\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{17/2}} \, dx\) [307]

Optimal result

Integrand size = 39, antiderivative size = 366 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=-\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{16 e^3 (d+e x)^{7/2}}+\frac {c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 e^3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{8 e^2 (d+e x)^{11/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}+\frac {3 c^5 d^5 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{128 e^{7/2} \left (c d^2-a e^2\right )^{5/2}} \] Output:

-1/16*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(e*x+d)^(7/2)+1/ 
64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(-a*e^2+c*d^2)/(e*x 
+d)^(5/2)+3/128*c^4*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^3/(-a*e^ 
2+c*d^2)^2/(e*x+d)^(3/2)-1/8*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e 
^2/(e*x+d)^(11/2)-1/5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e/(e*x+d)^(1 
5/2)+3/128*c^5*d^5*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/ 
(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/2))/e^(7/2)/(-a*e^2+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\frac {c^5 d^5 ((a e+c d x) (d+e x))^{5/2} \left (-\frac {\sqrt {e} \left (128 a^4 e^8+16 a^3 c d e^6 (-11 d+21 e x)+8 a^2 c^2 d^2 e^4 \left (d^2-64 d e x+31 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (5 d^3+23 d^2 e x-233 d e^2 x^2+5 e^3 x^3\right )+c^4 d^4 \left (15 d^4+70 d^3 e x+128 d^2 e^2 x^2-70 d e^3 x^3-15 e^4 x^4\right )\right )}{c^5 d^5 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 (d+e x)^5}+\frac {15 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2} (a e+c d x)^{5/2}}\right )}{640 e^{7/2} (d+e x)^{5/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x 
]
 

Output:

(c^5*d^5*((a*e + c*d*x)*(d + e*x))^(5/2)*(-((Sqrt[e]*(128*a^4*e^8 + 16*a^3 
*c*d*e^6*(-11*d + 21*e*x) + 8*a^2*c^2*d^2*e^4*(d^2 - 64*d*e*x + 31*e^2*x^2 
) + 2*a*c^3*d^3*e^2*(5*d^3 + 23*d^2*e*x - 233*d*e^2*x^2 + 5*e^3*x^3) + c^4 
*d^4*(15*d^4 + 70*d^3*e*x + 128*d^2*e^2*x^2 - 70*d*e^3*x^3 - 15*e^4*x^4))) 
/(c^5*d^5*(c*d^2 - a*e^2)^2*(a*e + c*d*x)^2*(d + e*x)^5)) + (15*ArcTan[(Sq 
rt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/((c*d^2 - a*e^2)^(5/2)*(a*e 
 + c*d*x)^(5/2))))/(640*e^(7/2)*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1130, 1130, 1130, 1135, 1135, 1136, 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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{13/2}}dx}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {c d \left (\frac {3 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{(d+e x)^{9/2}}dx}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \int \frac {1}{(d+e x)^{5/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \left (\frac {3 c d \int \frac {1}{(d+e x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \left (\frac {3 c d \left (\frac {c d \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \left (\frac {3 c d \left (\frac {c d e \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {c d \left (\frac {3 c d \left (\frac {c d \left (\frac {3 c d \left (\frac {c d \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}\right )}{6 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e (d+e x)^{15/2}}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x]
 

Output:

-1/5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(e*(d + e*x)^(15/2)) + 
(c*d*(-1/4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(e*(d + e*x)^(11/ 
2)) + (3*c*d*(-1/3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(d + e*x 
)^(7/2)) + (c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(2*(c*d^2 - a 
*e^2)*(d + e*x)^(5/2)) + (3*c*d*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^ 
2]/((c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (c*d*ArcTan[(Sqrt[e]*Sqrt[a*d*e + ( 
c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(Sqrt 
[e]*(c*d^2 - a*e^2)^(3/2))))/(4*(c*d^2 - a*e^2))))/(6*e)))/(8*e)))/(2*e)
 

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 1130

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + p + 1))), x] 
- Simp[c*(p/(e^2*(m + p + 1)))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p 
 - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && GtQ[p, 0] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + p + 1, 0] & 
& IntegerQ[2*p]
 

rule 1135

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e)))   Int 
[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && I 
ntegerQ[2*p]
 

rule 1136

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x 
_Symbol] :> Simp[2*e   Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + 
 b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, 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. \(899\) vs. \(2(322)=644\).

