\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{15/2}} \, dx\) [306]

Optimal result

Integrand size = 39, antiderivative size = 301 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^3 (d+e x)^{5/2}}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{24 e^2 (d+e x)^{9/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}+\frac {5 c^4 d^4 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 e^{7/2} \left (c d^2-a e^2\right )^{3/2}} \] Output:

-5/32*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)^(5/2)+5/ 
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)^(3/2)-5/24*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e^2/(e*x+d)^(9/ 
2)-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/e/(e*x+d)^(13/2)+5/64*c^4*d 
^4*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)^(3/2)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\frac {c^4 d^4 ((a e+c d x) (d+e x))^{5/2} \left (\frac {\sqrt {e} \left (48 a^3 e^6-8 a^2 c d e^4 (d-17 e x)-2 a c^2 d^2 e^2 \left (5 d^2+18 d e x-59 e^2 x^2\right )-c^3 d^3 \left (15 d^3+55 d^2 e x+73 d e^2 x^2-15 e^3 x^3\right )\right )}{c^4 d^4 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^4}+\frac {15 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2} (a e+c d x)^{5/2}}\right )}{192 e^{7/2} (d+e x)^{5/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x 
]
 

Output:

(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(5/2)*((Sqrt[e]*(48*a^3*e^6 - 8*a^2*c*d 
*e^4*(d - 17*e*x) - 2*a*c^2*d^2*e^2*(5*d^2 + 18*d*e*x - 59*e^2*x^2) - c^3* 
d^3*(15*d^3 + 55*d^2*e*x + 73*d*e^2*x^2 - 15*e^3*x^3)))/(c^4*d^4*(c*d^2 - 
a*e^2)*(a*e + c*d*x)^2*(d + e*x)^4) + (15*ArcTan[(Sqrt[e]*Sqrt[a*e + c*d*x 
])/Sqrt[c*d^2 - a*e^2]])/((c*d^2 - a*e^2)^(3/2)*(a*e + c*d*x)^(5/2))))/(19 
2*e^(7/2)*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1130, 1130, 1130, 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.

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

\(\Big \downarrow \) 1130

\(\displaystyle \frac {5 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{(d+e x)^{11/2}}dx}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {5 c d \left (\frac {c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{(d+e x)^{7/2}}dx}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {5 c d \left (\frac {c d \left (\frac {c d \int \frac {1}{(d+e x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {5 c d \left (\frac {c d \left (\frac {c d \left (\frac {c d \int \frac {1}{\sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 \left (c d^2-a e^2\right )}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {5 c d \left (\frac {c d \left (\frac {c d \left (\frac {c d e \int \frac {1}{\frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}+\left (c d^2-a e^2\right ) e}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}}{c d^2-a e^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {5 c d \left (\frac {c d \left (\frac {c d \left (\frac {c d \arctan \left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\sqrt {e} \left (c d^2-a e^2\right )^{3/2}}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}\right )}{4 e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 e (d+e x)^{5/2}}\right )}{2 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e (d+e x)^{9/2}}\right )}{8 e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{4 e (d+e x)^{13/2}}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x]
 

Output:

-1/4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(e*(d + e*x)^(13/2)) + 
(5*c*d*(-1/3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(e*(d + e*x)^(9 
/2)) + (c*d*(-1/2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(e*(d + e*x) 
^(5/2)) + (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*e)))/(2*e)))/(8*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. \(651\) vs. \(2(263)=526\).

Time = 1.07 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.17

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{4} x^{4}+60 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{3} x^{3}+90 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{2} x^{2}+60 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e x -15 c^{3} d^{3} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+15 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{8}-118 a \,c^{2} d^{2} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+73 c^{3} d^{4} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-136 a^{2} c d \,e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+36 a \,c^{2} d^{3} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+55 c^{3} d^{5} e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-48 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{3} e^{6}+8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4}+10 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{2}+15 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{3} d^{6}\right )}{192 \left (e x +d \right )^{\frac {9}{2}} \sqrt {c d x +a e}\, \left (a \,e^{2}-c \,d^{2}\right ) e^{3} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(652\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/(e*x+d)^(15/2),x,method=_RETUR 
NVERBOSE)
 

Output:

1/192*((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^4*d^4*e^4*x^4+60*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c* 
d^2)*e)^(1/2))*c^4*d^5*e^3*x^3+90*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^ 
2)*e)^(1/2))*c^4*d^6*e^2*x^2+60*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2) 
*e)^(1/2))*c^4*d^7*e*x-15*c^3*d^3*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^4*d^8- 
118*a*c^2*d^2*e^4*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+73*c^3*d^4 
*e^2*x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-136*a^2*c*d*e^5*x*(c*d* 
x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+36*a*c^2*d^3*e^3*x*(c*d*x+a*e)^(1/2)* 
((a*e^2-c*d^2)*e)^(1/2)+55*c^3*d^5*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e) 
^(1/2)-48*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*e^6+8*((a*e^2-c*d^ 
2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d^2*e^4+10*((a*e^2-c*d^2)*e)^(1/2)*(c* 
d*x+a*e)^(1/2)*a*c^2*d^4*e^2+15*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)* 
c^3*d^6)/(e*x+d)^(9/2)/(c*d*x+a*e)^(1/2)/(a*e^2-c*d^2)/e^3/((a*e^2-c*d^2)* 
e)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (263) = 526\).

