\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{x^7 (d+e x)} \, dx\) [58]

Optimal result

Integrand size = 40, antiderivative size = 369 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=-\frac {\left (c d^2-a e^2\right )^3 \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 a^3 d^4 e^3 x^2}+\frac {\left (c d^2-a e^2\right ) \left (5 c d^2+7 a e^2\right ) \left (2 a d e+\left (c d^2+a e^2\right ) x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{192 a^2 d^3 e^2 x^4}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{6 d x^6}-\frac {\left (\frac {5 c}{a e}-\frac {7 e}{d^2}\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{60 x^5}+\frac {\left (c d^2-a e^2\right )^5 \left (5 c d^2+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{512 a^{7/2} d^{9/2} e^{7/2}} \] Output:

-1/512*(-a*e^2+c*d^2)^3*(7*a*e^2+5*c*d^2)*(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+1/192*(-a*e^2+c*d^2)*(7* 
a*e^2+5*c*d^2)*(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) 
^(3/2)/a^2/d^3/e^2/x^4-1/6*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/d/x^6-1 
/60*(5*c/a/e-7*e/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^5+1/512*(- 
a*e^2+c*d^2)^5*(7*a*e^2+5*c*d^2)*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 = 1.47 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {\left (-c d^2+a e^2\right )^5 ((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {a} \sqrt {d} \sqrt {e} \left (75 c^5 d^{10} x^5-5 a c^4 d^8 e x^4 (10 d+49 e x)+10 a^2 c^3 d^6 e^2 x^3 \left (4 d^2+16 d e x+15 e^2 x^2\right )+6 a^3 c^2 d^4 e^3 x^2 \left (360 d^3+564 d^2 e x+58 d e^2 x^2-91 e^3 x^3\right )+a^4 c d^2 e^4 x \left (3200 d^4+4448 d^3 e x+216 d^2 e^2 x^2-272 d e^3 x^3+415 e^4 x^4\right )+a^5 e^5 \left (1280 d^5+1664 d^4 e x+48 d^3 e^2 x^2-56 d^2 e^3 x^3+70 d e^4 x^4-105 e^5 x^5\right )\right )}{\left (c d^2-a e^2\right )^5 x^6 (a e+c d x) (d+e x)}-\frac {15 \left (5 c d^2+7 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{7680 a^{7/2} d^{9/2} e^{7/2}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]
 

Output:

((-(c*d^2) + a*e^2)^5*((a*e + c*d*x)*(d + e*x))^(3/2)*((Sqrt[a]*Sqrt[d]*Sq 
rt[e]*(75*c^5*d^10*x^5 - 5*a*c^4*d^8*e*x^4*(10*d + 49*e*x) + 10*a^2*c^3*d^ 
6*e^2*x^3*(4*d^2 + 16*d*e*x + 15*e^2*x^2) + 6*a^3*c^2*d^4*e^3*x^2*(360*d^3 
 + 564*d^2*e*x + 58*d*e^2*x^2 - 91*e^3*x^3) + a^4*c*d^2*e^4*x*(3200*d^4 + 
4448*d^3*e*x + 216*d^2*e^2*x^2 - 272*d*e^3*x^3 + 415*e^4*x^4) + a^5*e^5*(1 
280*d^5 + 1664*d^4*e*x + 48*d^3*e^2*x^2 - 56*d^2*e^3*x^3 + 70*d*e^4*x^4 - 
105*e^5*x^5)))/((c*d^2 - a*e^2)^5*x^6*(a*e + c*d*x)*(d + e*x)) - (15*(5*c* 
d^2 + 7*a*e^2)*ArcTanh[(Sqrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d 
 + e*x])])/((a*e + c*d*x)^(3/2)*(d + e*x)^(3/2))))/(7680*a^(7/2)*d^(9/2)*e 
^(7/2))
 

Rubi [A] (verified)

Time = 1.13 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1215, 1237, 27, 1228, 1152, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int -\frac {a e \left (5 c d^2-2 c e x d-7 a e^2\right ) \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{3/2}}{2 x^6}dx}{6 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

\(\displaystyle \frac {-\frac {\left (\frac {5 c^2 d^4}{a}-7 a e^4+2 c d^2 e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{x^3}dx}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d e}-\frac {\left (\frac {5 c d}{a e}-\frac {7 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 x^5}}{12 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {-\frac {\left (\frac {5 c^2 d^4}{a}-7 a e^4+2 c d^2 e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (-\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d e}-\frac {\left (\frac {5 c d}{a e}-\frac {7 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 x^5}}{12 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {-\frac {\left (\frac {5 c^2 d^4}{a}-7 a e^4+2 c d^2 e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (c d^2-a e^2\right )^2 \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}}{4 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d e}-\frac {\left (\frac {5 c d}{a e}-\frac {7 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 x^5}}{12 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\left (\frac {5 c^2 d^4}{a}-7 a e^4+2 c d^2 e^2\right ) \left (-\frac {3 \left (c d^2-a e^2\right )^2 \left (\frac {\left (c d^2-a e^2\right )^2 \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 a^{3/2} d^{3/2} e^{3/2}}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 a d e x^2}\right )}{16 a d e}-\frac {\left (x \left (a e^2+c d^2\right )+2 a d e\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{8 a d e x^4}\right )}{2 d e}-\frac {\left (\frac {5 c d}{a e}-\frac {7 e}{d}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 x^5}}{12 d}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{6 d x^6}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x]
 

Output:

-1/6*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d*x^6) + (-1/5*(((5*c* 
d)/(a*e) - (7*e)/d)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2))/x^5 - ( 
((5*c^2*d^4)/a + 2*c*d^2*e^2 - 7*a*e^4)*(-1/8*((2*a*d*e + (c*d^2 + a*e^2)* 
x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(a*d*e*x^4) - (3*(c*d^2 
- a*e^2)^2*(-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))))/(16*a*d*e)))/(2*d*e))/(12*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. \(32290\) vs. \(2(337)=674\).