Time = 1.09 (sec) , antiderivative size = 900, normalized size of antiderivative = 2.46

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{5} d^{5} e^{5} x^{5}+75 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{5} d^{6} e^{4} x^{4}+150 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{5} d^{7} e^{3} x^{3}+150 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{5} d^{8} e^{2} x^{2}-15 c^{4} d^{4} e^{4} x^{4} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+75 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{5} d^{9} e x +10 a \,c^{3} d^{3} e^{5} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-70 c^{4} d^{5} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{5} d^{10}+248 a^{2} c^{2} d^{2} e^{6} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-466 a \,c^{3} d^{4} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+128 c^{4} d^{6} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+336 a^{3} c d \,e^{7} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-512 a^{2} c^{2} d^{3} e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+46 a \,c^{3} d^{5} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+70 c^{4} d^{7} e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+128 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{4} e^{8}-176 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{3} c \,d^{2} e^{6}+8 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} c^{2} d^{4} e^{4}+10 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,c^{3} d^{6} e^{2}+15 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{4} d^{8}\right )}{640 \left (e x +d \right )^{\frac {11}{2}} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, e^{3} \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\)	\(900\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(17/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-1/640*((e*x+d)*(c*d*x+a*e))^(1/2)*(15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2 
-c*d^2)*e)^(1/2))*c^5*d^5*e^5*x^5+75*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c 
*d^2)*e)^(1/2))*c^5*d^6*e^4*x^4+150*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c* 
d^2)*e)^(1/2))*c^5*d^7*e^3*x^3+150*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d 
^2)*e)^(1/2))*c^5*d^8*e^2*x^2-15*c^4*d^4*e^4*x^4*(c*d*x+a*e)^(1/2)*((a*e^2 
-c*d^2)*e)^(1/2)+75*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c 
^5*d^9*e*x+10*a*c^3*d^3*e^5*x^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)- 
70*c^4*d^5*e^3*x^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+15*arctanh(e* 
(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^5*d^10+248*a^2*c^2*d^2*e^6*x^ 
2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-466*a*c^3*d^4*e^4*x^2*(c*d*x+a 
*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+128*c^4*d^6*e^2*x^2*(c*d*x+a*e)^(1/2)*(( 
a*e^2-c*d^2)*e)^(1/2)+336*a^3*c*d*e^7*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e 
)^(1/2)-512*a^2*c^2*d^3*e^5*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+46 
*a*c^3*d^5*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+70*c^4*d^7*e*x* 
(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+128*(c*d*x+a*e)^(1/2)*((a*e^2-c* 
d^2)*e)^(1/2)*a^4*e^8-176*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^3*c* 
d^2*e^6+8*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^4*e^4+10*(c* 
d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^6*e^2+15*(c*d*x+a*e)^(1/2)* 
((a*e^2-c*d^2)*e)^(1/2)*c^4*d^8)/(e*x+d)^(11/2)/((a*e^2-c*d^2)*e)^(1/2)/e^ 
3/(a*e^2-c*d^2)^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. 832 vs. \(2 (322) = 644\).

Time = 0.17 (sec) , antiderivative size = 1686, normalized size of antiderivative = 4.61 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x, algori 
thm="fricas")
 

Output:

[-1/1280*(15*(c^5*d^5*e^6*x^6 + 6*c^5*d^6*e^5*x^5 + 15*c^5*d^7*e^4*x^4 + 2 
0*c^5*d^8*e^3*x^3 + 15*c^5*d^9*e^2*x^2 + 6*c^5*d^10*e*x + c^5*d^11)*sqrt(- 
c*d^2*e + a*e^3)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 - 2*sqr 
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d^2*e + a*e^3)*sqrt(e*x + 
 d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(15*c^5*d^10*e - 5*a*c^4*d^8*e^3 - 2*a 
^2*c^3*d^6*e^5 - 184*a^3*c^2*d^4*e^7 + 304*a^4*c*d^2*e^9 - 128*a^5*e^11 - 
15*(c^5*d^6*e^5 - a*c^4*d^4*e^7)*x^4 - 10*(7*c^5*d^7*e^4 - 8*a*c^4*d^5*e^6 
 + a^2*c^3*d^3*e^8)*x^3 + 2*(64*c^5*d^8*e^3 - 297*a*c^4*d^6*e^5 + 357*a^2* 
c^3*d^4*e^7 - 124*a^3*c^2*d^2*e^9)*x^2 + 2*(35*c^5*d^9*e^2 - 12*a*c^4*d^7* 
e^4 - 279*a^2*c^3*d^5*e^6 + 424*a^3*c^2*d^3*e^8 - 168*a^4*c*d*e^10)*x)*sqr 
t(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^3*d^12*e^4 - 3* 
a*c^2*d^10*e^6 + 3*a^2*c*d^8*e^8 - a^3*d^6*e^10 + (c^3*d^6*e^10 - 3*a*c^2* 
d^4*e^12 + 3*a^2*c*d^2*e^14 - a^3*e^16)*x^6 + 6*(c^3*d^7*e^9 - 3*a*c^2*d^5 
*e^11 + 3*a^2*c*d^3*e^13 - a^3*d*e^15)*x^5 + 15*(c^3*d^8*e^8 - 3*a*c^2*d^6 
*e^10 + 3*a^2*c*d^4*e^12 - a^3*d^2*e^14)*x^4 + 20*(c^3*d^9*e^7 - 3*a*c^2*d 
^7*e^9 + 3*a^2*c*d^5*e^11 - a^3*d^3*e^13)*x^3 + 15*(c^3*d^10*e^6 - 3*a*c^2 
*d^8*e^8 + 3*a^2*c*d^6*e^10 - a^3*d^4*e^12)*x^2 + 6*(c^3*d^11*e^5 - 3*a*c^ 
2*d^9*e^7 + 3*a^2*c*d^7*e^9 - a^3*d^5*e^11)*x), -1/640*(15*(c^5*d^5*e^6*x^ 
6 + 6*c^5*d^6*e^5*x^5 + 15*c^5*d^7*e^4*x^4 + 20*c^5*d^8*e^3*x^3 + 15*c^5*d 
^9*e^2*x^2 + 6*c^5*d^10*e*x + c^5*d^11)*sqrt(c*d^2*e - a*e^3)*arctan(-s...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(17/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {17}{2}}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x, algori 
thm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(17/2), 
x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (322) = 644\).

