 
-c*d*f)*g)^(1/2)*(c*d*x+a*e)^(1/2)*c^3*d^3*f^3)/(e*x+d)^(1/2)/((a*e*g-c*d* 
f)*g)^(1/2)/(g*x+f)^4/g^2/(a*e*g-c*d*f)^2/(c*d*x+a*e)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1099 vs. \(2 (297) = 594\).

Time = 0.57 (sec) , antiderivative size = 2239, normalized size of antiderivative = 6.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\text {Too large to display} \] 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)^5, 
x, algorithm="fricas")
 

Output:

[-1/128*(3*(c^4*d^4*e*g^4*x^5 + c^4*d^5*f^4 + (4*c^4*d^4*e*f*g^3 + c^4*d^5 
*g^4)*x^4 + 2*(3*c^4*d^4*e*f^2*g^2 + 2*c^4*d^5*f*g^3)*x^3 + 2*(2*c^4*d^4*e 
*f^3*g + 3*c^4*d^5*f^2*g^2)*x^2 + (c^4*d^4*e*f^4 + 4*c^4*d^5*f^3*g)*x)*sqr 
t(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d*e*f - 
 (c*d^2 + 2*a*e^2)*g)*x - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq 
rt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x)) + 2 
*(3*c^4*d^4*f^4*g - a*c^3*d^3*e*f^3*g^2 - 26*a^2*c^2*d^2*e^2*f^2*g^3 + 40* 
a^3*c*d*e^3*f*g^4 - 16*a^4*e^4*g^5 - 3*(c^4*d^4*f*g^4 - a*c^3*d^3*e*g^5)*x 
^3 - (11*c^4*d^4*f^2*g^3 - 13*a*c^3*d^3*e*f*g^4 + 2*a^2*c^2*d^2*e^2*g^5)*x 
^2 + (11*c^4*d^4*f^3*g^2 - 55*a*c^3*d^3*e*f^2*g^3 + 68*a^2*c^2*d^2*e^2*f*g 
^4 - 24*a^3*c*d*e^3*g^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sq 
rt(e*x + d))/(c^3*d^4*f^7*g^3 - 3*a*c^2*d^3*e*f^6*g^4 + 3*a^2*c*d^2*e^2*f^ 
5*g^5 - a^3*d*e^3*f^4*g^6 + (c^3*d^3*e*f^3*g^7 - 3*a*c^2*d^2*e^2*f^2*g^8 + 
 3*a^2*c*d*e^3*f*g^9 - a^3*e^4*g^10)*x^5 + (4*c^3*d^3*e*f^4*g^6 - a^3*d*e^ 
3*g^10 + (c^3*d^4 - 12*a*c^2*d^2*e^2)*f^3*g^7 - 3*(a*c^2*d^3*e - 4*a^2*c*d 
*e^3)*f^2*g^8 + (3*a^2*c*d^2*e^2 - 4*a^3*e^4)*f*g^9)*x^4 + 2*(3*c^3*d^3*e* 
f^5*g^5 - 2*a^3*d*e^3*f*g^9 + (2*c^3*d^4 - 9*a*c^2*d^2*e^2)*f^4*g^6 - 3*(2 
*a*c^2*d^3*e - 3*a^2*c*d*e^3)*f^3*g^7 + 3*(2*a^2*c*d^2*e^2 - a^3*e^4)*f^2* 
g^8)*x^3 + 2*(2*c^3*d^3*e*f^6*g^4 - 3*a^3*d*e^3*f^2*g^8 + 3*(c^3*d^4 - 2*a 
*c^2*d^2*e^2)*f^5*g^5 - 3*(3*a*c^2*d^3*e - 2*a^2*c*d*e^3)*f^4*g^6 + (9*...
 

Sympy [F(-1)]

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

Output:

Timed out
 

Maxima [F]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 604 vs. \(2 (297) = 594\).

