\(\int \frac {1}{(d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [320]

Optimal result

Integrand size = 39, antiderivative size = 269 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {1}{2 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 c d}{4 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {15 c^2 d^2 \sqrt {d+e x}}{4 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {15 c^2 d^2 \sqrt {e} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c d^2-a e^2} \left (-2 a^2 e^4+a c d e^2 (9 d+5 e x)+c^2 d^2 \left (8 d^2+25 d e x+15 e^2 x^2\right )\right )-15 c^2 d^2 \sqrt {e} \sqrt {a e+c d x} (d+e x)^2 \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2} (d+e x)^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2) 
),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {1135, 1135, 1132, 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.

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

Int[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
 

Output:

1/(2*(c*d^2 - a*e^2)*(d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d* 
e*x^2]) + (5*c*d*(1/((c*d^2 - a*e^2)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a 
*e^2)*x + c*d*e*x^2]) + (3*c*d*((-2*Sqrt[d + e*x])/((c*d^2 - a*e^2)*Sqrt[a 
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (2*Sqrt[e]*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]) 
])/(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 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 1132

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 
 1)*(b^2 - 4*a*c))), x] - Simp[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(b^2 - 
 4*a*c)))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; Free 
Q[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && LtQ 
[0, m, 1] && 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 [A] (verified)

Time = 1.27 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (15 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{2} e^{3} x^{2}+30 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{3} e^{2} x +15 \sqrt {c d x +a e}\, \operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) c^{2} d^{4} e -15 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}-5 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a c d \,e^{3} x -25 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{3} e x +2 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a^{2} e^{4}-9 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a c \,d^{2} e^{2}-8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{4}\right )}{4 \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(374\)

Input:

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2),x,method=_RETU 
RNVERBOSE)
 

Output:

-1/4/(e*x+d)^(5/2)*((e*x+d)*(c*d*x+a*e))^(1/2)*(15*(c*d*x+a*e)^(1/2)*arcta 
nh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^2*d^2*e^3*x^2+30*(c*d*x+ 
a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^2*d^3*e^ 
2*x+15*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/ 
2))*c^2*d^4*e-15*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^2*e^2*x^2-5*((a*e^2-c*d^2)* 
e)^(1/2)*a*c*d*e^3*x-25*((a*e^2-c*d^2)*e)^(1/2)*c^2*d^3*e*x+2*((a*e^2-c*d^ 
2)*e)^(1/2)*a^2*e^4-9*((a*e^2-c*d^2)*e)^(1/2)*a*c*d^2*e^2-8*((a*e^2-c*d^2) 
*e)^(1/2)*c^2*d^4)/(c*d*x+a*e)/(a*e^2-c*d^2)^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. 559 vs. \(2 (237) = 474\).

Time = 0.15 (sec) , antiderivative size = 1140, normalized size of antiderivative = 4.24 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algor 
ithm="fricas")
 

Output:

[-1/8*(15*(c^3*d^3*e^3*x^4 + a*c^2*d^5*e + (3*c^3*d^4*e^2 + a*c^2*d^2*e^4) 
*x^3 + 3*(c^3*d^5*e + a*c^2*d^3*e^3)*x^2 + (c^3*d^6 + 3*a*c^2*d^4*e^2)*x)* 
sqrt(-e/(c*d^2 - a*e^2))*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)*(c*d^2 - a*e^2)*sqrt(e*x 
+ d)*sqrt(-e/(c*d^2 - a*e^2)))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(15*c^2*d^2* 
e^2*x^2 + 8*c^2*d^4 + 9*a*c*d^2*e^2 - 2*a^2*e^4 + 5*(5*c^2*d^3*e + a*c*d*e 
^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^3*d 
^9*e - 3*a^2*c^2*d^7*e^3 + 3*a^3*c*d^5*e^5 - a^4*d^3*e^7 + (c^4*d^7*e^3 - 
3*a*c^3*d^5*e^5 + 3*a^2*c^2*d^3*e^7 - a^3*c*d*e^9)*x^4 + (3*c^4*d^8*e^2 - 
8*a*c^3*d^6*e^4 + 6*a^2*c^2*d^4*e^6 - a^4*e^10)*x^3 + 3*(c^4*d^9*e - 2*a*c 
^3*d^7*e^3 + 2*a^3*c*d^3*e^7 - a^4*d*e^9)*x^2 + (c^4*d^10 - 6*a^2*c^2*d^6* 
e^4 + 8*a^3*c*d^4*e^6 - 3*a^4*d^2*e^8)*x), -1/4*(15*(c^3*d^3*e^3*x^4 + a*c 
^2*d^5*e + (3*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^3 + 3*(c^3*d^5*e + a*c^2*d^3* 
e^3)*x^2 + (c^3*d^6 + 3*a*c^2*d^4*e^2)*x)*sqrt(e/(c*d^2 - a*e^2))*arctan(- 
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)* 
sqrt(e/(c*d^2 - a*e^2))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) + ( 
15*c^2*d^2*e^2*x^2 + 8*c^2*d^4 + 9*a*c*d^2*e^2 - 2*a^2*e^4 + 5*(5*c^2*d^3* 
e + a*c*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d 
))/(a*c^3*d^9*e - 3*a^2*c^2*d^7*e^3 + 3*a^3*c*d^5*e^5 - a^4*d^3*e^7 + (c^4 
*d^7*e^3 - 3*a*c^3*d^5*e^5 + 3*a^2*c^2*d^3*e^7 - a^3*c*d*e^9)*x^4 + (3*...
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
 