Time = 4.56 (sec) , antiderivative size = 32291, normalized size of antiderivative = 87.51

method	result	size
default	\(\text {Expression too large to display}\)	\(32291\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)/x^7/(e*x+d),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [A] (verification not implemented)

Time = 33.00 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx =\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x, algorithm 
="fricas")
 

Output:

[-1/30720*(15*(5*c^6*d^12 - 18*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 20*a^ 
3*c^3*d^6*e^6 - 45*a^4*c^2*d^4*e^8 + 30*a^5*c*d^2*e^10 - 7*a^6*e^12)*sqrt( 
a*d*e)*x^6*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*(1280*a^6*d^6*e^6 + 
(75*a*c^5*d^11*e - 245*a^2*c^4*d^9*e^3 + 150*a^3*c^3*d^7*e^5 - 546*a^4*c^2 
*d^5*e^7 + 415*a^5*c*d^3*e^9 - 105*a^6*d*e^11)*x^5 - 2*(25*a^2*c^4*d^10*e^ 
2 - 80*a^3*c^3*d^8*e^4 - 174*a^4*c^2*d^6*e^6 + 136*a^5*c*d^4*e^8 - 35*a^6* 
d^2*e^10)*x^4 + 8*(5*a^3*c^3*d^9*e^3 + 423*a^4*c^2*d^7*e^5 + 27*a^5*c*d^5* 
e^7 - 7*a^6*d^3*e^9)*x^3 + 16*(135*a^4*c^2*d^8*e^4 + 278*a^5*c*d^6*e^6 + 3 
*a^6*d^4*e^8)*x^2 + 128*(25*a^5*c*d^7*e^5 + 13*a^6*d^5*e^7)*x)*sqrt(c*d*e* 
x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^4*d^5*e^4*x^6), -1/15360*(15*(5*c^6*d 
^12 - 18*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 20*a^3*c^3*d^6*e^6 - 45*a^4 
*c^2*d^4*e^8 + 30*a^5*c*d^2*e^10 - 7*a^6*e^12)*sqrt(-a*d*e)*x^6*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*(1280*a^6*d^6*e^6 + (75*a*c^5*d^11*e - 245*a^2*c^4*d^9*e^3 + 150*a^3* 
c^3*d^7*e^5 - 546*a^4*c^2*d^5*e^7 + 415*a^5*c*d^3*e^9 - 105*a^6*d*e^11)*x^ 
5 - 2*(25*a^2*c^4*d^10*e^2 - 80*a^3*c^3*d^8*e^4 - 174*a^4*c^2*d^6*e^6 + 13 
6*a^5*c*d^4*e^8 - 35*a^6*d^2*e^10)*x^4 + 8*(5*a^3*c^3*d^9*e^3 + 423*a^4...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/x**7/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )} x^{7}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x, algorithm 
="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/((e*x + d)*x^7), x 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3388 vs. \(2 (337) = 674\).

Time = 0.31 (sec) , antiderivative size = 3388, normalized size of antiderivative = 9.18 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x, algorithm 
="giac")
 

Output:

-1/512*(5*c^6*d^12 - 18*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 20*a^3*c^3*d 
^6*e^6 - 45*a^4*c^2*d^4*e^8 + 30*a^5*c*d^2*e^10 - 7*a^6*e^12)*arctan(-(sqr 
t(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))/(s 
qrt(-a*d*e)*a^3*d^4*e^3) + 1/7680*(75*(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^6*d^17*e^5 - 270*(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^5*d^15*e^7 + 225*(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^4*d^13*e^9 + 15660*( 
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^3*d^11* 
e^11 + 14685*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)) 
*a^9*c^2*d^9*e^13 + 450*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2* 
x + a*d*e))*a^10*c*d^7*e^15 - 105*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2* 
x + a*e^2*x + a*d*e))*a^11*d^5*e^17 + 3072*sqrt(c*d*e)*a^9*c^2*d^10*e^12 - 
 425*(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 
^6*d^16*e^4 + 1530*(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^5*d^14*e^6 + 75525*(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^4*d^12*e^8 + 203100*(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^3*d^10*e^10 + 142065*(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*c^2*d^8*e^12 
+ 28170*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^3*a^ 
9*c*d^6*e^14 + 595*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{x^7\,\left (d+e\,x\right )} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)),x)
                                                                                    
                                                                                    
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)/(x^7*(d + e*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d e \,x^{2}\right )}^{\frac {5}{2}}}{x^{7} \left (e x +d \right )}d x \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x)
 

Output:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/x^7/(e*x+d),x)