\(\int \frac {(d+e x) \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^6} \, dx\) [10]

Optimal result

Integrand size = 38, antiderivative size = 387 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx=\frac {\left (c d^2-a e^2\right ) \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^4 d^3 e^4 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 a e x^5}+\frac {\left (7 c d^2-3 a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 a^2 d e^2 x^4}-\frac {\left (35 c^2 d^4-12 a c d^2 e^2-15 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^3 d^2 e^3 x^3}-\frac {\left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{9/2} d^{7/2} e^{9/2}} \] Output:

1/128*(-a*e^2+c*d^2)*(3*a^2*e^4+6*a*c*d^2*e^2+7*c^2*d^4)*(2*a*d*e+(a*e^2+c 
*d^2)*x)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^4/d^3/e^4/x^2-1/5*(a*d* 
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a/e/x^5+1/40*(-3*a*e^2+7*c*d^2)*(a*d*e+ 
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^2/d/e^2/x^4-1/240*(-15*a^2*e^4-12*a*c*d 
^2*e^2+35*c^2*d^4)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/a^3/d^2/e^3/x^3 
-1/128*(-a*e^2+c*d^2)^3*(3*a^2*e^4+6*a*c*d^2*e^2+7*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^(9/2) 
/d^(7/2)/e^(9/2)
 

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (105 c^4 d^8 x^4-10 a c^3 d^6 e x^3 (7 d+19 e x)+2 a^2 c^2 d^4 e^2 x^2 \left (28 d^2+61 d e x+18 e^2 x^2\right )-6 a^3 c d^2 e^3 x \left (8 d^3+16 d^2 e x+3 d e^2 x^2-5 e^3 x^3\right )-3 a^4 e^4 \left (128 d^4+176 d^3 e x+8 d^2 e^2 x^2-10 d e^3 x^3+15 e^4 x^4\right )\right )}{x^5}-\frac {15 \left (c d^2-a e^2\right )^3 \left (7 c^2 d^4+6 a c d^2 e^2+3 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 a^{9/2} d^{7/2} e^{9/2}} \] Input:

Integrate[((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x^6,x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(105*c^4*d^8*x^4 
- 10*a*c^3*d^6*e*x^3*(7*d + 19*e*x) + 2*a^2*c^2*d^4*e^2*x^2*(28*d^2 + 61*d 
*e*x + 18*e^2*x^2) - 6*a^3*c*d^2*e^3*x*(8*d^3 + 16*d^2*e*x + 3*d*e^2*x^2 - 
 5*e^3*x^3) - 3*a^4*e^4*(128*d^4 + 176*d^3*e*x + 8*d^2*e^2*x^2 - 10*d*e^3* 
x^3 + 15*e^4*x^4)))/x^5 - (15*(c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2 
 + 3*a^2*e^4)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d 
+ e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(1920*a^(9/2)*d^(7/2)*e^(9/2 
))
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1237, 27, 1237, 27, 1228, 1152, 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.

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

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int \frac {d \left (7 c d^2+4 c e x d-3 a e^2\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 x^5}dx}{5 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\left (7 c d^2+4 c e x d-3 a e^2\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^5}dx}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {\left (35 c^2 d^4-12 a c e^2 d^2+2 c e \left (7 c d^2-3 a e^2\right ) x d-15 a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{2 x^4}dx}{4 a d e}-\frac {\left (\frac {7 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 x^4}}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\left (35 c^2 d^4-12 a c e^2 d^2+2 c e \left (7 c d^2-3 a e^2\right ) x d-15 a^2 e^4\right ) \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^4}dx}{8 a d e}-\frac {\left (\frac {7 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 x^4}}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^3}dx}{2 a d e}-\frac {\left (\frac {35 c^2 d^4}{a}-15 a e^4-12 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 d e x^3}}{8 a d e}-\frac {\left (\frac {7 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 x^4}}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 1152

