\(\int \frac {d+e x}{x^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [79]

Optimal result

Integrand size = 38, antiderivative size = 279 \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 a e x^3}+\frac {\left (5 c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 a^2 d e^2 x^2}-\frac {\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^3 d^2 e^3 x}+\frac {\left (c d^2-a e^2\right ) \left (5 c^2 d^4+2 a c d^2 e^2+a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^{7/2} d^{5/2} e^{7/2}} \] Output:

-1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a/e/x^3+1/12*(-a*e^2+5*c*d^2) 
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^2/d/e^2/x^2-1/24*(-3*a*e^2+5*c* 
d^2)*(a*e^2+3*c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^3/d^2/e^3/x 
+1/8*(-a*e^2+c*d^2)*(a^2*e^4+2*a*c*d^2*e^2+5*c^2*d^4)*arctanh(a^(1/2)*e^(1 
/2)*(e*x+d)/d^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/a^(7/2)/d^(5/ 
2)/e^(7/2)
 

Mathematica [A] (verified)

Time = 10.19 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.92 \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-\sqrt {a} \sqrt {d} \sqrt {e} (d+e x) \left (15 c^3 d^5 x^3+a c^2 d^3 e x^2 (5 d-4 e x)+a^3 e^3 \left (8 d^2+2 d e x-3 e^2 x^2\right )-a^2 c d e^2 x \left (2 d^2+2 d e x+3 e^2 x^2\right )\right )+3 \left (5 c^3 d^6-3 a c^2 d^4 e^2-a^2 c d^2 e^4-a^3 e^6\right ) x^3 \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{24 a^{7/2} d^{5/2} e^{7/2} x^3 \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(d + e*x)/(x^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
 

Output:

(-(Sqrt[a]*Sqrt[d]*Sqrt[e]*(d + e*x)*(15*c^3*d^5*x^3 + a*c^2*d^3*e*x^2*(5* 
d - 4*e*x) + a^3*e^3*(8*d^2 + 2*d*e*x - 3*e^2*x^2) - a^2*c*d*e^2*x*(2*d^2 
+ 2*d*e*x + 3*e^2*x^2))) + 3*(5*c^3*d^6 - 3*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 
- a^3*e^6)*x^3*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*Sqrt[a*e + 
 c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(24*a^(7/2)*d^(5/2)*e^(7/2)*x^3 
*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1237, 27, 1237, 27, 1228, 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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {5 c d^2+4 c e x d-a e^2}{x^3 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{6 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a e x^3}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+2 c d e \left (5 c d^2-a e^2\right ) x}{2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a e x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\left (5 c d^2-3 a e^2\right ) \left (3 c d^2+a e^2\right )+2 c d e \left (5 c d^2-a e^2\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{4 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a e x^3}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {3 \left (c d^2-a e^2\right ) \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{2 a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a e x^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{a d e}-\frac {\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a e x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {3 \left (c d^2-a e^2\right ) \left (a^2 e^4+2 a c d^2 e^2+5 c^2 d^4\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (5 c d^2-3 a e^2\right ) \left (a e^2+3 c d^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{a d e x}}{4 a d e}-\frac {\left (\frac {5 c d}{a e}-\frac {e}{d}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 x^2}}{6 a e}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a e x^3}\)

Input:

Int[(d + e*x)/(x^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]
 

Output:

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

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 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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1237

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(732\) vs. \(2(251)=502\).

Time = 2.80 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.63

method	result	size
default	\(d \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{3 a d e \,x^{3}}-\frac {5 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 a d e \,x^{2}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )}{4 a d e}+\frac {c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a \sqrt {a d e}}\right )}{6 a d e}-\frac {2 c \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )}{3 a}\right )+e \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 a d e \,x^{2}}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )}{4 a d e}+\frac {c \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a \sqrt {a d e}}\right )\)	\(733\)

Input:

int((e*x+d)/x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 562, normalized size of antiderivative = 2.01 \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {3 \, {\left (5 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {a d e} x^{3} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} d^{3} e^{3} + {\left (15 \, a c^{2} d^{5} e - 4 \, a^{2} c d^{3} e^{3} - 3 \, a^{3} d e^{5}\right )} x^{2} - 2 \, {\left (5 \, a^{2} c d^{4} e^{2} - a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, a^{4} d^{3} e^{4} x^{3}}, -\frac {3 \, {\left (5 \, c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-a d e} x^{3} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} d^{3} e^{3} + {\left (15 \, a c^{2} d^{5} e - 4 \, a^{2} c d^{3} e^{3} - 3 \, a^{3} d e^{5}\right )} x^{2} - 2 \, {\left (5 \, a^{2} c d^{4} e^{2} - a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, a^{4} d^{3} e^{4} x^{3}}\right ] \] Input:

integrate((e*x+d)/x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm 
="fricas")
 

