3.4.21 \(\int \frac {x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\)
Optimal. Leaf size=259 \[ \frac {8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 \left (x \left (-2 a^3 e^6+a^2 c d^2 e^4+c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right )}{15 e \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2 x^2}{5 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 259, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used =
{854, 777, 613} \begin {gather*} \frac {8 \left (5 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {8 \left (x \left (a^2 c d^2 e^4-2 a^3 e^6+c^3 d^6\right )+a d e \left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )\right )}{15 e \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}+\frac {2 x^2}{5 (d+e x) \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}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
(2*x^2)/(5*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (8*(a*d*e*(c*d^2 - a*e^2
)*(c*d^2 + 3*a*e^2) + (c^3*d^6 + a^2*c*d^2*e^4 - 2*a^3*e^6)*x))/(15*e*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^
2)*x + c*d*e*x^2)^(3/2)) + (8*(c^2*d^4 + 10*a*c*d^2*e^2 + 5*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(15*e*(c*d^2
- a*e^2)^5*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])
Rule 613
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 777
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Rule 854
Int[(((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> -Si
mp[((2*c*d - b*e)*(f + g*x)^n*(a + b*x + c*x^2)^(p + 1))/(e*p*(b^2 - 4*a*c)*(d + e*x)), x] - Dist[1/(d*e*p*(b^
2 - 4*a*c)), Int[(f + g*x)^(n - 1)*(a + b*x + c*x^2)^p*Simp[b*(a*e*g*n - c*d*f*(2*p + 1)) - 2*a*c*(d*g*n - e*f
*(2*p + 1)) - c*g*(b*d - 2*a*e)*(n + 2*p + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && IGtQ[n, 0] && ILtQ[n + 2*p,
0]
Rubi steps
\begin {align*} \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}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {2 \int \frac {x \left (-2 a d e^2 \left (c d^2-a e^2\right )+2 c d^2 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 )^{5/2}} \, dx}{5 d e \left (c d^2-a e^2\right )^2}\\ &=\frac {2 x^2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {8 \left (a d e \left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (c^3 d^6+a^2 c d^2 e^4-2 a^3 e^6\right ) x\right )}{15 e \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {\left (4 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{15 e \left (c d^2-a e^2\right )^3}\\ &=\frac {2 x^2}{5 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {8 \left (a d e \left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right )+\left (c^3 d^6+a^2 c d^2 e^4-2 a^3 e^6\right ) x\right )}{15 e \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {8 \left (c^2 d^4+10 a c d^2 e^2+5 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{15 e \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 235, normalized size = 0.91 \begin {gather*} \frac {2 \left (a^4 e^6 \left (8 d^2+20 d e x+15 e^2 x^2\right )+4 a^3 c d e^4 \left (20 d^3+53 d^2 e x+45 d e^2 x^2+15 e^3 x^3\right )+2 a^2 c^2 d^2 e^2 \left (20 d^4+110 d^3 e x+189 d^2 e^2 x^2+110 d e^3 x^3+20 e^4 x^4\right )+4 a c^3 d^4 e x \left (15 d^3+45 d^2 e x+53 d e^2 x^2+20 e^3 x^3\right )+c^4 d^6 x^2 \left (15 d^2+20 d e x+8 e^2 x^2\right )\right )}{15 (d+e x) \left (c d^2-a e^2\right )^5 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