\(\displaystyle -\frac {-\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 a d e}-\frac {\left (\frac {35 c^2 d^4}{a}-15 a e^4-12 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 d e x^3}}{8 a d e}-\frac {\left (\frac {7 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 x^4}}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 a d e}-\frac {\left (\frac {35 c^2 d^4}{a}-15 a e^4-12 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 d e x^3}}{8 a d e}-\frac {\left (\frac {7 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 x^4}}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {-\frac {5 \left (c d^2-a e^2\right ) \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{2 a d e}-\frac {\left (\frac {35 c^2 d^4}{a}-15 a e^4-12 c d^2 e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 d e x^3}}{8 a d e}-\frac {\left (\frac {7 c d}{a e}-\frac {3 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 x^4}}{10 a e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 a e x^5}\)

Input:

Int[((d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/x^6,x]
 

Output:

-1/5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(a*e*x^5) - (-1/4*(((7* 
c*d)/(a*e) - (3*e)/d)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/x^4 - 
 (-1/3*(((35*c^2*d^4)/a - 12*c*d^2*e^2 - 15*a*e^4)*(a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2)^(3/2))/(d*e*x^3) - (5*(c*d^2 - a*e^2)*(7*c^2*d^4 + 6*a*c* 
d^2*e^2 + 3*a^2*e^4)*(-1/4*((2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c* 
d^2 + a*e^2)*x + c*d*e*x^2])/(a*d*e*x^2) + ((c*d^2 - a*e^2)^2*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])])/(8*a^(3/2)*d^(3/2)*e^(3/2))))/(2*a*d*e))/(8*a*d* 
e))/(10*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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 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. \(3582\) vs. \(2(355)=710\).

Time = 3.39 (sec) , antiderivative size = 3583, normalized size of antiderivative = 9.26

method	result	size
default	\(\text {Expression too large to display}\)	\(3583\)

Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/x^6,x,method=_RETURNVE 
RBOSE)
 

Output:

d*(-1/5/a/d/e/x^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-7/10*(a*e^2+c*d^ 
2)/a/d/e*(-1/4/a/d/e/x^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-5/8*(a*e^ 
2+c*d^2)/a/d/e*(-1/3/a/d/e/x^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)/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)^(3/ 
2)-1/4*(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)^( 
3/2)+1/2*(a*e^2+c*d^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 
))+2*c/a*(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)))+1/2*c/a*((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))))-1/4*c/a* 
(-1/2/a/d/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)-1/4*(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)^(3/2)+1/2*(a*e^2+c*d^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*...
 

Fricas [A] (verification not implemented)

Time = 10.54 (sec) , antiderivative size = 874, normalized size of antiderivative = 2.26 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^6,x, algorithm 
="fricas")
 

Output:

[-1/7680*(15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^ 
2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a^5*e^10)*sqrt(a*d*e)*x^5*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*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 8*(a*c*d^3* 
e + a^2*d*e^3)*x)/x^2) + 4*(384*a^5*d^5*e^5 - (105*a*c^4*d^9*e - 190*a^2*c 
^3*d^7*e^3 + 36*a^3*c^2*d^5*e^5 + 30*a^4*c*d^3*e^7 - 45*a^5*d*e^9)*x^4 + 2 
*(35*a^2*c^3*d^8*e^2 - 61*a^3*c^2*d^6*e^4 + 9*a^4*c*d^4*e^6 - 15*a^5*d^2*e 
^8)*x^3 - 8*(7*a^3*c^2*d^7*e^3 - 12*a^4*c*d^5*e^5 - 3*a^5*d^3*e^7)*x^2 + 4 
8*(a^4*c*d^6*e^4 + 11*a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a* 
e^2)*x))/(a^5*d^4*e^5*x^5), 1/3840*(15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6* 
a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a^5*e^10)*sqrt(- 
a*d*e)*x^5*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*(384*a^5*d^5*e^5 - (105*a*c^4*d^9*e - 190*a^2*c^ 
3*d^7*e^3 + 36*a^3*c^2*d^5*e^5 + 30*a^4*c*d^3*e^7 - 45*a^5*d*e^9)*x^4 + 2* 
(35*a^2*c^3*d^8*e^2 - 61*a^3*c^2*d^6*e^4 + 9*a^4*c*d^4*e^6 - 15*a^5*d^2*e^ 
8)*x^3 - 8*(7*a^3*c^2*d^7*e^3 - 12*a^4*c*d^5*e^5 - 3*a^5*d^3*e^7)*x^2 + 48 
*(a^4*c*d^6*e^4 + 11*a^5*d^4*e^6)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e 
^2)*x))/(a^5*d^4*e^5*x^5)]
 

Sympy [F]

\[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx=\int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )}{x^{6}}\, dx \] Input:

integrate((e*x+d)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**6,x)
 

Output:

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^6,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. 2352 vs. \(2 (355) = 710\).