Output:

Integral(1/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algor 
ithm="maxima")
 

Output:

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)^(3/2) 
), x)
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {1}{4} \, {\left (\frac {15 \, c^{2} d^{2} e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} + \frac {8 \, c^{2} d^{2} e}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}} + \frac {9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} - 9 \, \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} + 7 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e}{{\left (c^{3} d^{6} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{2} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{4} {\left | e \right |} - a^{3} e^{6} {\left | e \right |}\right )} {\left (e x + d\right )}^{2} c^{2} d^{2} e^{2}}\right )} e \] Input:

integrate(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algor 
ithm="giac")
 

Output:

-1/4*(15*c^2*d^2*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^3*d^6*abs(e) - 3*a*c^2*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4 
*abs(e) - a^3*e^6*abs(e))*sqrt(c*d^2*e - a*e^3)) + 8*c^2*d^2*e/((c^3*d^6*a 
bs(e) - 3*a*c^2*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4*abs(e) - a^3*e^6*abs(e))* 
sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)) + (9*sqrt((e*x + d)*c*d*e - c*d^2 
*e + a*e^3)*c^3*d^4*e^2 - 9*sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*a*c^2* 
d^2*e^4 + 7*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^2*d^2*e)/((c^3*d^6 
*abs(e) - 3*a*c^2*d^4*e^2*abs(e) + 3*a^2*c*d^2*e^4*abs(e) - a^3*e^6*abs(e) 
)*(e*x + d)^2*c^2*d^2*e^2))*e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \] Input:

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
 

Output:

int(1/((d + e*x)^(3/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.87 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-15 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{2} d^{4}-30 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{2} d^{3} e x -15 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c^{2} d^{2} e^{2} x^{2}-2 a^{3} e^{6}+11 a^{2} c \,d^{2} e^{4}+5 a^{2} c d \,e^{5} x -a \,c^{2} d^{4} e^{2}+20 a \,c^{2} d^{3} e^{3} x +15 a \,c^{2} d^{2} e^{4} x^{2}-8 c^{3} d^{6}-25 c^{3} d^{5} e x -15 c^{3} d^{4} e^{2} x^{2}}{4 \sqrt {c d x +a e}\, \left (a^{4} e^{10} x^{2}-4 a^{3} c \,d^{2} e^{8} x^{2}+6 a^{2} c^{2} d^{4} e^{6} x^{2}-4 a \,c^{3} d^{6} e^{4} x^{2}+c^{4} d^{8} e^{2} x^{2}+2 a^{4} d \,e^{9} x -8 a^{3} c \,d^{3} e^{7} x +12 a^{2} c^{2} d^{5} e^{5} x -8 a \,c^{3} d^{7} e^{3} x +2 c^{4} d^{9} e x +a^{4} d^{2} e^{8}-4 a^{3} c \,d^{4} e^{6}+6 a^{2} c^{2} d^{6} e^{4}-4 a \,c^{3} d^{8} e^{2}+c^{4} d^{10}\right )} \] Input:

int(1/(e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
 

Output:

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