\(\int \frac {(d+e x)^{3/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [197]

Optimal result

Integrand size = 37, antiderivative size = 176 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}-\frac {15 \sqrt {c} \sqrt {d} e^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \] Output:

15/4*e^2/(-a*e^2+c*d^2)^3/(e*x+d)^(1/2)-1/2/(-a*e^2+c*d^2)/(c*d*x+a*e)^2/( 
e*x+d)^(1/2)+5/4*e/(-a*e^2+c*d^2)^2/(c*d*x+a*e)/(e*x+d)^(1/2)-15/4*c^(1/2) 
*d^(1/2)*e^2*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^(1/2))/( 
-a*e^2+c*d^2)^(7/2)
 

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {1}{4} \left (\frac {8 a^2 e^4+a c d e^2 (9 d+25 e x)+c^2 d^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{\left (c d^2-a e^2\right )^3 (a e+c d x)^2 \sqrt {d+e x}}-\frac {15 \sqrt {c} \sqrt {d} e^2 \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{7/2}}\right ) \] Input:

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

Output:

((8*a^2*e^4 + a*c*d*e^2*(9*d + 25*e*x) + c^2*d^2*(-2*d^2 + 5*d*e*x + 15*e^ 
2*x^2))/((c*d^2 - a*e^2)^3*(a*e + c*d*x)^2*Sqrt[d + e*x]) - (15*Sqrt[c]*Sq 
rt[d]*e^2*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[-(c*d^2) + a*e^2]])/ 
(-(c*d^2) + a*e^2)^(7/2))/4
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {1121, 52, 52, 61, 73, 221}

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.

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

\(\Big \downarrow \) 1121

\(\displaystyle \int \frac {1}{(d+e x)^{3/2} (a e+c d x)^3}dx\)

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*1/((c*d^2 - a*e^2)*(a*e + c*d*x)^2*Sqrt[d + e*x]) - (5*e*(-(1/((c*d^2 
 - a*e^2)*(a*e + c*d*x)*Sqrt[d + e*x])) - (3*e*(2/((c*d^2 - a*e^2)*Sqrt[d 
+ e*x]) - (2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[ 
c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(3/2)))/(2*(c*d^2 - a*e^2))))/(4*(c*d^2 - 
 a*e^2))
 

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1121

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

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(2 e^{2} \left (-\frac {1}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {e x +d}}-\frac {c d \left (\frac {\frac {7 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9 a \,e^{2}}{8}-\frac {9 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {15 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{8 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}\right )\)	\(152\)
default	\(2 e^{2} \left (-\frac {1}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {e x +d}}-\frac {c d \left (\frac {\frac {7 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9 a \,e^{2}}{8}-\frac {9 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+a \,e^{2}-c \,d^{2}\right )^{2}}+\frac {15 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{8 \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3}}\right )\)	\(152\)
pseudoelliptic	\(-\frac {15 \left (c d \,e^{2} \sqrt {e x +d}\, \left (c d x +a e \right )^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}\right )+\frac {8 \left (\left (\frac {15}{8} d^{2} e^{2} x^{2}+\frac {5}{8} d^{3} e x -\frac {1}{4} d^{4}\right ) c^{2}+\frac {9 e^{2} a \left (\frac {25 e x}{9}+d \right ) d c}{8}+a^{2} e^{4}\right ) \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}}{15}\right )}{4 \sqrt {e x +d}\, \sqrt {c d \left (a \,e^{2}-c \,d^{2}\right )}\, \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d x +a e \right )^{2}}\)	\(172\)

Input:

int((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^3,x,method=_RETURNVERB 
OSE)
 

Output:

2*e^2*(-1/(a*e^2-c*d^2)^3/(e*x+d)^(1/2)-1/(a*e^2-c*d^2)^3*c*d*((7/8*c*d*(e 
*x+d)^(3/2)+(9/8*a*e^2-9/8*c*d^2)*(e*x+d)^(1/2))/(c*d*(e*x+d)+a*e^2-c*d^2) 
^2+15/8/(c*d*(a*e^2-c*d^2))^(1/2)*arctan(c*d*(e*x+d)^(1/2)/(c*d*(a*e^2-c*d 
^2))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (148) = 296\).

