\(\int \frac {(d+e x)^3}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [258]

Optimal result

Integrand size = 37, antiderivative size = 163 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^2}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2}+\frac {3 \sqrt {e} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{c^{5/2} d^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-\sqrt {c} \sqrt {d} (d+e x) \left (-3 a e^2+c d (2 d-e x)\right )+3 \sqrt {e} \left (c d^2-a e^2\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{c^{5/2} d^{5/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1124, 27, 1160, 1092, 219}

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.

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

\(\Big \downarrow \) 1124

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(-2*(c*d^2 - a*e^2)*(d + e*x))/(c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c 
*d*e*x^2]) + (e*(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] + (3*(c*d^2 - 
 a*e^2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqr 
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[c]*Sqrt[d]*Sqrt[e])))/ 
(c^2*d^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1124

Int[((d_.) + (e_.)*(x_))^(m_.)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x 
_Symbol] :> Simp[-2*e*(2*c*d - b*e)^(m - 2)*((d + e*x)/(c^(m - 1)*Sqrt[a + 
b*x + c*x^2])), x] + Simp[e^2/c^(m - 1)   Int[(1/Sqrt[a + b*x + c*x^2])*Exp 
andToSum[((2*c*d - b*e)^(m - 1) - c^(m - 1)*(d + e*x)^(m - 1))/(c*d - b*e - 
 c*e*x), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b*d*e + a*e 
^2, 0] && IGtQ[m, 0]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(959\) vs. \(2(145)=290\).

Time = 1.23 (sec) , antiderivative size = 960, normalized size of antiderivative = 5.89

method	result	size
default	\(\frac {2 d^{3} \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}+e^{3} \left (\frac {x^{2}}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {x}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{d e c \sqrt {d e c}}\right )}{2 d e c}-\frac {2 a \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{c}\right )+3 d \,e^{2} \left (-\frac {x}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )}{2 d e c}+\frac {\ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{d e c \sqrt {d e c}}\right )+3 d^{2} e \left (-\frac {1}{d e c \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (2 c d x e +a \,e^{2}+c \,d^{2}\right )}{d e c \left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}\right )\)	\(960\)

Input:

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

Output:

2*d^3*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^ 
2+c*d^2)*x+c*d*x^2*e)^(1/2)+e^3*(x^2/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-3/2*(a*e^2+c*d^2)/d/e/c*(-x/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2* 
e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2 
+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+1/d/e/c*ln((1/2*a*e^2+ 
1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/ 
(d*e*c)^(1/2))-2*a/c*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)-(a* 
e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/( 
a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)))+3*d*e^2*(-x/d/e/c/(a*d*e+(a*e^2+c 
*d^2)*x+c*d*x^2*e)^(1/2)-1/2*(a*e^2+c*d^2)/d/e/c*(-1/d/e/c/(a*d*e+(a*e^2+c 
*d^2)*x+c*d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a* 
c*d^2*e^2-(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))+1/d/e/ 
c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c* 
d*x^2*e)^(1/2))/(d*e*c)^(1/2))+3*d^2*e*(-1/d/e/c/(a*d*e+(a*e^2+c*d^2)*x+c* 
d*x^2*e)^(1/2)-(a*e^2+c*d^2)/d/e/c*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2- 
(a*e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.74 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt {\frac {e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x - 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{4 \, {\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, -\frac {3 \, {\left (a c d^{2} e - a^{2} e^{3} + {\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{2 \, {\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%{[%%%{2,[3,3,4]%%%},0]:[1,0,%%%{-1,[1,1,1]%%%}]%%},[2,2] 
%%%}+%%%{
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.41 \[ \int \frac {(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx=\frac {-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{2}+12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{2}+9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,e^{2}-9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c \,d^{2}+12 \sqrt {e x +d}\, a c d \,e^{2}-8 \sqrt {e x +d}\, c^{2} d^{3}+4 \sqrt {e x +d}\, c^{2} d^{2} e x}{4 \sqrt {c d x +a e}\, c^{3} d^{3}} \] Input:

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

Output:

( - 12*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c 
*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*e**2 + 12* 
sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*log((sqrt(e)*sqrt(a*e + c*d*x) + 
 sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c*d**2 + 9*sqrt(e)* 
sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*e**2 - 9*sqrt(e)*sqrt(d)*sqrt(c)*sqrt( 
a*e + c*d*x)*c*d**2 + 12*sqrt(d + e*x)*a*c*d*e**2 - 8*sqrt(d + e*x)*c**2*d 
**3 + 4*sqrt(d + e*x)*c**2*d**2*e*x)/(4*sqrt(a*e + c*d*x)*c**3*d**3)