Time = 0.18 (sec) , antiderivative size = 2352, normalized size of antiderivative = 6.08 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^6,x, algorithm 
="giac")
 

Output:

1/128*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e 
^6 + 3*a^4*c*d^2*e^8 - 3*a^5*e^10)*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))/(sqrt(-a*d*e)*a^4*d^3*e^4) - 
1/1920*(105*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))* 
a^4*c^5*d^14*e^4 + 3615*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2* 
x + a*d*e))*a^5*c^4*d^12*e^6 + 7770*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^ 
2*x + a*e^2*x + a*d*e))*a^6*c^3*d^10*e^8 + 3870*(sqrt(c*d*e)*x - sqrt(c*d* 
e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^7*c^2*d^8*e^10 + 45*(sqrt(c*d*e)*x - 
 sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^8*c*d^6*e^12 - 45*(sqrt(c* 
d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*a^9*d^4*e^14 + 1280* 
sqrt(c*d*e)*a^6*c^3*d^11*e^7 + 768*sqrt(c*d*e)*a^7*c^2*d^9*e^9 + 790*(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^5*d^13*e^ 
3 + 12570*(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*c^4*d^11*e^5 + 41820*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2 
*x + a*d*e))^3*a^5*c^3*d^9*e^7 + 34420*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c 
*d^2*x + a*e^2*x + a*d*e))^3*a^6*c^2*d^7*e^9 + 7470*(sqrt(c*d*e)*x - sqrt( 
c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^7*c*d^5*e^11 + 210*(sqrt(c*d*e 
)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^8*d^3*e^13 + 3840*s 
qrt(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*c^4*d^12*e^4 + 16640*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx=\int \frac {\left (d+e\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^6} \,d x \] Input:

int(((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/x^6,x)
 

Output:

int(((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/x^6, x)
 

Reduce [B] (verification not implemented)

Time = 40.97 (sec) , antiderivative size = 1910, normalized size of antiderivative = 4.94 \[ \int \frac {(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^6} \, dx =\text {Too large to display} \] Input:

int((e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^6,x)
 

Output:

( - 768*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**5*e**7 - 1056*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a**6*d**4*e**8*x - 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**6*d**3*e**9*x**2 + 60*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**2*e**10*x 
**3 - 90*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d*e**11*x**4 - 768*sqrt(d + 
e*x)*sqrt(a*e + c*d*x)*a**5*c*d**7*e**5 - 1152*sqrt(d + e*x)*sqrt(a*e + c* 
d*x)*a**5*c*d**6*e**6*x - 240*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**5* 
e**7*x**2 + 24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**4*e**8*x**3 - 30* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**3*e**9*x**4 - 96*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**4*c**2*d**8*e**4*x - 80*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**4*c**2*d**7*e**5*x**2 + 208*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2 
*d**6*e**6*x**3 + 132*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**5*e**7* 
x**4 + 112*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**9*e**3*x**2 + 104* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**8*e**4*x**3 - 308*sqrt(d + e* 
x)*sqrt(a*e + c*d*x)*a**3*c**3*d**7*e**5*x**4 - 140*sqrt(d + e*x)*sqrt(a*e 
 + c*d*x)*a**2*c**4*d**10*e**2*x**3 - 170*sqrt(d + e*x)*sqrt(a*e + c*d*x)* 
a**2*c**4*d**9*e**3*x**4 + 210*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**1 
1*e*x**4 - 45*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**6*e**12*x**5 + 75*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)*sqr...