Time = 0.13 (sec) , antiderivative size = 870, normalized size of antiderivative = 4.94 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/8*(15*(c^2*d^2*e^3*x^3 + a^2*d*e^4 + (c^2*d^3*e^2 + 2*a*c*d*e^4)*x^2 + 
 (2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(c*d/(c*d^2 - a*e^2))*log((c*d*e*x + 2*c 
*d^2 - a*e^2 + 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a*e^2)))/ 
(c*d*x + a*e)) - 2*(15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 9*a*c*d^2*e^2 + 8*a^2 
*e^4 + 5*(c^2*d^3*e + 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(a^2*c^3*d^7*e^2 - 3* 
a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 - a^5*d*e^8 + (c^5*d^8*e - 3*a*c^4*d^6*e 
^3 + 3*a^2*c^3*d^4*e^5 - a^3*c^2*d^2*e^7)*x^3 + (c^5*d^9 - a*c^4*d^7*e^2 - 
 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^3*e^6 - 2*a^4*c*d*e^8)*x^2 + (2*a*c^4*d^8 
*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^2*d^4*e^5 + a^4*c*d^2*e^7 - a^5*e^9)*x), 
1/4*(15*(c^2*d^2*e^3*x^3 + a^2*d*e^4 + (c^2*d^3*e^2 + 2*a*c*d*e^4)*x^2 + ( 
2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(-c*d/(c*d^2 - a*e^2))*arctan(sqrt(e*x + d 
)*sqrt(-c*d/(c*d^2 - a*e^2))) + (15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 9*a*c*d^ 
2*e^2 + 8*a^2*e^4 + 5*(c^2*d^3*e + 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(a^2*c^3 
*d^7*e^2 - 3*a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 - a^5*d*e^8 + (c^5*d^8*e - 
3*a*c^4*d^6*e^3 + 3*a^2*c^3*d^4*e^5 - a^3*c^2*d^2*e^7)*x^3 + (c^5*d^9 - a* 
c^4*d^7*e^2 - 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^3*e^6 - 2*a^4*c*d*e^8)*x^2 + 
 (2*a*c^4*d^8*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^2*d^4*e^5 + a^4*c*d^2*e^7 - 
a^5*e^9)*x)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm=" 
maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.53 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {15 \, c d e^{2} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, e^{2}}{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {e x + d}} + \frac {7 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} - 9 \, \sqrt {e x + d} c^{2} d^{3} e^{2} + 9 \, \sqrt {e x + d} a c d e^{4}}{4 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}^{2}} \] Input:

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

Output:

15/4*c*d*e^2*arctan(sqrt(e*x + d)*c*d/sqrt(-c^2*d^3 + a*c*d*e^2))/((c^3*d^ 
6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(-c^2*d^3 + a*c*d*e^2 
)) + 2*e^2/((c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(e 
*x + d)) + 1/4*(7*(e*x + d)^(3/2)*c^2*d^2*e^2 - 9*sqrt(e*x + d)*c^2*d^3*e^ 
2 + 9*sqrt(e*x + d)*a*c*d*e^4)/((c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e 
^4 - a^3*e^6)*((e*x + d)*c*d - c*d^2 + a*e^2)^2)
 

Mupad [B] (verification not implemented)

Time = 5.29 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.43 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {\frac {2\,e^2}{a\,e^2-c\,d^2}+\frac {25\,c\,d\,e^2\,\left (d+e\,x\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {15\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2}{4\,{\left (a\,e^2-c\,d^2\right )}^3}}{\sqrt {d+e\,x}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}+c^2\,d^2\,{\left (d+e\,x\right )}^{5/2}}-\frac {15\,\sqrt {c}\,\sqrt {d}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^{7/2}}\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^{7/2}} \] Input:

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

Output:

- ((2*e^2)/(a*e^2 - c*d^2) + (25*c*d*e^2*(d + e*x))/(4*(a*e^2 - c*d^2)^2) 
+ (15*c^2*d^2*e^2*(d + e*x)^2)/(4*(a*e^2 - c*d^2)^3))/((d + e*x)^(1/2)*(a^ 
2*e^4 + c^2*d^4 - 2*a*c*d^2*e^2) - (2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(3/ 
2) + c^2*d^2*(d + e*x)^(5/2)) - (15*c^(1/2)*d^(1/2)*e^2*atan((c^(1/2)*d^(1 
/2)*(d + e*x)^(1/2)*(a^3*e^6 - c^3*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4 
))/(a*e^2 - c*d^2)^(7/2)))/(4*(a*e^2 - c*d^2)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.94 \[ \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {-15 \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}-30 \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c d \,e^{3} x -15 \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {a \,e^{2}-c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {d}\, \sqrt {c}\, \sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{2} e^{2} x^{2}-8 a^{3} e^{6}-a^{2} c \,d^{2} e^{4}-25 a^{2} c d \,e^{5} x +11 a \,c^{2} d^{4} e^{2}+20 a \,c^{2} d^{3} e^{3} x -15 a \,c^{2} d^{2} e^{4} x^{2}-2 c^{3} d^{6}+5 c^{3} d^{5} e x +15 c^{3} d^{4} e^{2} x^{2}}{4 \sqrt {e x +d}\, \left (a^{4} c^{2} d^{2} e^{8} x^{2}-4 a^{3} c^{3} d^{4} e^{6} x^{2}+6 a^{2} c^{4} d^{6} e^{4} x^{2}-4 a \,c^{5} d^{8} e^{2} x^{2}+c^{6} d^{10} x^{2}+2 a^{5} c d \,e^{9} x -8 a^{4} c^{2} d^{3} e^{7} x +12 a^{3} c^{3} d^{5} e^{5} x -8 a^{2} c^{4} d^{7} e^{3} x +2 a \,c^{5} d^{9} e x +a^{6} e^{10}-4 a^{5} c \,d^{2} e^{8}+6 a^{4} c^{2} d^{4} e^{6}-4 a^{3} c^{3} d^{6} e^{4}+a^{2} c^{4} d^{8} e^{2}\right )} \] Input:

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

Output:

( - 15*sqrt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + 
e*x)*c*d)/(sqrt(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a**2*e**4 - 30*sqrt(d)* 
sqrt(c)*sqrt(d + e*x)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt 
(d)*sqrt(c)*sqrt(a*e**2 - c*d**2)))*a*c*d*e**3*x - 15*sqrt(d)*sqrt(c)*sqrt 
(d + e*x)*sqrt(a*e**2 - c*d**2)*atan((sqrt(d + e*x)*c*d)/(sqrt(d)*sqrt(c)* 
sqrt(a*e**2 - c*d**2)))*c**2*d**2*e**2*x**2 - 8*a**3*e**6 - a**2*c*d**2*e* 
*4 - 25*a**2*c*d*e**5*x + 11*a*c**2*d**4*e**2 + 20*a*c**2*d**3*e**3*x - 15 
*a*c**2*d**2*e**4*x**2 - 2*c**3*d**6 + 5*c**3*d**5*e*x + 15*c**3*d**4*e**2 
*x**2)/(4*sqrt(d + e*x)*(a**6*e**10 - 4*a**5*c*d**2*e**8 + 2*a**5*c*d*e**9 
*x + 6*a**4*c**2*d**4*e**6 - 8*a**4*c**2*d**3*e**7*x + a**4*c**2*d**2*e**8 
*x**2 - 4*a**3*c**3*d**6*e**4 + 12*a**3*c**3*d**5*e**5*x - 4*a**3*c**3*d** 
4*e**6*x**2 + a**2*c**4*d**8*e**2 - 8*a**2*c**4*d**7*e**3*x + 6*a**2*c**4* 
d**6*e**4*x**2 + 2*a*c**5*d**9*e*x - 4*a*c**5*d**8*e**2*x**2 + c**6*d**10* 
x**2))