(2*(c^4*d^6*x^2*(15*d^2 + 20*d*e*x + 8*e^2*x^2) + a^4*e^6*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 4*a^3*c*d*e^4*(20*
d^3 + 53*d^2*e*x + 45*d*e^2*x^2 + 15*e^3*x^3) + 4*a*c^3*d^4*e*x*(15*d^3 + 45*d^2*e*x + 53*d*e^2*x^2 + 20*e^3*x
^3) + 2*a^2*c^2*d^2*e^2*(20*d^4 + 110*d^3*e*x + 189*d^2*e^2*x^2 + 110*d*e^3*x^3 + 20*e^4*x^4)))/(15*(c*d^2 - a
*e^2)^5*(d + e*x)*((a*e + c*d*x)*(d + e*x))^(3/2))
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.07, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[x^2/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
$Aborted
________________________________________________________________________________________
fricas [B] time = 27.13, size = 820, normalized size = 3.17 \begin {gather*} \frac {2 \, {\left (40 \, a^{2} c^{2} d^{6} e^{2} + 80 \, a^{3} c d^{4} e^{4} + 8 \, a^{4} d^{2} e^{6} + 8 \, {\left (c^{4} d^{6} e^{2} + 10 \, a c^{3} d^{4} e^{4} + 5 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{4} + 4 \, {\left (5 \, c^{4} d^{7} e + 53 \, a c^{3} d^{5} e^{3} + 55 \, a^{2} c^{2} d^{3} e^{5} + 15 \, a^{3} c d e^{7}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{8} + 60 \, a c^{3} d^{6} e^{2} + 126 \, a^{2} c^{2} d^{4} e^{4} + 60 \, a^{3} c d^{2} e^{6} + 5 \, a^{4} e^{8}\right )} x^{2} + 4 \, {\left (15 \, a c^{3} d^{7} e + 55 \, a^{2} c^{2} d^{5} e^{3} + 53 \, a^{3} c d^{3} e^{5} + 5 \, a^{4} d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15 \, {\left (a^{2} c^{5} d^{13} e^{2} - 5 \, a^{3} c^{4} d^{11} e^{4} + 10 \, a^{4} c^{3} d^{9} e^{6} - 10 \, a^{5} c^{2} d^{7} e^{8} + 5 \, a^{6} c d^{5} e^{10} - a^{7} d^{3} e^{12} + {\left (c^{7} d^{12} e^{3} - 5 \, a c^{6} d^{10} e^{5} + 10 \, a^{2} c^{5} d^{8} e^{7} - 10 \, a^{3} c^{4} d^{6} e^{9} + 5 \, a^{4} c^{3} d^{4} e^{11} - a^{5} c^{2} d^{2} e^{13}\right )} x^{5} + {\left (3 \, c^{7} d^{13} e^{2} - 13 \, a c^{6} d^{11} e^{4} + 20 \, a^{2} c^{5} d^{9} e^{6} - 10 \, a^{3} c^{4} d^{7} e^{8} - 5 \, a^{4} c^{3} d^{5} e^{10} + 7 \, a^{5} c^{2} d^{3} e^{12} - 2 \, a^{6} c d e^{14}\right )} x^{4} + {\left (3 \, c^{7} d^{14} e - 9 \, a c^{6} d^{12} e^{3} + a^{2} c^{5} d^{10} e^{5} + 25 \, a^{3} c^{4} d^{8} e^{7} - 35 \, a^{4} c^{3} d^{6} e^{9} + 17 \, a^{5} c^{2} d^{4} e^{11} - a^{6} c d^{2} e^{13} - a^{7} e^{15}\right )} x^{3} + {\left (c^{7} d^{15} + a c^{6} d^{13} e^{2} - 17 \, a^{2} c^{5} d^{11} e^{4} + 35 \, a^{3} c^{4} d^{9} e^{6} - 25 \, a^{4} c^{3} d^{7} e^{8} - a^{5} c^{2} d^{5} e^{10} + 9 \, a^{6} c d^{3} e^{12} - 3 \, a^{7} d e^{14}\right )} x^{2} + {\left (2 \, a c^{6} d^{14} e - 7 \, a^{2} c^{5} d^{12} e^{3} + 5 \, a^{3} c^{4} d^{10} e^{5} + 10 \, a^{4} c^{3} d^{8} e^{7} - 20 \, a^{5} c^{2} d^{6} e^{9} + 13 \, a^{6} c d^{4} e^{11} - 3 \, a^{7} d^{2} e^{13}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
