\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{13/2}} \, dx\) [305]

Optimal result

Integrand size = 39, antiderivative size = 236 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^3 (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 e^2 (d+e x)^{7/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}+\frac {5 c^3 d^3 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{8 e^{7/2} \sqrt {c d^2-a e^2}} \] Output:

-5/8*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)^(3/2)-5/1 
2*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(7/2)-1/3*(a*d*e 
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e/(e*x+d)^(11/2)+5/8*c^3*d^3*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)^(1/2)
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {e} \left (8 a^2 e^4+2 a c d e^2 (5 d+13 e x)+c^2 d^2 \left (15 d^2+40 d e x+33 e^2 x^2\right )\right )+\frac {15 c^3 d^3 (d+e x)^3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\sqrt {c d^2-a e^2} \sqrt {a e+c d x}}\right )}{24 e^{7/2} (d+e x)^{7/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x 
]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[e]*(8*a^2*e^4 + 2*a*c*d*e^2*(5*d + 
13*e*x) + c^2*d^2*(15*d^2 + 40*d*e*x + 33*e^2*x^2))) + (15*c^3*d^3*(d + e* 
x)^3*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt[c*d^2 
- a*e^2]*Sqrt[a*e + c*d*x])))/(24*e^(7/2)*(d + e*x)^(7/2))
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1130, 1130, 1130, 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.

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

\(\Big \downarrow \) 1130

\(\displaystyle \frac {5 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{9/2}}dx}{6 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 e (d+e x)^{11/2}}\)

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x]
 

Output:

-1/3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(e*(d + e*x)^(11/2)) + 
(5*c*d*(-1/2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(e*(d + e*x)^(7 
/2)) + (3*c*d*(-(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(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])])/(e^(3/2)*Sqrt[c*d^2 - a*e^2])))/(4 
*e)))/(6*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 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. \(432\) vs. \(2(204)=408\).

Time = 1.08 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.83

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{3} e^{3} x^{3}+45 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{4} e^{2} x^{2}+45 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{5} e x +15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{3} d^{6}+33 c^{2} d^{2} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+26 a c d \,e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+40 c^{2} d^{3} e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}+10 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}+15 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {c d x +a e}\, e^{3} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(433\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(13/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-1/24*((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^3*d^3*e^3*x^3+45*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c* 
d^2)*e)^(1/2))*c^3*d^4*e^2*x^2+45*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^ 
2)*e)^(1/2))*c^3*d^5*e*x+15*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^ 
(1/2))*c^3*d^6+33*c^2*d^2*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2 
)+26*a*c*d*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+40*c^2*d^3*e*x* 
(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+8*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x 
+a*e)^(1/2)*a^2*e^4+10*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a*c*d^2*e 
^2+15*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c^2*d^4)/(e*x+d)^(7/2)/(c* 
d*x+a*e)^(1/2)/e^3/((a*e^2-c*d^2)*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 836, normalized size of antiderivative = 3.54 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),x, algori 
thm="fricas")
 

Output:

[-1/48*(15*(c^3*d^3*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^ 
3*d^6*e*x + c^3*d^7)*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*sqrt(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^3*d^6 
*e - 5*a*c^2*d^4*e^3 - 2*a^2*c*d^2*e^5 - 8*a^3*e^7 + 33*(c^3*d^4*e^3 - a*c 
^2*d^2*e^5)*x^2 + 2*(20*c^3*d^5*e^2 - 7*a*c^2*d^3*e^4 - 13*a^2*c*d*e^6)*x) 
*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c*d^6*e^4 - a 
*d^4*e^6 + (c*d^2*e^8 - a*e^10)*x^4 + 4*(c*d^3*e^7 - a*d*e^9)*x^3 + 6*(c*d 
^4*e^6 - a*d^2*e^8)*x^2 + 4*(c*d^5*e^5 - a*d^3*e^7)*x), -1/24*(15*(c^3*d^3 
*e^4*x^4 + 4*c^3*d^4*e^3*x^3 + 6*c^3*d^5*e^2*x^2 + 4*c^3*d^6*e*x + c^3*d^7 
)*sqrt(c*d^2*e - a*e^3)*arctan(-sqrt(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)/(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3) 
*x)) + (15*c^3*d^6*e - 5*a*c^2*d^4*e^3 - 2*a^2*c*d^2*e^5 - 8*a^3*e^7 + 33* 
(c^3*d^4*e^3 - a*c^2*d^2*e^5)*x^2 + 2*(20*c^3*d^5*e^2 - 7*a*c^2*d^3*e^4 - 
13*a^2*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + 
d))/(c*d^6*e^4 - a*d^4*e^6 + (c*d^2*e^8 - a*e^10)*x^4 + 4*(c*d^3*e^7 - a*d 
*e^9)*x^3 + 6*(c*d^4*e^6 - a*d^2*e^8)*x^2 + 4*(c*d^5*e^5 - a*d^3*e^7)*x)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),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)^(13/2), 
x)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {c^{3} d^{3} {\left (\frac {15 \, {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}} e^{3}} - \frac {15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{2} d^{4} e^{2} {\left | e \right |} - 30 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c d^{2} e^{4} {\left | e \right |} + 15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} e^{6} {\left | e \right |} + 40 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c d^{2} e {\left | e \right |} - 40 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} {\left | e \right |} + 33 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} {\left | e \right |}}{{\left (e x + d\right )}^{3} c^{3} d^{3} e^{6}}\right )}}{24 \, e} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),x, algori 
thm="giac")
 

