\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^{13/2}} \, dx\) [296]

Optimal result

Integrand size = 39, antiderivative size = 315 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=-\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e^2 (d+e x)^{7/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{32 e^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 e^2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{4 e (d+e x)^{11/2}}+\frac {3 c^4 d^4 \arctan \left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 e^{5/2} \left (c d^2-a e^2\right )^{5/2}} \] Output:

-1/8*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^2/(e*x+d)^(7/2)+1/32*c^ 
2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^2/(-a*e^2+c*d^2)/(e*x+d)^( 
5/2)+3/64*c^3*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/e^2/(-a*e^2+c*d^ 
2)^2/(e*x+d)^(3/2)-1/4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/e/(e*x+d)^( 
11/2)+3/64*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^(5/2)/(-a*e^2+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.77 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {c^4 d^4 ((a e+c d x) (d+e x))^{3/2} \left (-\frac {\sqrt {e} \left (16 a^3 e^6-24 a^2 c d e^4 (d-e x)+2 a c^2 d^2 e^2 \left (d^2-22 d e x+e^2 x^2\right )+c^3 d^3 \left (3 d^3+11 d^2 e x-11 d e^2 x^2-3 e^3 x^3\right )\right )}{c^4 d^4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^4}+\frac {3 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2} (a e+c d x)^{3/2}}\right )}{64 e^{5/2} (d+e x)^{3/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(13/2),x 
]
 

Output:

(c^4*d^4*((a*e + c*d*x)*(d + e*x))^(3/2)*(-((Sqrt[e]*(16*a^3*e^6 - 24*a^2* 
c*d*e^4*(d - e*x) + 2*a*c^2*d^2*e^2*(d^2 - 22*d*e*x + e^2*x^2) + c^3*d^3*( 
3*d^3 + 11*d^2*e*x - 11*d*e^2*x^2 - 3*e^3*x^3)))/(c^4*d^4*(c*d^2 - a*e^2)^ 
2*(a*e + c*d*x)*(d + e*x)^4)) + (3*ArcTan[(Sqrt[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)^(3/2))))/(64*e^(5/2 
)*(d + e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {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.

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1130

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^(13/2),x]
 

Output:

-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)
 

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. \(2(277)=554\).

Time = 1.08 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.07

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{4} x^{4}+12 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{3} x^{3}+18 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{2} x^{2}+12 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e x -3 c^{3} d^{3} e^{3} x^{3} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{4} d^{8}+2 a \,c^{2} d^{2} e^{4} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-11 c^{3} d^{4} e^{2} x^{2} \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+24 a^{2} c d \,e^{5} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}-44 a \,c^{2} d^{3} e^{3} x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+11 c^{3} d^{5} e x \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}+16 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{3} e^{6}-24 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4}+2 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{2}+3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{3} d^{6}\right )}{64 \left (e x +d \right )^{\frac {9}{2}} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, e^{2} \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d x +a e}}\)	\(652\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/(e*x+d)^(13/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-1/64*((e*x+d)*(c*d*x+a*e))^(1/2)*(3*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+12*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+18*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+12*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)* 
e)^(1/2))*c^4*d^7*e*x-3*c^3*d^3*e^3*x^3*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e 
)^(1/2)+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*d^8+2*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)-11*c^3*d^4*e^2* 
x^2*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+24*a^2*c*d*e^5*x*(c*d*x+a*e) 
^(1/2)*((a*e^2-c*d^2)*e)^(1/2)-44*a*c^2*d^3*e^3*x*(c*d*x+a*e)^(1/2)*((a*e^ 
2-c*d^2)*e)^(1/2)+11*c^3*d^5*e*x*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2) 
+16*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*a^3*e^6-24*((a*e^2-c*d^2)*e) 
^(1/2)*(c*d*x+a*e)^(1/2)*a^2*c*d^2*e^4+2*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a* 
e)^(1/2)*a*c^2*d^4*e^2+3*((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)/((a*e^2-c*d^2)*e)^(1/2)/e^2/(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. 682 vs. \(2 (277) = 554\).

