\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^6 (d+e x)} \, dx\) [37]

Optimal result

Integrand size = 40, antiderivative size = 378 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=-\frac {\left (c d^2-a e^2\right ) \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 a^3 d^4 e^3 x^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 d x^5}-\frac {\left (\frac {3 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 x^4}+\frac {\left (15 c^2 d^4+12 a c d^2 e^2-35 a^2 e^4\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{240 a^2 d^3 e^2 x^3}+\frac {\left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 a^{7/2} d^{9/2} e^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (45 c^4 d^8 x^4-30 a c^3 d^6 e x^3 (d+e x)+6 a^2 c^2 d^4 e^2 x^2 \left (4 d^2+3 d e x-6 e^2 x^2\right )+2 a^3 c d^2 e^3 x \left (264 d^3+48 d^2 e x-61 d e^2 x^2+95 e^3 x^3\right )+a^4 e^4 \left (384 d^4+48 d^3 e x-56 d^2 e^2 x^2+70 d e^3 x^3-105 e^4 x^4\right )\right )}{x^5}+\frac {15 \left (c d^2-a e^2\right )^3 \left (3 c^2 d^4+6 a c d^2 e^2+7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} \sqrt {d+e x}}{\sqrt {d} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{1920 a^{7/2} d^{9/2} e^{7/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]
 

Output:

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-((Sqrt[a]*Sqrt[d]*Sqrt[e]*(45*c^4*d^8*x^4 
 - 30*a*c^3*d^6*e*x^3*(d + e*x) + 6*a^2*c^2*d^4*e^2*x^2*(4*d^2 + 3*d*e*x - 
 6*e^2*x^2) + 2*a^3*c*d^2*e^3*x*(264*d^3 + 48*d^2*e*x - 61*d*e^2*x^2 + 95* 
e^3*x^3) + a^4*e^4*(384*d^4 + 48*d^3*e*x - 56*d^2*e^2*x^2 + 70*d*e^3*x^3 - 
 105*e^4*x^4)))/x^5) + (15*(c*d^2 - a*e^2)^3*(3*c^2*d^4 + 6*a*c*d^2*e^2 + 
7*a^2*e^4)*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d 
*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(1920*a^(7/2)*d^(9/2)*e^(7/2))
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {1215, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x]
 

Output:

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

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 1215

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

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. \(10574\) vs. \(2(346)=692\).

Time = 4.76 (sec) , antiderivative size = 10575, normalized size of antiderivative = 27.98

method	result	size
default	\(\text {Expression too large to display}\)	\(10575\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/x^6/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 10.97 (sec) , antiderivative size = 872, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),x, algorithm 
="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**6/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{6}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),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)*x^6), x 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2352 vs. \(2 (346) = 692\).

Time = 0.21 (sec) , antiderivative size = 2352, normalized size of antiderivative = 6.22 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),x, algorithm 
="giac")
 

Output:

-1/128*(3*c^5*d^10 - 3*a*c^4*d^8*e^2 - 2*a^2*c^3*d^6*e^4 - 6*a^3*c^2*d^4*e 
^6 + 15*a^4*c*d^2*e^8 - 7*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^3*d^4*e^3) + 
 1/1920*(45*(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 - 45*(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 - 3870*(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 - 7770*(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 - 3615*(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 - 105*(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 - 768* 
sqrt(c*d*e)*a^7*c^2*d^9*e^9 - 1280*sqrt(c*d*e)*a^8*c*d^7*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^3*c^5*d^13*e^3 
 - 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^ 
4*c^4*d^11*e^5 - 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^5*c^3*d^9*e^7 - 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^6*c^2*d^7*e^9 - 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^7*c*d^5*e^11 - 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^8*d^3*e^13 - 7680*sq 
rt(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^5*c^3*d^10*e^6 - 23040*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^6\,\left (d+e\,x\right )} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)),x)
                                                                                    
                                                                                    
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^6*(d + e*x)), x)
 

Reduce [B] (verification not implemented)

Time = 45.78 (sec) , antiderivative size = 1910, normalized size of antiderivative = 5.05 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^6 (d+e x)} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^6/(e*x+d),x)
 

Output:

( - 768*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**5*e**7 - 96*sqrt(d + e*x)* 
sqrt(a*e + c*d*x)*a**6*d**4*e**8*x + 112*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
**6*d**3*e**9*x**2 - 140*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*d**2*e**10*x 
**3 + 210*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 - 80*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**5* 
e**7*x**2 + 104*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**4*e**8*x**3 - 17 
0*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**5*c*d**3*e**9*x**4 - 1056*sqrt(d + e* 
x)*sqrt(a*e + c*d*x)*a**4*c**2*d**8*e**4*x - 240*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 - 308*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**5* 
e**7*x**4 - 48*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**9*e**3*x**2 + 
24*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**8*e**4*x**3 + 132*sqrt(d + 
 e*x)*sqrt(a*e + c*d*x)*a**3*c**3*d**7*e**5*x**4 + 60*sqrt(d + e*x)*sqrt(a 
*e + c*d*x)*a**2*c**4*d**10*e**2*x**3 - 30*sqrt(d + e*x)*sqrt(a*e + c*d*x) 
*a**2*c**4*d**9*e**3*x**4 - 90*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**1 
1*e*x**4 + 105*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**6*e**12*x**5 - 120*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)*s...