\(\int \frac {1}{\sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [319]

Optimal result

Integrand size = 39, antiderivative size = 196 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {1}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d \sqrt {e} \arctan \left (\frac {\sqrt {c d^2-a e^2} \sqrt {d+e x}}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c d^2-a e^2} \left (a e^2+c d (2 d+3 e x)\right )-3 c d \sqrt {e} \sqrt {a e+c d x} (d+e x) \arctan \left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2} \sqrt {d+e x} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {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.

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

\(\Big \downarrow \) 1135

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

\(\Big \downarrow \) 1132

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

\(\Big \downarrow \) 1136

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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))
 

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 = 1.10 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {\sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c d \,e^{2} x +3 \,\operatorname {arctanh}\left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right ) \sqrt {c d x +a e}\, c \,d^{2} e -3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c d e x -\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,e^{2}-2 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c \,d^{2}\right )}{\left (e x +d \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\)	\(225\)

Input:

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

Output:

((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*d*x+a*e)^(1/2)*c*d*e^2*x+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^ 
2-c*d^2)*e)^(1/2))*(c*d*x+a*e)^(1/2)*c*d^2*e-3*((a*e^2-c*d^2)*e)^(1/2)*c*d 
*e*x-((a*e^2-c*d^2)*e)^(1/2)*a*e^2-2*((a*e^2-c*d^2)*e)^(1/2)*c*d^2)/(e*x+d 
)^(3/2)/(c*d*x+a*e)/(a*e^2-c*d^2)^2/((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. 371 vs. \(2 (176) = 352\).

Time = 0.12 (sec) , antiderivative size = 764, normalized size of antiderivative = 3.90 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{3} + a c d^{3} e + {\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x^{2} + {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2}\right )} x\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (-\frac {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 + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (a c^{2} d^{6} e - 2 \, a^{2} c d^{4} e^{3} + a^{3} d^{2} e^{5} + {\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{3} + {\left (2 \, c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + a^{3} e^{7}\right )} x^{2} + {\left (c^{3} d^{7} - 3 \, a^{2} c d^{3} e^{4} + 2 \, a^{3} d e^{6}\right )} x\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{3} + a c d^{3} e + {\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x^{2} + {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2}\right )} x\right )} \sqrt {\frac {e}{c d^{2} - a e^{2}}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {e}{c d^{2} - a e^{2}}}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt {e x + d}}{a c^{2} d^{6} e - 2 \, a^{2} c d^{4} e^{3} + a^{3} d^{2} e^{5} + {\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{3} + {\left (2 \, c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + a^{3} e^{7}\right )} x^{2} + {\left (c^{3} d^{7} - 3 \, a^{2} c d^{3} e^{4} + 2 \, a^{3} d e^{6}\right )} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

integrate(1/(e*x+d)**(1/2)/(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)*sqrt(d + e*x)), x)
 

Maxima [F]

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

integrate(1/(e*x+d)^(1/2)/(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)*sqrt(e*x + d)), 
 x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-{\left (\frac {3 \, c d e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right )}{{\left (c^{2} d^{4} {\left | e \right |} - 2 \, a c d^{2} e^{2} {\left | e \right |} + a^{2} e^{4} {\left | e \right |}\right )} \sqrt {c d^{2} e - a e^{3}}} + \frac {2 \, c^{2} d^{3} e^{2} - 2 \, a c d e^{4} + 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )} c d e}{{\left (c^{2} d^{4} {\left | e \right |} - 2 \, a c d^{2} e^{2} {\left | e \right |} + a^{2} e^{4} {\left | e \right |}\right )} {\left (\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} c d^{2} e - \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} a e^{3} + {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}\right )}}\right )} e \] Input:

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

Output:

-(3*c*d*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^2*d^4*abs(e) - 2*a*c*d^2*e^2*abs(e) + a^2*e^4*abs(e))*sqrt(c*d^2 
*e - a*e^3)) + (2*c^2*d^3*e^2 - 2*a*c*d*e^4 + 3*((e*x + d)*c*d*e - c*d^2*e 
 + a*e^3)*c*d*e)/((c^2*d^4*abs(e) - 2*a*c*d^2*e^2*abs(e) + a^2*e^4*abs(e)) 
*(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*c*d^2*e - sqrt((e*x + d)*c*d*e - 
 c*d^2*e + a*e^3)*a*e^3 + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2))))*e
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {3 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c \,d^{2}+3 \sqrt {e}\, \sqrt {c d x +a e}\, \sqrt {-a \,e^{2}+c \,d^{2}}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {e}\, \sqrt {-a \,e^{2}+c \,d^{2}}}\right ) c d e x -a^{2} e^{4}-a c \,d^{2} e^{2}-3 a c d \,e^{3} x +2 c^{2} d^{4}+3 c^{2} d^{3} e x}{\sqrt {c d x +a e}\, \left (a^{3} e^{7} x -3 a^{2} c \,d^{2} e^{5} x +3 a \,c^{2} d^{4} e^{3} x -c^{3} d^{6} e x +a^{3} d \,e^{6}-3 a^{2} c \,d^{3} e^{4}+3 a \,c^{2} d^{5} e^{2}-c^{3} d^{7}\right )} \] Input:

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

Output:

(3*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*d**2 + 3*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*d*e*x - a**2*e**4 - a*c*d**2*e**2 - 3*a*c*d*e**3*x + 
 2*c**2*d**4 + 3*c**2*d**3*e*x)/(sqrt(a*e + c*d*x)*(a**3*d*e**6 + a**3*e** 
7*x - 3*a**2*c*d**3*e**4 - 3*a**2*c*d**2*e**5*x + 3*a*c**2*d**5*e**2 + 3*a 
*c**2*d**4*e**3*x - c**3*d**7 - c**3*d**6*e*x))