Output:

[-1/96*(3*(5*c^3*d^6 - 3*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a^3*e^6)*sqrt(a*d 
*e)*x^3*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*s 
qrt(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)*s 
qrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*(8*a^3*d^3*e^3 + (15*a* 
c^2*d^5*e - 4*a^2*c*d^3*e^3 - 3*a^3*d*e^5)*x^2 - 2*(5*a^2*c*d^4*e^2 - a^3* 
d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^3*e^4*x^3) 
, -1/48*(3*(5*c^3*d^6 - 3*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a^3*e^6)*sqrt(-a 
*d*e)*x^3*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*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a*c*d^ 
3*e + a^2*d*e^3)*x)) + 2*(8*a^3*d^3*e^3 + (15*a*c^2*d^5*e - 4*a^2*c*d^3*e^ 
3 - 3*a^3*d*e^5)*x^2 - 2*(5*a^2*c*d^4*e^2 - a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 
 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^3*e^4*x^3)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {d + e x}{x^{4} \sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \] Input:

integrate((e*x+d)/x**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)
 

Output:

Integral((d + e*x)/(x**4*sqrt((d + e*x)*(a*e + c*d*x))), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)/x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 914 vs. \(2 (251) = 502\).

Time = 0.15 (sec) , antiderivative size = 914, normalized size of antiderivative = 3.28 \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)/x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm 
="giac")
 

Output:

-1/8*(5*c^3*d^6 - 3*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - a^3*e^6)*arctan(-(sqrt 
(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))/sqrt(-a*d*e))/(sq 
rt(-a*d*e)*a^3*d^2*e^3) - 1/24*(33*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2 
*x + a*e^2*x + a*d*e))*a^2*c^3*d^8*e^2 + 105*(sqrt(c*d*e)*x - sqrt(c*d*e*x 
^2 + c*d^2*x + a*e^2*x + a*d*e))*a^3*c^2*d^6*e^4 + 51*(sqrt(c*d*e)*x - sqr 
t(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^4*c*d^4*e^6 + 3*(sqrt(c*d*e)*x 
 - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^5*d^2*e^8 + 48*sqrt(c*d* 
e)*a^3*c^2*d^7*e^3 + 16*sqrt(c*d*e)*a^4*c*d^5*e^5 - 40*(sqrt(c*d*e)*x - sq 
rt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a*c^3*d^7*e + 24*(sqrt(c*d*e) 
*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^2*c^2*d^5*e^3 + 72*( 
sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^3*c*d^3*e 
^5 + 8*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^4 
*d*e^7 + 144*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2 
*x + a*d*e))^2*a^3*c*d^4*e^4 + 48*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e* 
x^2 + c*d^2*x + a*e^2*x + a*d*e))^2*a^4*d^2*e^6 + 15*(sqrt(c*d*e)*x - sqrt 
(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*c^3*d^6 - 9*(sqrt(c*d*e)*x - sq 
rt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a*c^2*d^4*e^2 - 3*(sqrt(c*d*e 
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^2*c*d^2*e^4 - 3*(sq 
rt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^5*a^3*e^6)/((a* 
d*e - (sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {d+e\,x}{x^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:

int((d + e*x)/(x^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)),x)
 

Output:

int((d + e*x)/(x^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 1316, normalized size of antiderivative = 4.72 \[ \int \frac {d+e x}{x^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)/x^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
 

Output:

( - 16*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*d**3*e**5 - 4*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**4*d**2*e**6*x + 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4* 
d*e**7*x**2 - 16*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**5*e**3 + 16*sqr 
t(d + e*x)*sqrt(a*e + c*d*x)*a**3*c*d**4*e**4*x + 14*sqrt(d + e*x)*sqrt(a* 
e + c*d*x)*a**3*c*d**3*e**5*x**2 + 20*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2 
*c**2*d**6*e**2*x - 22*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**2*d**5*e**3 
*x**2 - 30*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**3*d**7*e*x**2 + 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**4*e**8*x**3 + 6*sq 
rt(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**3*c*d**2*e** 
6*x**3 + 12*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 
**2*c**2*d**4*e**4*x**3 - 6*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*c**3*d**6*e**2*x**3 - 15*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) + sq 
rt(d)*sqrt(c)*sqrt(d + e*x))*c**4*d**8*x**3 + 3*sqrt(e)*sqrt(d)*sqrt(a)*lo 
g(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**4*e**8*x**3 + 6*sqrt(e)*sqrt(d)*s...