Time = 0.13 (sec) , antiderivative size = 1204, normalized size of antiderivative = 4.00 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),x, algori 
thm="fricas")
 

Output:

[1/384*(15*(c^4*d^4*e^5*x^5 + 5*c^4*d^5*e^4*x^4 + 10*c^4*d^6*e^3*x^3 + 10* 
c^4*d^7*e^2*x^2 + 5*c^4*d^8*e*x + c^4*d^9)*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^4*d^8*e - 5*a*c^3*d^6*e^3 - 2*a^2*c^2*d^4*e^5 - 56*a^3*c* 
d^2*e^7 + 48*a^4*e^9 - 15*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 + (73*c^4*d^6* 
e^3 - 191*a*c^3*d^4*e^5 + 118*a^2*c^2*d^2*e^7)*x^2 + (55*c^4*d^7*e^2 - 19* 
a*c^3*d^5*e^4 - 172*a^2*c^2*d^3*e^6 + 136*a^3*c*d*e^8)*x)*sqrt(c*d*e*x^2 + 
 a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(c^2*d^9*e^4 - 2*a*c*d^7*e^6 + 
a^2*d^5*e^8 + (c^2*d^4*e^9 - 2*a*c*d^2*e^11 + a^2*e^13)*x^5 + 5*(c^2*d^5*e 
^8 - 2*a*c*d^3*e^10 + a^2*d*e^12)*x^4 + 10*(c^2*d^6*e^7 - 2*a*c*d^4*e^9 + 
a^2*d^2*e^11)*x^3 + 10*(c^2*d^7*e^6 - 2*a*c*d^5*e^8 + a^2*d^3*e^10)*x^2 + 
5*(c^2*d^8*e^5 - 2*a*c*d^6*e^7 + a^2*d^4*e^9)*x), -1/192*(15*(c^4*d^4*e^5* 
x^5 + 5*c^4*d^5*e^4*x^4 + 10*c^4*d^6*e^3*x^3 + 10*c^4*d^7*e^2*x^2 + 5*c^4* 
d^8*e*x + c^4*d^9)*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^4*d^8*e - 5*a*c^3*d^6*e^3 - 2*a^2*c^2*d^4*e^ 
5 - 56*a^3*c*d^2*e^7 + 48*a^4*e^9 - 15*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 + 
 (73*c^4*d^6*e^3 - 191*a*c^3*d^4*e^5 + 118*a^2*c^2*d^2*e^7)*x^2 + (55*c^4* 
d^7*e^2 - 19*a*c^3*d^5*e^4 - 172*a^2*c^2*d^3*e^6 + 136*a^3*c*d*e^8)*x)*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(15/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {15}{2}}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),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)^(15/2), 
x)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 506, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\frac {\frac {15 \, c^{5} d^{5} e {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{\sqrt {c d^{2} e - a e^{3}} {\left (c d^{2} - a e^{2}\right )}} - \frac {15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{8} d^{11} e^{4} {\left | e \right |} - 45 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{7} d^{9} e^{6} {\left | e \right |} + 45 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{6} d^{7} e^{8} {\left | e \right |} - 15 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{5} d^{5} e^{10} {\left | e \right |} + 55 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{7} d^{9} e^{3} {\left | e \right |} - 110 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{6} d^{7} e^{5} {\left | e \right |} + 55 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{5} d^{5} e^{7} {\left | e \right |} + 73 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{6} d^{7} e^{2} {\left | e \right |} - 73 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{4} {\left | e \right |} - 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{5} d^{5} e {\left | e \right |}}{{\left (c d^{2} - a e^{2}\right )} {\left (e x + d\right )}^{4} c^{4} d^{4} e^{4}}}{192 \, c d e^{5}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),x, algori 
thm="giac")
 

Output:

1/192*(15*c^5*d^5*e*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))/(sqrt(c*d^2*e - a*e^3)*(c*d^2 - a*e^2)) - (15*sqrt( 
(e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^8*d^11*e^4*abs(e) - 45*sqrt((e*x + d) 
*c*d*e - c*d^2*e + a*e^3)*a*c^7*d^9*e^6*abs(e) + 45*sqrt((e*x + d)*c*d*e - 
 c*d^2*e + a*e^3)*a^2*c^6*d^7*e^8*abs(e) - 15*sqrt((e*x + d)*c*d*e - c*d^2 
*e + a*e^3)*a^3*c^5*d^5*e^10*abs(e) + 55*((e*x + d)*c*d*e - c*d^2*e + a*e^ 
3)^(3/2)*c^7*d^9*e^3*abs(e) - 110*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2 
)*a*c^6*d^7*e^5*abs(e) + 55*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2* 
c^5*d^5*e^7*abs(e) + 73*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^6*d^7* 
e^2*abs(e) - 73*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^5*d^5*e^4*ab 
s(e) - 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^5*d^5*e*abs(e))/((c* 
d^2 - a*e^2)*(e*x + d)^4*c^4*d^4*e^4))/(c*d*e^5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (d+e\,x\right )}^{15/2}} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(d + e*x)^(15/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 782, normalized size of antiderivative = 2.60 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\frac {15 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{4} d^{8}+60 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{4} d^{7} e x +90 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{4} d^{6} e^{2} x^{2}+60 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{4} d^{5} e^{3} x^{3}+15 \sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{4} d^{4} e^{4} x^{4}-48 \sqrt {c d x +a e}\, a^{4} e^{9}+56 \sqrt {c d x +a e}\, a^{3} c \,d^{2} e^{7}-136 \sqrt {c d x +a e}\, a^{3} c d \,e^{8} x +2 \sqrt {c d x +a e}\, a^{2} c^{2} d^{4} e^{5}+172 \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{6} x -118 \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{7} x^{2}+5 \sqrt {c d x +a e}\, a \,c^{3} d^{6} e^{3}+19 \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{4} x +191 \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{5} x^{2}-15 \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{6} x^{3}-15 \sqrt {c d x +a e}\, c^{4} d^{8} e -55 \sqrt {c d x +a e}\, c^{4} d^{7} e^{2} x -73 \sqrt {c d x +a e}\, c^{4} d^{6} e^{3} x^{2}+15 \sqrt {c d x +a e}\, c^{4} d^{5} e^{4} x^{3}}{192 e^{4} \left (a^{2} e^{8} x^{4}-2 a c \,d^{2} e^{6} x^{4}+c^{2} d^{4} e^{4} x^{4}+4 a^{2} d \,e^{7} x^{3}-8 a c \,d^{3} e^{5} x^{3}+4 c^{2} d^{5} e^{3} x^{3}+6 a^{2} d^{2} e^{6} x^{2}-12 a c \,d^{4} e^{4} x^{2}+6 c^{2} d^{6} e^{2} x^{2}+4 a^{2} d^{3} e^{5} x -8 a c \,d^{5} e^{3} x +4 c^{2} d^{7} e x +a^{2} d^{4} e^{4}-2 a c \,d^{6} e^{2}+c^{2} d^{8}\right )} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/(e*x+d)^(15/2),x)
 

Output:

(15*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqrt(e)*s 
qrt( - a*e**2 + c*d**2)))*c**4*d**8 + 60*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**4*d**7*e 
*x + 90*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**4*d**6*e**2*x**2 + 60*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**4*d**5*e**3*x**3 + 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**4*d**4*e**4*x**4 - 48 
*sqrt(a*e + c*d*x)*a**4*e**9 + 56*sqrt(a*e + c*d*x)*a**3*c*d**2*e**7 - 136 
*sqrt(a*e + c*d*x)*a**3*c*d*e**8*x + 2*sqrt(a*e + c*d*x)*a**2*c**2*d**4*e* 
*5 + 172*sqrt(a*e + c*d*x)*a**2*c**2*d**3*e**6*x - 118*sqrt(a*e + c*d*x)*a 
**2*c**2*d**2*e**7*x**2 + 5*sqrt(a*e + c*d*x)*a*c**3*d**6*e**3 + 19*sqrt(a 
*e + c*d*x)*a*c**3*d**5*e**4*x + 191*sqrt(a*e + c*d*x)*a*c**3*d**4*e**5*x* 
*2 - 15*sqrt(a*e + c*d*x)*a*c**3*d**3*e**6*x**3 - 15*sqrt(a*e + c*d*x)*c** 
4*d**8*e - 55*sqrt(a*e + c*d*x)*c**4*d**7*e**2*x - 73*sqrt(a*e + c*d*x)*c* 
*4*d**6*e**3*x**2 + 15*sqrt(a*e + c*d*x)*c**4*d**5*e**4*x**3)/(192*e**4*(a 
**2*d**4*e**4 + 4*a**2*d**3*e**5*x + 6*a**2*d**2*e**6*x**2 + 4*a**2*d*e**7 
*x**3 + a**2*e**8*x**4 - 2*a*c*d**6*e**2 - 8*a*c*d**5*e**3*x - 12*a*c*d**4 
*e**4*x**2 - 8*a*c*d**3*e**5*x**3 - 2*a*c*d**2*e**6*x**4 + c**2*d**8 + 4*c 
**2*d**7*e*x + 6*c**2*d**6*e**2*x**2 + 4*c**2*d**5*e**3*x**3 + c**2*d**...