\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x (d+e x)^3} \, dx\) [43]

Optimal result

Integrand size = 40, antiderivative size = 193 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x (d+e x)^3} \, dx=-\frac {2 \left (\frac {c d}{e}-\frac {a e}{d}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x}-\frac {2 a^{3/2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{d^{3/2}}+\frac {2 c^{3/2} d^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{e^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.48 (sec) , antiderivative size = 538, normalized size of antiderivative = 2.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x (d+e x)^3} \, dx=-\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (c d^{5/2} \sqrt {e} \left (c d^2-a e^2\right ) \sqrt {a e+c d x}+a^{3/2} e^3 \left (\sqrt {a} e-i \sqrt {c d^2-a e^2}\right ) \sqrt {c d^2-2 a e^2-2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d^2-2 a e^2-2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {d+e x}}{\sqrt {d} \sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )+a^{3/2} e^3 \left (\sqrt {a} e+i \sqrt {c d^2-a e^2}\right ) \sqrt {c d^2-2 a e^2+2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {d+e x} \arctan \left (\frac {\sqrt {c d^2-2 a e^2+2 i \sqrt {a} e \sqrt {c d^2-a e^2}} \sqrt {d+e x}}{\sqrt {d} \sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )+2 c^{5/2} d^5 \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {-\frac {c d^2}{e}+a e}-\sqrt {a e+c d x}\right )}\right )\right )}{c d^{7/2} e^{3/2} (a e+c d x)^{3/2} (d+e x)^2} \] Input:

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

Output:

(-2*((a*e + c*d*x)*(d + e*x))^(3/2)*(c*d^(5/2)*Sqrt[e]*(c*d^2 - a*e^2)*Sqr 
t[a*e + c*d*x] + a^(3/2)*e^3*(Sqrt[a]*e - I*Sqrt[c*d^2 - a*e^2])*Sqrt[c*d^ 
2 - 2*a*e^2 - (2*I)*Sqrt[a]*e*Sqrt[c*d^2 - a*e^2]]*Sqrt[d + e*x]*ArcTan[(S 
qrt[c*d^2 - 2*a*e^2 - (2*I)*Sqrt[a]*e*Sqrt[c*d^2 - a*e^2]]*Sqrt[d + e*x])/ 
(Sqrt[d]*Sqrt[e]*(Sqrt[-((c*d^2)/e) + a*e] - Sqrt[a*e + c*d*x]))] + a^(3/2 
)*e^3*(Sqrt[a]*e + I*Sqrt[c*d^2 - a*e^2])*Sqrt[c*d^2 - 2*a*e^2 + (2*I)*Sqr 
t[a]*e*Sqrt[c*d^2 - a*e^2]]*Sqrt[d + e*x]*ArcTan[(Sqrt[c*d^2 - 2*a*e^2 + ( 
2*I)*Sqrt[a]*e*Sqrt[c*d^2 - a*e^2]]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[e]*(Sqrt[ 
-((c*d^2)/e) + a*e] - Sqrt[a*e + c*d*x]))] + 2*c^(5/2)*d^5*Sqrt[d + e*x]*A 
rcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*(Sqrt[-((c*d^2)/e) + a*e] 
- Sqrt[a*e + c*d*x]))]))/(c*d^(7/2)*e^(3/2)*(a*e + c*d*x)^(3/2)*(d + e*x)^ 
2)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1214, 25, 27, 1269, 1092, 219, 1154, 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.

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

\(\Big \downarrow \) 1214

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1214

Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 3)*(Sqrt[a + b*x + c*x^2] 
/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[ExpandToS 
um[(((-d)^n*(-2*c*d + b*e)^(-m - 1))/(e^n*x^n) - ((-c)*d + b*e + c*e*x)^(-m 
 - 1))/(d + e*x), x]/(Sqrt[a + b*x + c*x^2]/x^n), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && ILtQ[n, 0] && 
EqQ[m + p, -3/2]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1315\) vs. \(2(163)=326\).

Time = 3.33 (sec) , antiderivative size = 1316, normalized size of antiderivative = 6.82

method	result	size
default	\(\text {Expression too large to display}\)	\(1316\)

Input:

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

Output:

1/d^3*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+1/2*(a*e^2+c*d^2)*(1/4* 
(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8* 
(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/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))+a*d*e*( 
(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)*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)-a*d*e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2 
)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/x)))-1/d^3*(1/3*(d*e*c*(x+d/e)^ 
2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1/4*(2*d*e*c*(x+d/e)+a*e 
^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c 
*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e))/(d*e*c)^(1/2)+(d*e*c* 
(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^(1/2)))-1/e/d^2*(2/(a*e^2- 
c*d^2)/(x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a* 
e^2-c*d^2)*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c 
*d^2)*(1/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d 
^2)*(x+d/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c 
*(x+d/e))/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d* 
e*c)^(1/2))))-1/e^2/d*(-2/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/e)^2+(a*e^2- 
c*d^2)*(x+d/e))^(5/2)+4*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2)/(x+d/e)^2*(d* 
e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a*e^2-c*d^2)*(1/3*(...
 

Fricas [A] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 1277, normalized size of antiderivative = 6.62 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x (d+e x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x (d+e x)^3} \, dx=\frac {2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a d \,e^{3}-2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c \,d^{3} e +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a d \,e^{3}+\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{4} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a d \,e^{3}+\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{4} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) a d \,e^{3}-\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) a \,e^{4} x +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \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^{4}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \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^{3} e x +2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a \,d^{2} e^{2}+2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a d \,e^{3} x -2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c \,d^{4}-2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c \,d^{3} e x}{d^{2} e^{2} \left (e x +d \right )} \] Input:

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

Output:

(2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*d*e**3 - 2*sqrt(d + e*x)*sqrt(a*e + c 
*d*x)*c*d**3*e + sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - s 
qrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e* 
x))*a*d*e**3 + sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqr 
t(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x) 
)*a*e**4*x + sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt( 
2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))* 
a*d*e**3 + sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2* 
sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a* 
e**4*x - sqrt(e)*sqrt(d)*sqrt(a)*log(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d + e* 
x)*sqrt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e*x)*a*d*e**3 - sqrt( 
e)*sqrt(d)*sqrt(a)*log(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e + 
c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e*x)*a*e**4*x + 2*sqrt(e)*sqrt(d)*s 
qrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqr 
t(a*e**2 - c*d**2))*c*d**4 + 2*sqrt(e)*sqrt(d)*sqrt(c)*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**3 
*e*x + 2*sqrt(e)*sqrt(d)*sqrt(c)*a*d**2*e**2 + 2*sqrt(e)*sqrt(d)*sqrt(c)*a 
*d*e**3*x - 2*sqrt(e)*sqrt(d)*sqrt(c)*c*d**4 - 2*sqrt(e)*sqrt(d)*sqrt(c)*c 
*d**3*e*x)/(d**2*e**2*(d + e*x))