Output:

1/24*c^3*d^3*(15*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sqr 
t(c*d^2*e - a*e^3))/(sqrt(c*d^2*e - a*e^3)*e^3) - (15*sqrt((e*x + d)*c*d*e 
 - c*d^2*e + a*e^3)*c^2*d^4*e^2*abs(e) - 30*sqrt((e*x + d)*c*d*e - c*d^2*e 
 + a*e^3)*a*c*d^2*e^4*abs(e) + 15*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)* 
a^2*e^6*abs(e) + 40*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c*d^2*e*abs( 
e) - 40*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a*e^3*abs(e) + 33*((e*x 
+ d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*abs(e))/((e*x + d)^3*c^3*d^3*e^6))/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(13/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.12 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{13/2}} \, dx=\frac {-15 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{6}-45 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{5} e x -45 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{4} e^{2} x^{2}-15 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{3} d^{3} e^{3} x^{3}-8 \sqrt {c d x +a e}\, a^{3} e^{7}-2 \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{5}-26 \sqrt {c d x +a e}\, a^{2} c d \,e^{6} x -5 \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{3}-14 \sqrt {c d x +a e}\, a \,c^{2} d^{3} e^{4} x -33 \sqrt {c d x +a e}\, a \,c^{2} d^{2} e^{5} x^{2}+15 \sqrt {c d x +a e}\, c^{3} d^{6} e +40 \sqrt {c d x +a e}\, c^{3} d^{5} e^{2} x +33 \sqrt {c d x +a e}\, c^{3} d^{4} e^{3} x^{2}}{24 e^{4} \left (a \,e^{5} x^{3}-c \,d^{2} e^{3} x^{3}+3 a d \,e^{4} x^{2}-3 c \,d^{3} e^{2} x^{2}+3 a \,d^{2} e^{3} x -3 c \,d^{4} e x +a \,d^{3} e^{2}-c \,d^{5}\right )} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(13/2),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**3*d**6 - 45*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**3*d** 
5*e*x - 45*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sq 
rt(e)*sqrt( - a*e**2 + c*d**2)))*c**3*d**4*e**2*x**2 - 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**3*d**3*e**3*x**3 - 8*sqrt(a*e + c*d*x)*a**3*e**7 - 2*sqrt(a*e + c 
*d*x)*a**2*c*d**2*e**5 - 26*sqrt(a*e + c*d*x)*a**2*c*d*e**6*x - 5*sqrt(a*e 
 + c*d*x)*a*c**2*d**4*e**3 - 14*sqrt(a*e + c*d*x)*a*c**2*d**3*e**4*x - 33* 
sqrt(a*e + c*d*x)*a*c**2*d**2*e**5*x**2 + 15*sqrt(a*e + c*d*x)*c**3*d**6*e 
 + 40*sqrt(a*e + c*d*x)*c**3*d**5*e**2*x + 33*sqrt(a*e + c*d*x)*c**3*d**4* 
e**3*x**2)/(24*e**4*(a*d**3*e**2 + 3*a*d**2*e**3*x + 3*a*d*e**4*x**2 + a*e 
**5*x**3 - c*d**5 - 3*c*d**4*e*x - 3*c*d**3*e**2*x**2 - c*d**2*e**3*x**3))