Time = 0.18 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx=\frac {3 \, c^{4} d^{4} {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{64 \, {\left (c^{2} d^{2} f^{2} g^{2} - 2 \, a c d e f g^{3} + a^{2} e^{2} g^{4}\right )} \sqrt {c d f g - a e g^{2}} e} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{7} e^{6} f^{3} {\left | e \right |} - 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{6} e^{7} f^{2} g {\left | e \right |} + 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{5} d^{5} e^{8} f g^{2} {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{4} d^{4} e^{9} g^{3} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{6} e^{4} f^{2} g {\left | e \right |} - 22 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{5} d^{5} e^{5} f g^{2} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{6} g^{3} {\left | e \right |} - 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{5} d^{5} e^{2} f g^{2} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{4} d^{4} e^{3} g^{3} {\left | e \right |} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{4} d^{4} g^{3} {\left | e \right |}}{64 \, {\left (c^{2} d^{2} f^{2} g^{2} - 2 \, a c d e f g^{3} + a^{2} e^{2} g^{4}\right )} {\left (c d e^{2} f - a e^{3} g + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} g\right )}^{4}} \] 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)^5, 
x, algorithm="giac")
 

Output:

3/64*c^4*d^4*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt 
(c*d*f*g - a*e*g^2)*e))/((c^2*d^2*f^2*g^2 - 2*a*c*d*e*f*g^3 + a^2*e^2*g^4) 
*sqrt(c*d*f*g - a*e*g^2)*e) - 1/64*(3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e 
^3)*c^7*d^7*e^6*f^3*abs(e) - 9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c 
^6*d^6*e^7*f^2*g*abs(e) + 9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^ 
5*d^5*e^8*f*g^2*abs(e) - 3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^4 
*d^4*e^9*g^3*abs(e) + 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^6*d^6 
*e^4*f^2*g*abs(e) - 22*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^5*d^5 
*e^5*f*g^2*abs(e) + 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^4*d 
^4*e^6*g^3*abs(e) - 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^5*d^5*e 
^2*f*g^2*abs(e) + 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^4*d^4*e 
^3*g^3*abs(e) - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^4*d^4*g^3*ab 
s(e))/((c^2*d^2*f^2*g^2 - 2*a*c*d*e*f*g^3 + a^2*e^2*g^4)*(c*d*e^2*f - a*e^ 
3*g + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g)^4)
 

Mupad [F(-1)]

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

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

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 962, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^5} \, dx =\text {Too large to display} \] 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)^5,x)
 

Output:

( - 3*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*s 
qrt( - a*e*g + c*d*f)))*c**4*d**4*f**4 - 12*sqrt(g)*sqrt( - a*e*g + c*d*f) 
*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**4*d**4*f* 
*3*g*x - 18*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqr 
t(g)*sqrt( - a*e*g + c*d*f)))*c**4*d**4*f**2*g**2*x**2 - 12*sqrt(g)*sqrt( 
- a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f 
)))*c**4*d**4*f*g**3*x**3 - 3*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a* 
e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**4*d**4*g**4*x**4 - 16*s 
qrt(a*e + c*d*x)*a**4*e**4*g**5 + 40*sqrt(a*e + c*d*x)*a**3*c*d*e**3*f*g** 
4 - 24*sqrt(a*e + c*d*x)*a**3*c*d*e**3*g**5*x - 26*sqrt(a*e + c*d*x)*a**2* 
c**2*d**2*e**2*f**2*g**3 + 68*sqrt(a*e + c*d*x)*a**2*c**2*d**2*e**2*f*g**4 
*x - 2*sqrt(a*e + c*d*x)*a**2*c**2*d**2*e**2*g**5*x**2 - sqrt(a*e + c*d*x) 
*a*c**3*d**3*e*f**3*g**2 - 55*sqrt(a*e + c*d*x)*a*c**3*d**3*e*f**2*g**3*x 
+ 13*sqrt(a*e + c*d*x)*a*c**3*d**3*e*f*g**4*x**2 + 3*sqrt(a*e + c*d*x)*a*c 
**3*d**3*e*g**5*x**3 + 3*sqrt(a*e + c*d*x)*c**4*d**4*f**4*g + 11*sqrt(a*e 
+ c*d*x)*c**4*d**4*f**3*g**2*x - 11*sqrt(a*e + c*d*x)*c**4*d**4*f**2*g**3* 
x**2 - 3*sqrt(a*e + c*d*x)*c**4*d**4*f*g**4*x**3)/(64*g**3*(a**3*e**3*f**4 
*g**3 + 4*a**3*e**3*f**3*g**4*x + 6*a**3*e**3*f**2*g**5*x**2 + 4*a**3*e**3 
*f*g**6*x**3 + a**3*e**3*g**7*x**4 - 3*a**2*c*d*e**2*f**5*g**2 - 12*a**2*c 
*d*e**2*f**4*g**3*x - 18*a**2*c*d*e**2*f**3*g**4*x**2 - 12*a**2*c*d*e**...