\(\int \frac {x^2 (d+e x)}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [140]

Optimal result

Integrand size = 38, antiderivative size = 123 \[ \int \frac {x^2 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 x^2 (d+e x)}{3 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 d \left (2 a d e+\left (c d^2+a e^2\right ) x\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \] Output:

-2/3*x^2*(e*x+d)/(-a*e^2+c*d^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+8/ 
3*d*(2*a*d*e+(a*e^2+c*d^2)*x)/(-a*e^2+c*d^2)^3/(a*d*e+(a*e^2+c*d^2)*x+c*d* 
e*x^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 (d+e x) \left (3 c^2 d^4 x^2+6 a c d^2 e x (2 d+e x)+a^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{3 \left (c d^2-a e^2\right )^3 ((a e+c d x) (d+e x))^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1227, 1158}

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.

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

\(\Big \downarrow \) 1227

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

\(\Big \downarrow \) 1158

\(\displaystyle \frac {8 d \left (x \left (a e^2+c d^2\right )+2 a d e\right )}{3 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 x^2 (d+e x)}{3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}\)

Input:

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

Output:

(-2*x^2*(d + e*x))/(3*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x 
^2)^(3/2)) + (8*d*(2*a*d*e + (c*d^2 + a*e^2)*x))/(3*(c*d^2 - a*e^2)^3*Sqrt 
[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
 

Defintions of rubi rules used

rule 1158

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x 
+ c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1227

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( 
e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m 
 - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
&& EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.24

method	result	size
gosper	\(-\frac {2 \left (e x +d \right )^{2} \left (c d x +a e \right ) \left (-a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}+3 c^{2} d^{4} x^{2}+4 a^{2} d \,e^{3} x +12 a c \,d^{3} e x +8 d^{2} e^{2} a^{2}\right )}{3 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(152\)
trager	\(-\frac {2 \left (-a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}+3 c^{2} d^{4} x^{2}+4 a^{2} d \,e^{3} x +12 a c \,d^{3} e x +8 d^{2} e^{2} a^{2}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (c d x +a e \right )^{2} \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )}\)	\(153\)
orering	\(-\frac {2 \left (-a^{2} e^{4} x^{2}+6 a c \,d^{2} e^{2} x^{2}+3 c^{2} d^{4} x^{2}+4 a^{2} d \,e^{3} x +12 a c \,d^{3} e x +8 d^{2} e^{2} a^{2}\right ) \left (e x +d \right )^{2} \left (c d x +a e \right )}{3 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {5}{2}}}\)	\(153\)
default	\(\text {Expression too large to display}\)	\(1155\)

Input:

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

Output:

-2/3*(e*x+d)^2*(c*d*x+a*e)*(-a^2*e^4*x^2+6*a*c*d^2*e^2*x^2+3*c^2*d^4*x^2+4 
*a^2*d*e^3*x+12*a*c*d^3*e*x+8*a^2*d^2*e^2)/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^ 
2*d^4*e^2-c^3*d^6)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (115) = 230\).

Time = 1.87 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.59 \[ \int \frac {x^2 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, a^{2} d^{2} e^{2} + {\left (3 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} x^{2} + 4 \, {\left (3 \, a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3 \, {\left (a^{2} c^{3} d^{7} e^{2} - 3 \, a^{3} c^{2} d^{5} e^{4} + 3 \, a^{4} c d^{3} e^{6} - a^{5} d e^{8} + {\left (c^{5} d^{8} e - 3 \, a c^{4} d^{6} e^{3} + 3 \, a^{2} c^{3} d^{4} e^{5} - a^{3} c^{2} d^{2} e^{7}\right )} x^{3} + {\left (c^{5} d^{9} - a c^{4} d^{7} e^{2} - 3 \, a^{2} c^{3} d^{5} e^{4} + 5 \, a^{3} c^{2} d^{3} e^{6} - 2 \, a^{4} c d e^{8}\right )} x^{2} + {\left (2 \, a c^{4} d^{8} e - 5 \, a^{2} c^{3} d^{6} e^{3} + 3 \, a^{3} c^{2} d^{4} e^{5} + a^{4} c d^{2} e^{7} - a^{5} e^{9}\right )} x\right )}} \] Input:

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

Output:

2/3*(8*a^2*d^2*e^2 + (3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*x^2 + 4*(3*a*c* 
d^3*e + a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c^3 
*d^7*e^2 - 3*a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 - a^5*d*e^8 + (c^5*d^8*e - 
3*a*c^4*d^6*e^3 + 3*a^2*c^3*d^4*e^5 - a^3*c^2*d^2*e^7)*x^3 + (c^5*d^9 - a* 
c^4*d^7*e^2 - 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^3*e^6 - 2*a^4*c*d*e^8)*x^2 + 
 (2*a*c^4*d^8*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^2*d^4*e^5 + a^4*c*d^2*e^7 - 
a^5*e^9)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.71 (sec) , antiderivative size = 1114, normalized size of antiderivative = 9.06 \[ \int \frac {x^2 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*a^2*e^3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(3*(c^5*d^8*x - 
3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a*c^4*d^7*e - a^4*c*d*e^7 - 3*a*c^ 
4*d^6*e^2*x + 3*a^2*c^3*d^4*e^4*x - a^3*c^2*d^2*e^6*x)) - (2*a^2*e^2*(a*d* 
e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2))/(3*c^5*d^7*x^2 + 3*a^2*c^3*d^5*e 
^2 - 6*a^3*c^2*d^3*e^4 + 3*a^4*c*d*e^6 + 6*a*c^4*d^6*e*x + 3*a^2*c^3*d^3*e 
^4*x^2 - 12*a^2*c^3*d^4*e^3*x + 6*a^3*c^2*d^2*e^5*x - 6*a*c^4*d^5*e^2*x^2) 
 + (2*a^3*d*e^6)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e 
 - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) 
 + (2*a^3*e^7*x)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9*e 
 - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9)) 
 + (22*a*c^2*d^5*e^2)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^ 
5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c* 
d*e^9)) - (28*a^2*c*d^3*e^4)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1 
/2)*(c^5*d^9*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + 
 a^4*c*d*e^9)) - (4*a*c*d^2*e*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2 
))/(3*(c^5*d^8*x - 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a*c^4*d^7*e - a 
^4*c*d*e^7 - 3*a*c^4*d^6*e^2*x + 3*a^2*c^3*d^4*e^4*x - a^3*c^2*d^2*e^6*x)) 
 + (2*c^3*d^6*e*x)/((a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1/2)*(c^5*d^9 
*e - 4*a*c^4*d^7*e^3 + 6*a^2*c^3*d^5*e^5 - 4*a^3*c^2*d^3*e^7 + a^4*c*d*e^9 
)) + (10*a*c^2*d^4*e^3*x)/(3*(a*d*e + a*e^2*x + c*d^2*x + c*d*e*x^2)^(1...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 498, normalized size of antiderivative = 4.05 \[ \int \frac {x^2 (d+e x)}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{3} d \,e^{5}}{3}+\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{3} e^{6} x}{3}+\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} c \,d^{2} e^{4} x}{3}+\frac {4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a^{2} c d \,e^{5} x^{2}}{3}+4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,c^{2} d^{5} e +4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, a \,c^{2} d^{4} e^{2} x +4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{3} d^{6} x +4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {c d x +a e}\, c^{3} d^{5} e \,x^{2}-\frac {16 \sqrt {e x +d}\, a^{2} c^{2} d^{4} e^{3}}{3}-\frac {8 \sqrt {e x +d}\, a^{2} c^{2} d^{3} e^{4} x}{3}+\frac {2 \sqrt {e x +d}\, a^{2} c^{2} d^{2} e^{5} x^{2}}{3}-8 \sqrt {e x +d}\, a \,c^{3} d^{5} e^{2} x -4 \sqrt {e x +d}\, a \,c^{3} d^{4} e^{3} x^{2}-2 \sqrt {e x +d}\, c^{4} d^{6} e \,x^{2}}{\sqrt {c d x +a e}\, c^{2} d^{2} e \left (a^{3} c d \,e^{7} x^{2}-3 a^{2} c^{2} d^{3} e^{5} x^{2}+3 a \,c^{3} d^{5} e^{3} x^{2}-c^{4} d^{7} e \,x^{2}+a^{4} e^{8} x -2 a^{3} c \,d^{2} e^{6} x +2 a \,c^{3} d^{6} e^{2} x -c^{4} d^{8} x +a^{4} d \,e^{7}-3 a^{3} c \,d^{3} e^{5}+3 a^{2} c^{2} d^{5} e^{3}-a \,c^{3} d^{7} e \right )} \] Input:

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

Output:

(2*(2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*d*e**5 + 2*sqrt(e)*sq 
rt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a**3*e**6*x + 2*sqrt(e)*sqrt(d)*sqrt(c)*sq 
rt(a*e + c*d*x)*a**2*c*d**2*e**4*x + 2*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + 
c*d*x)*a**2*c*d*e**5*x**2 + 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a* 
c**2*d**5*e + 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*a*c**2*d**4*e**2 
*x + 6*sqrt(e)*sqrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**6*x + 6*sqrt(e)*s 
qrt(d)*sqrt(c)*sqrt(a*e + c*d*x)*c**3*d**5*e*x**2 - 8*sqrt(d + e*x)*a**2*c 
**2*d**4*e**3 - 4*sqrt(d + e*x)*a**2*c**2*d**3*e**4*x + sqrt(d + e*x)*a**2 
*c**2*d**2*e**5*x**2 - 12*sqrt(d + e*x)*a*c**3*d**5*e**2*x - 6*sqrt(d + e* 
x)*a*c**3*d**4*e**3*x**2 - 3*sqrt(d + e*x)*c**4*d**6*e*x**2))/(3*sqrt(a*e 
+ c*d*x)*c**2*d**2*e*(a**4*d*e**7 + a**4*e**8*x - 3*a**3*c*d**3*e**5 - 2*a 
**3*c*d**2*e**6*x + a**3*c*d*e**7*x**2 + 3*a**2*c**2*d**5*e**3 - 3*a**2*c* 
*2*d**3*e**5*x**2 - a*c**3*d**7*e + 2*a*c**3*d**6*e**2*x + 3*a*c**3*d**5*e 
**3*x**2 - c**4*d**8*x - c**4*d**7*e*x**2))