Time = 0.15 (sec) , antiderivative size = 1386, normalized size of antiderivative = 4.40 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(13/2),x, algori 
thm="fricas")
 

Output:

[-1/128*(3*(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*(3*c^4*d^8*e - a*c^3*d^6*e^3 - 26*a^2*c^2*d^4*e^5 + 40*a^3*c*d^ 
2*e^7 - 16*a^4*e^9 - 3*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 - (11*c^4*d^6*e^3 
 - 13*a*c^3*d^4*e^5 + 2*a^2*c^2*d^2*e^7)*x^2 + (11*c^4*d^7*e^2 - 55*a*c^3* 
d^5*e^4 + 68*a^2*c^2*d^3*e^6 - 24*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^3*d^11*e^3 - 3*a*c^2*d^9*e^5 + 3*a^2 
*c*d^7*e^7 - a^3*d^5*e^9 + (c^3*d^6*e^8 - 3*a*c^2*d^4*e^10 + 3*a^2*c*d^2*e 
^12 - a^3*e^14)*x^5 + 5*(c^3*d^7*e^7 - 3*a*c^2*d^5*e^9 + 3*a^2*c*d^3*e^11 
- a^3*d*e^13)*x^4 + 10*(c^3*d^8*e^6 - 3*a*c^2*d^6*e^8 + 3*a^2*c*d^4*e^10 - 
 a^3*d^2*e^12)*x^3 + 10*(c^3*d^9*e^5 - 3*a*c^2*d^7*e^7 + 3*a^2*c*d^5*e^9 - 
 a^3*d^3*e^11)*x^2 + 5*(c^3*d^10*e^4 - 3*a*c^2*d^8*e^6 + 3*a^2*c*d^6*e^8 - 
 a^3*d^4*e^10)*x), -1/64*(3*(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)) + (3*c^4*d^ 
8*e - a*c^3*d^6*e^3 - 26*a^2*c^2*d^4*e^5 + 40*a^3*c*d^2*e^7 - 16*a^4*e^9 - 
 3*(c^4*d^5*e^4 - a*c^3*d^3*e^6)*x^3 - (11*c^4*d^6*e^3 - 13*a*c^3*d^4*e...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(13/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {13}{2}}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(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)^(3/2)/(e*x + d)^(13/2), 
x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\frac {\frac {3 \, c^{5} d^{5} e {\left | e \right |} \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {c d^{2} e - a e^{3}}} - \frac {3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{8} d^{11} e^{4} {\left | e \right |} - 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{7} d^{9} e^{6} {\left | e \right |} + 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{6} d^{7} e^{8} {\left | e \right |} - 3 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{5} d^{5} e^{10} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{7} d^{9} e^{3} {\left | e \right |} - 22 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{6} d^{7} e^{5} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{5} d^{5} e^{7} {\left | e \right |} - 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{6} d^{7} e^{2} {\left | e \right |} + 11 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{4} {\left | e \right |} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{5} d^{5} e {\left | e \right |}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{4} c^{4} d^{4} e^{4}}}{64 \, c d e^{4}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(13/2),x, algori 
thm="giac")
 

Output:

1/64*(3*c^5*d^5*e*abs(e)*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)/sq 
rt(c*d^2*e - a*e^3))/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(c*d^2*e - a 
*e^3)) - (3*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c^8*d^11*e^4*abs(e) - 
9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^7*d^9*e^6*abs(e) + 9*sqrt((e 
*x + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^6*d^7*e^8*abs(e) - 3*sqrt((e*x + d) 
*c*d*e - c*d^2*e + a*e^3)*a^3*c^5*d^5*e^10*abs(e) + 11*((e*x + d)*c*d*e - 
c*d^2*e + a*e^3)^(3/2)*c^7*d^9*e^3*abs(e) - 22*((e*x + d)*c*d*e - c*d^2*e 
+ a*e^3)^(3/2)*a*c^6*d^7*e^5*abs(e) + 11*((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) - 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^( 
5/2)*c^6*d^7*e^2*abs(e) + 11*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c 
^5*d^5*e^4*abs(e) - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^5*d^5*e* 
abs(e))/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^4*c^4*d^4*e^4))/(c* 
d*e^4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{{\left (d+e\,x\right )}^{13/2}} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^(13/2),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(d + e*x)^(13/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 864, normalized size of antiderivative = 2.74 \[ \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)^{13/2}} \, dx=\frac {-3 \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}-12 \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 -18 \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}-12 \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}-3 \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}-16 \sqrt {c d x +a e}\, a^{4} e^{9}+40 \sqrt {c d x +a e}\, a^{3} c \,d^{2} e^{7}-24 \sqrt {c d x +a e}\, a^{3} c d \,e^{8} x -26 \sqrt {c d x +a e}\, a^{2} c^{2} d^{4} e^{5}+68 \sqrt {c d x +a e}\, a^{2} c^{2} d^{3} e^{6} x -2 \sqrt {c d x +a e}\, a^{2} c^{2} d^{2} e^{7} x^{2}-\sqrt {c d x +a e}\, a \,c^{3} d^{6} e^{3}-55 \sqrt {c d x +a e}\, a \,c^{3} d^{5} e^{4} x +13 \sqrt {c d x +a e}\, a \,c^{3} d^{4} e^{5} x^{2}+3 \sqrt {c d x +a e}\, a \,c^{3} d^{3} e^{6} x^{3}+3 \sqrt {c d x +a e}\, c^{4} d^{8} e +11 \sqrt {c d x +a e}\, c^{4} d^{7} e^{2} x -11 \sqrt {c d x +a e}\, c^{4} d^{6} e^{3} x^{2}-3 \sqrt {c d x +a e}\, c^{4} d^{5} e^{4} x^{3}}{64 e^{3} \left (a^{3} e^{10} x^{4}-3 a^{2} c \,d^{2} e^{8} x^{4}+3 a \,c^{2} d^{4} e^{6} x^{4}-c^{3} d^{6} e^{4} x^{4}+4 a^{3} d \,e^{9} x^{3}-12 a^{2} c \,d^{3} e^{7} x^{3}+12 a \,c^{2} d^{5} e^{5} x^{3}-4 c^{3} d^{7} e^{3} x^{3}+6 a^{3} d^{2} e^{8} x^{2}-18 a^{2} c \,d^{4} e^{6} x^{2}+18 a \,c^{2} d^{6} e^{4} x^{2}-6 c^{3} d^{8} e^{2} x^{2}+4 a^{3} d^{3} e^{7} x -12 a^{2} c \,d^{5} e^{5} x +12 a \,c^{2} d^{7} e^{3} x -4 c^{3} d^{9} e x +a^{3} d^{4} e^{6}-3 a^{2} c \,d^{6} e^{4}+3 a \,c^{2} d^{8} e^{2}-c^{3} d^{10}\right )} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(13/2),x)
 

Output:

( - 3*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**8 - 12*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 - 18*sqrt(e)*sqrt( - a*e**2 + c*d**2)*atan((sqrt(a*e + c*d*x)*e)/(sqr 
t(e)*sqrt( - a*e**2 + c*d**2)))*c**4*d**6*e**2*x**2 - 12*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 - 3*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 - 1 
6*sqrt(a*e + c*d*x)*a**4*e**9 + 40*sqrt(a*e + c*d*x)*a**3*c*d**2*e**7 - 24 
*sqrt(a*e + c*d*x)*a**3*c*d*e**8*x - 26*sqrt(a*e + c*d*x)*a**2*c**2*d**4*e 
**5 + 68*sqrt(a*e + c*d*x)*a**2*c**2*d**3*e**6*x - 2*sqrt(a*e + c*d*x)*a** 
2*c**2*d**2*e**7*x**2 - sqrt(a*e + c*d*x)*a*c**3*d**6*e**3 - 55*sqrt(a*e + 
 c*d*x)*a*c**3*d**5*e**4*x + 13*sqrt(a*e + c*d*x)*a*c**3*d**4*e**5*x**2 + 
3*sqrt(a*e + c*d*x)*a*c**3*d**3*e**6*x**3 + 3*sqrt(a*e + c*d*x)*c**4*d**8* 
e + 11*sqrt(a*e + c*d*x)*c**4*d**7*e**2*x - 11*sqrt(a*e + c*d*x)*c**4*d**6 
*e**3*x**2 - 3*sqrt(a*e + c*d*x)*c**4*d**5*e**4*x**3)/(64*e**3*(a**3*d**4* 
e**6 + 4*a**3*d**3*e**7*x + 6*a**3*d**2*e**8*x**2 + 4*a**3*d*e**9*x**3 + a 
**3*e**10*x**4 - 3*a**2*c*d**6*e**4 - 12*a**2*c*d**5*e**5*x - 18*a**2*c*d* 
*4*e**6*x**2 - 12*a**2*c*d**3*e**7*x**3 - 3*a**2*c*d**2*e**8*x**4 + 3*a*c* 
*2*d**8*e**2 + 12*a*c**2*d**7*e**3*x + 18*a*c**2*d**6*e**4*x**2 + 12*a*...