\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx\) [33]

Optimal result

Integrand size = 46, antiderivative size = 463 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 g^3 \sqrt {d+e x} (f+g x)^4}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{192 g^3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{768 g^3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^5 d^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 g^3 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^5}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}+\frac {5 c^6 d^6 \arctan \left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{512 g^{7/2} (c d f-a e g)^{7/2}} \] Output:

-1/32*c^2*d^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)/(g 
*x+f)^4+1/192*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+ 
c*d*f)/(e*x+d)^(1/2)/(g*x+f)^3+5/768*c^4*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e* 
x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^2/(e*x+d)^(1/2)/(g*x+f)^2+5/512*c^5*d^5*(a*d 
*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/g^3/(-a*e*g+c*d*f)^3/(e*x+d)^(1/2)/(g* 
x+f)-1/12*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/g^2/(e*x+d)^(3/2)/(g 
*x+f)^5-1/6*(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 
)^6+5/512*c^6*d^6*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^(7/2)/(-a*e*g+c*d*f)^(7/2)
 

Mathematica [A] (verified)

Time = 4.49 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\frac {c^6 d^6 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {g} \left (256 a^5 e^5 g^5+640 a^4 c d e^4 g^4 (-f+g x)+16 a^3 c^2 d^2 e^3 g^3 \left (27 f^2-106 f g x+27 g^2 x^2\right )+8 a^2 c^3 d^3 e^2 g^2 \left (-f^3+159 f^2 g x-159 f g^2 x^2+g^3 x^3\right )-2 a c^4 d^4 e g \left (5 f^4+28 f^3 g x-594 f^2 g^2 x^2+28 f g^3 x^3+5 g^4 x^4\right )+c^5 d^5 \left (-15 f^5-85 f^4 g x-198 f^3 g^2 x^2+198 f^2 g^3 x^3+85 f g^4 x^4+15 g^5 x^5\right )\right )}{c^6 d^6 (c d f-a e g)^3 (a e+c d x)^2 (f+g x)^6}+\frac {15 \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{(c d f-a e g)^{7/2} (a e+c d x)^{5/2}}\right )}{1536 g^{7/2} (d+e x)^{5/2}} \] 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)^7),x]
 

Output:

(c^6*d^6*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[g]*(256*a^5*e^5*g^5 + 640* 
a^4*c*d*e^4*g^4*(-f + g*x) + 16*a^3*c^2*d^2*e^3*g^3*(27*f^2 - 106*f*g*x + 
27*g^2*x^2) + 8*a^2*c^3*d^3*e^2*g^2*(-f^3 + 159*f^2*g*x - 159*f*g^2*x^2 + 
g^3*x^3) - 2*a*c^4*d^4*e*g*(5*f^4 + 28*f^3*g*x - 594*f^2*g^2*x^2 + 28*f*g^ 
3*x^3 + 5*g^4*x^4) + c^5*d^5*(-15*f^5 - 85*f^4*g*x - 198*f^3*g^2*x^2 + 198 
*f^2*g^3*x^3 + 85*f*g^4*x^4 + 15*g^5*x^5)))/(c^6*d^6*(c*d*f - a*e*g)^3*(a* 
e + c*d*x)^2*(f + g*x)^6) + (15*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt[c* 
d*f - a*e*g]])/((c*d*f - a*e*g)^(7/2)*(a*e + c*d*x)^(5/2))))/(1536*g^(7/2) 
*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1249, 1249, 1249, 1254, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx\)

\(\Big \downarrow \) 1249

\(\displaystyle \frac {5 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^6}dx}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 1249

\(\displaystyle \frac {5 c d \left (\frac {3 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x} (f+g x)^5}dx}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 1249

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 1254

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \left (\frac {c d \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 1255

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \left (\frac {c d e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d f-a e g}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {5 c d \left (\frac {3 c d \left (\frac {c d \left (\frac {5 c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} (c d f-a e g)^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} (f+g x) (c d f-a e g)}\right )}{4 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}\right )}{6 (c d f-a e g)}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}\right )}{8 g}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}\right )}{10 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 g (d+e x)^{3/2} (f+g x)^5}\right )}{12 g}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 g (d+e x)^{5/2} (f+g x)^6}\)

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)^7),x]
 

Output:

-1/6*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(g*(d + e*x)^(5/2)*(f + 
 g*x)^6) + (5*c*d*(-1/5*(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)^5) + (3*c*d*(-1/4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x 
+ c*d*e*x^2]/(g*Sqrt[d + e*x]*(f + g*x)^4) + (c*d*(Sqrt[a*d*e + (c*d^2 + a 
*e^2)*x + c*d*e*x^2]/(3*(c*d*f - a*e*g)*Sqrt[d + e*x]*(f + g*x)^3) + (5*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*(c*d*f - a*e*g)) 
))/(8*g)))/(10*g)))/(12*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. \(1250\) vs. \(2(413)=826\).