2/15*(40*a^2*c^2*d^6*e^2 + 80*a^3*c*d^4*e^4 + 8*a^4*d^2*e^6 + 8*(c^4*d^6*e^2 + 10*a*c^3*d^4*e^4 + 5*a^2*c^2*d^
2*e^6)*x^4 + 4*(5*c^4*d^7*e + 53*a*c^3*d^5*e^3 + 55*a^2*c^2*d^3*e^5 + 15*a^3*c*d*e^7)*x^3 + 3*(5*c^4*d^8 + 60*
a*c^3*d^6*e^2 + 126*a^2*c^2*d^4*e^4 + 60*a^3*c*d^2*e^6 + 5*a^4*e^8)*x^2 + 4*(15*a*c^3*d^7*e + 55*a^2*c^2*d^5*e
^3 + 53*a^3*c*d^3*e^5 + 5*a^4*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^2*c^5*d^13*e^2 - 5*a^3*
c^4*d^11*e^4 + 10*a^4*c^3*d^9*e^6 - 10*a^5*c^2*d^7*e^8 + 5*a^6*c*d^5*e^10 - a^7*d^3*e^12 + (c^7*d^12*e^3 - 5*a
*c^6*d^10*e^5 + 10*a^2*c^5*d^8*e^7 - 10*a^3*c^4*d^6*e^9 + 5*a^4*c^3*d^4*e^11 - a^5*c^2*d^2*e^13)*x^5 + (3*c^7*
d^13*e^2 - 13*a*c^6*d^11*e^4 + 20*a^2*c^5*d^9*e^6 - 10*a^3*c^4*d^7*e^8 - 5*a^4*c^3*d^5*e^10 + 7*a^5*c^2*d^3*e^
12 - 2*a^6*c*d*e^14)*x^4 + (3*c^7*d^14*e - 9*a*c^6*d^12*e^3 + a^2*c^5*d^10*e^5 + 25*a^3*c^4*d^8*e^7 - 35*a^4*c
^3*d^6*e^9 + 17*a^5*c^2*d^4*e^11 - a^6*c*d^2*e^13 - a^7*e^15)*x^3 + (c^7*d^15 + a*c^6*d^13*e^2 - 17*a^2*c^5*d^
11*e^4 + 35*a^3*c^4*d^9*e^6 - 25*a^4*c^3*d^7*e^8 - a^5*c^2*d^5*e^10 + 9*a^6*c*d^3*e^12 - 3*a^7*d*e^14)*x^2 + (
2*a*c^6*d^14*e - 7*a^2*c^5*d^12*e^3 + 5*a^3*c^4*d^10*e^5 + 10*a^4*c^3*d^8*e^7 - 20*a^5*c^2*d^6*e^9 + 13*a^6*c*
d^4*e^11 - 3*a^7*d^2*e^13)*x)
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.5Unable to transpose Error: Bad Argument Value
________________________________________________________________________________________
maple [A] time = 0.01, size = 366, normalized size = 1.41 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (40 a^{2} c^{2} d^{2} e^{6} x^{4}+80 a \,c^{3} d^{4} e^{4} x^{4}+8 c^{4} d^{6} e^{2} x^{4}+60 a^{3} c d \,e^{7} x^{3}+220 a^{2} c^{2} d^{3} e^{5} x^{3}+212 a \,c^{3} d^{5} e^{3} x^{3}+20 c^{4} d^{7} e \,x^{3}+15 a^{4} e^{8} x^{2}+180 a^{3} c \,d^{2} e^{6} x^{2}+378 a^{2} c^{2} d^{4} e^{4} x^{2}+180 a \,c^{3} d^{6} e^{2} x^{2}+15 c^{4} d^{8} x^{2}+20 a^{4} d \,e^{7} x +212 a^{3} c \,d^{3} e^{5} x +220 a^{2} c^{2} d^{5} e^{3} x +60 a \,c^{3} d^{7} e x +8 a^{4} d^{2} e^{6}+80 a^{3} c \,d^{4} e^{4}+40 a^{2} c^{2} d^{6} e^{2}\right )}{15 \left (a^{5} e^{10}-5 a^{4} c \,d^{2} e^{8}+10 a^{3} c^{2} d^{4} e^{6}-10 a^{2} c^{3} d^{6} e^{4}+5 a \,c^{4} d^{8} e^{2}-c^{5} d^{10}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(e*x+d)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(5/2),x)
[Out]
-2/15*(c*d*x+a*e)*(40*a^2*c^2*d^2*e^6*x^4+80*a*c^3*d^4*e^4*x^4+8*c^4*d^6*e^2*x^4+60*a^3*c*d*e^7*x^3+220*a^2*c^
2*d^3*e^5*x^3+212*a*c^3*d^5*e^3*x^3+20*c^4*d^7*e*x^3+15*a^4*e^8*x^2+180*a^3*c*d^2*e^6*x^2+378*a^2*c^2*d^4*e^4*
x^2+180*a*c^3*d^6*e^2*x^2+15*c^4*d^8*x^2+20*a^4*d*e^7*x+212*a^3*c*d^3*e^5*x+220*a^2*c^2*d^5*e^3*x+60*a*c^3*d^7
*e*x+8*a^4*d^2*e^6+80*a^3*c*d^4*e^4+40*a^2*c^2*d^6*e^2)/(a^5*e^10-5*a^4*c*d^2*e^8+10*a^3*c^2*d^4*e^6-10*a^2*c^
3*d^6*e^4+5*a*c^4*d^8*e^2-c^5*d^10)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?