Time = 0.22 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x, algori 
thm="giac")
 

Output:

1/640*c^5*d^5*e*(15*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/ 
sqrt(c*d^2*e - a*e^3))/((c^2*d^4*e^5 - 2*a*c*d^2*e^7 + a^2*e^9)*sqrt(c*d^2 
*e - a*e^3)) - (15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^4*d^8*e^4*abs 
(e) - 60*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^3*d^6*e^6*abs(e) + 90 
*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^2*d^4*e^8*abs(e) - 60*sqrt( 
(e*x + d)*c*d*e - c*d^2*e + a*e^3)*a^3*c*d^2*e^10*abs(e) + 15*sqrt((e*x + 
d)*c*d*e - c*d^2*e + a*e^3)*a^4*e^12*abs(e) + 70*((e*x + d)*c*d*e - c*d^2* 
e + a*e^3)^(3/2)*c^3*d^6*e^3*abs(e) - 210*((e*x + d)*c*d*e - c*d^2*e + a*e 
^3)^(3/2)*a*c^2*d^4*e^5*abs(e) + 210*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^( 
3/2)*a^2*c*d^2*e^7*abs(e) - 70*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a 
^3*e^9*abs(e) + 128*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^2*d^4*e^2* 
abs(e) - 256*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c*d^2*e^4*abs(e) 
+ 128*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a^2*e^6*abs(e) - 70*((e*x 
+ d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c*d^2*e*abs(e) + 70*((e*x + d)*c*d*e - 
 c*d^2*e + a*e^3)^(7/2)*a*e^3*abs(e) - 15*((e*x + d)*c*d*e - c*d^2*e + a*e 
^3)^(9/2)*abs(e))/((c^2*d^4*e^5 - 2*a*c*d^2*e^7 + a^2*e^9)*(e*x + d)^5*c^5 
*d^5*e^5))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{17/2}} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(17/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1128, normalized size of antiderivative = 3.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{17/2}} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(17/2),x)
 

Output:

( - 15*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e 
)*sqrt( - a*e**2 + c*d**2)))*c**5*d**10 - 75*sqrt(e)*sqrt( - a*e**2 + c*d* 
*2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c**5*d* 
*9*e*x - 150*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/( 
sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c**5*d**8*e**2*x**2 - 150*sqrt(e)*sqrt( 
 - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c 
*d**2)))*c**5*d**7*e**3*x**3 - 75*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((s 
qrt(a*e + c*d*x)*e)/(sqrt(e)*sqrt( - a*e**2 + c*d**2)))*c**5*d**6*e**4*x** 
4 - 15*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e 
)*sqrt( - a*e**2 + c*d**2)))*c**5*d**5*e**5*x**5 - 128*sqrt(a*e + c*d*x)*a 
**5*e**11 + 304*sqrt(a*e + c*d*x)*a**4*c*d**2*e**9 - 336*sqrt(a*e + c*d*x) 
*a**4*c*d*e**10*x - 184*sqrt(a*e + c*d*x)*a**3*c**2*d**4*e**7 + 848*sqrt(a 
*e + c*d*x)*a**3*c**2*d**3*e**8*x - 248*sqrt(a*e + c*d*x)*a**3*c**2*d**2*e 
**9*x**2 - 2*sqrt(a*e + c*d*x)*a**2*c**3*d**6*e**5 - 558*sqrt(a*e + c*d*x) 
*a**2*c**3*d**5*e**6*x + 714*sqrt(a*e + c*d*x)*a**2*c**3*d**4*e**7*x**2 - 
10*sqrt(a*e + c*d*x)*a**2*c**3*d**3*e**8*x**3 - 5*sqrt(a*e + c*d*x)*a*c**4 
*d**8*e**3 - 24*sqrt(a*e + c*d*x)*a*c**4*d**7*e**4*x - 594*sqrt(a*e + c*d* 
x)*a*c**4*d**6*e**5*x**2 + 80*sqrt(a*e + c*d*x)*a*c**4*d**5*e**6*x**3 + 15 
*sqrt(a*e + c*d*x)*a*c**4*d**4*e**7*x**4 + 15*sqrt(a*e + c*d*x)*c**5*d**10 
*e + 70*sqrt(a*e + c*d*x)*c**5*d**9*e**2*x + 128*sqrt(a*e + c*d*x)*c**5...