Time = 2.87 (sec) , antiderivative size = 1251, normalized size of antiderivative = 2.70

method	result	size
default	\(\text {Expression too large to display}\)	\(1251\)

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)^7,x,meth 
od=_RETURNVERBOSE)
 

Output:

1/1536*((e*x+d)*(c*d*x+a*e))^(1/2)*(-85*c^5*d^5*f*g^4*x^4*(c*d*x+a*e)^(1/2 
)*((a*e*g-c*d*f)*g)^(1/2)+15*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g) 
^(1/2))*c^6*d^6*f^6+8*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^2*c^3*d^ 
3*e^2*f^3*g^2-15*c^5*d^5*g^5*x^5*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2) 
+90*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*f*g^5*x^5 
+10*a*c^4*d^4*e*g^5*x^4*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+15*arcta 
nh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))*c^6*d^6*g^6*x^6+1272*a^2*c 
^3*d^3*e^2*f*g^4*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-1188*a*c^4* 
d^4*e*f^2*g^3*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+1696*a^3*c^2*d 
^2*e^3*f*g^4*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-1272*a^2*c^3*d^3* 
e^2*f^2*g^3*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+56*a*c^4*d^4*e*f^3 
*g^2*x*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+56*a*c^4*d^4*e*f*g^4*x^3* 
(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-8*a^2*c^3*d^3*e^2*g^5*x^3*(c*d*x 
+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-432*a^3*c^2*d^2*e^3*g^5*x^2*(c*d*x+a*e 
)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)-640*a^4*c*d*e^4*g^5*x*(c*d*x+a*e)^(1/2)*(( 
a*e*g-c*d*f)*g)^(1/2)-256*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*a^5*e^ 
5*g^5+15*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)*c^5*d^5*f^5-198*c^5*d^5 
*f^2*g^3*x^3*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+198*c^5*d^5*f^3*g^2 
*x^2*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+85*c^5*d^5*f^4*g*x*(c*d*x+a 
*e)^(1/2)*((a*e*g-c*d*f)*g)^(1/2)+640*(c*d*x+a*e)^(1/2)*((a*e*g-c*d*f)*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1916 vs. \(2 (413) = 826\).

Time = 3.67 (sec) , antiderivative size = 3873, normalized size of antiderivative = 8.37 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Too large to display} \] 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)^7, 
x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Timed out} \] 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)**7,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{7}} \,d x } \] 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)^7, 
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)^7), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (413) = 826\).

Time = 0.24 (sec) , antiderivative size = 1132, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\text {Too large to display} \] 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)^7, 
x, algorithm="giac")
 

Output:

5/512*c^6*d^6*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqr 
t(c*d*f*g - a*e*g^2)*e))/((c^3*d^3*f^3*g^3 - 3*a*c^2*d^2*e*f^2*g^4 + 3*a^2 
*c*d*e^2*f*g^5 - a^3*e^3*g^6)*sqrt(c*d*f*g - a*e*g^2)*e) - 1/1536*(15*sqrt 
((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^11*d^11*e^10*f^5*abs(e) - 75*sqrt((e 
*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^10*d^10*e^11*f^4*g*abs(e) + 150*sqrt( 
(e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^9*d^9*e^12*f^3*g^2*abs(e) - 150*s 
qrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c^8*d^8*e^13*f^2*g^3*abs(e) + 7 
5*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^4*c^7*d^7*e^14*f*g^4*abs(e) - 
15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^5*c^6*d^6*e^15*g^5*abs(e) + 8 
5*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^10*d^10*e^8*f^4*g*abs(e) - 3 
40*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*c^9*d^9*e^9*f^3*g^2*abs(e) 
+ 510*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^8*d^8*e^10*f^2*g^3*a 
bs(e) - 340*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*c^7*d^7*e^11*f*g 
^4*abs(e) + 85*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^4*c^6*d^6*e^12* 
g^5*abs(e) + 198*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^9*d^9*e^6*f^3 
*g^2*abs(e) - 594*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^8*d^8*e^7* 
f^2*g^3*abs(e) + 594*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*c^7*d^7 
*e^8*f*g^4*abs(e) - 198*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^3*c^6* 
d^6*e^9*g^5*abs(e) - 198*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^8*d^8 
*e^4*f^2*g^3*abs(e) + 396*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*c...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^7\,{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/((f + g*x)^7*(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)^7*(d + e*x)^( 
5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 1754, normalized size of antiderivative = 3.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^7} \, dx =\text {Too large to display} \] 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)^7,x)
 

Output:

(15*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqr 
t( - a*e*g + c*d*f)))*c**6*d**6*f**6 + 90*sqrt(g)*sqrt( - a*e*g + c*d*f)*a 
tan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**6*d**6*f**5 
*g*x + 225*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**6*d**6*f**4*g**2*x**2 + 300*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**6*d**6*f**3*g**3*x**3 + 225*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sq 
rt(a*e + c*d*x)*g)/(sqrt(g)*sqrt( - a*e*g + c*d*f)))*c**6*d**6*f**2*g**4*x 
**4 + 90*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**6*d**6*f*g**5*x**5 + 15*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* 
*6*d**6*g**6*x**6 - 256*sqrt(a*e + c*d*x)*a**6*e**6*g**7 + 896*sqrt(a*e + 
c*d*x)*a**5*c*d*e**5*f*g**6 - 640*sqrt(a*e + c*d*x)*a**5*c*d*e**5*g**7*x - 
 1072*sqrt(a*e + c*d*x)*a**4*c**2*d**2*e**4*f**2*g**5 + 2336*sqrt(a*e + c* 
d*x)*a**4*c**2*d**2*e**4*f*g**6*x - 432*sqrt(a*e + c*d*x)*a**4*c**2*d**2*e 
**4*g**7*x**2 + 440*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**3*f**3*g**4 - 2968 
*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e**3*f**2*g**5*x + 1704*sqrt(a*e + c*d*x 
)*a**3*c**3*d**3*e**3*f*g**6*x**2 - 8*sqrt(a*e + c*d*x)*a**3*c**3*d**3*e** 
3*g**7*x**3 + 2*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**2*f**4*g**3 + 1328*sqr 
t(a*e + c*d*x)*a**2*c**4*d**4*e**2*f**3*g**4*x - 2460*sqrt(a*e + c*d*x)...