________________________________________________________________________________________
mupad [B] time = 4.33, size = 3099, normalized size = 11.97
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/((d + e*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2)),x)
[Out]
(((6*a*e^2 - 10*c*d^2)/(15*(a*e^2 - c*d^2)^4) - (4*c*d^2)/(5*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e +
c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((e*(2*a*e^3 - 2*c*d^2*e))/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (4
*c*d^2*e^2)/(5*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e + (e*(2*c*d^3 + 2*a*d*e^2))/(5*(a*e^2 - c*d^2)^3*(
3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + ((x*(a*e^2 + c*d^2) + a*d*
e + c*d*e*x^2)^(1/2)*(x*((((12*c^3*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))
- (4*c^3*d^3*e^2*(a*e^2 + c*d^2))/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(a*e^2 +
c*d^2))/(c*d*e) - (6*c^2*d^2*e*(a*e^2 + c*d^2))/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5)) + (8*a*c^3*d^4*e^3)/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (2*c^2*d^2*e*(46*a
^2*e^4 + 4*c^2*d^4 + 66*a*c*d^2*e^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a
*((12*c^3*d^3*e^2)/(5*(a*e^2 - c*d^2)^2*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^3*d^3*e^2*(a*e^2 +
c*d^2))/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - (c*d*(a*e^2 + c*d^2)*(46*a^2*
e^4 + 4*c^2*d^4 + 66*a*c*d^2*e^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/((a*e
+ c*d*x)*(d + e*x)) + ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x*((a*(((a*e^2 + c*d^2)*((4*c^4*d^4*e^3*
(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^4*d^4*e^3*(5*a*e^2
- c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (2*c^2*d^2*e^2*(10*c^
3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) -
(8*a*c^4*d^5*e^4)/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^3*d^3*e^2*(a*e^2
+ c*d^2)*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + ((a*e^2 +
c*d^2)*((a*((4*c^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)
) - (4*c^4*d^4*e^3*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c -
((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((4*c^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^
2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^4*d^4*e^3*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e
^3 + a^2*c*d*e^5))))/(c*d*e) - (2*c^2*d^2*e^2*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e^2 - c*d
^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^4*d^5*e^4)/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*
c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^3*d^3*e^2*(a*e^2 + c*d^2)*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5
*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(12*a^3*e^5 - 36*a^2*c*d^2*e^3))/(15*(a*e^2 -
c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(a*e^2 + c*d^2)*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 -
8*a^2*c*d*e^4))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (8*a^3*c^2*d^3
*e^6)/(5*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(12*a^3*e^5 - 36*a^2*c*d^2*e^
3)*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) + (a*((a*((4*c^4*d^4*e
^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^4*d^4*e^3*(5*a*e
^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^
2 + c*d^2)*((4*c^4*d^4*e^3*(a*e^2 + c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))
- (4*c^4*d^4*e^3*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d
*e) - (2*c^2*d^2*e^2*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^
2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^4*d^5*e^4)/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5
)) + (2*c^3*d^3*e^2*(a*e^2 + c*d^2)*(5*a*e^2 - c*d^2))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^
2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(12*a^3*e^5 - 36*a^2*c*d^2*e^3))/(15*(a*e^2 - c*d^2)^3*(c^3*d^5*e - 2*a
*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(a*e^2 + c*d^2)*(10*c^3*d^5 + 6*a*c^2*d^3*e^2 - 8*a^2*c*d*e^4))/(15*(a*e
^2 - c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (4*a^3*c*d^2*e^5*(a*e^2 + c*d^2))/(5*(a*e^2 -
c*d^2)^3*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/((a*e + c*d*x)^2*(d + e*x)^2) - (2*d^2*e*(x*(a*e^2 +
c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d + e*x)^3*(5*a^3*e^7 - 5*c^3*d^6*e + 15*a*c^2*d^4*e^3 - 15*a^2*c*d^2*e^5
))
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
Timed out
________________________________________________________________________________________