3.4.22 \(\int \frac {x^2}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}} \, dx\)
Optimal. Leaf size=341 \[ -\frac {128 c d \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {8 \left (x \left (3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right )+2 a d e \left (2 a e^2+c d^2\right )\right )}{35 e \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}+\frac {2 x^2}{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 341, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used =
{854, 777, 614, 613} \begin {gather*} -\frac {128 c d \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {16 \left (7 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right ) \left (a e^2+c d^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {8 \left (x \left (3 a^2 e^4+a c d^2 e^2+2 c^2 d^4\right )+2 a d e \left (2 a e^2+c d^2\right )\right )}{35 e \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}+\frac {2 x^2}{7 (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 )^{5/2}} \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)^(7/2)),x]
[Out]
(2*x^2)/(7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)) - (8*(2*a*d*e*(c*d^2 + 2*a
*e^2) + (2*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4)*x))/(35*e*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x
^2)^(5/2)) + (16*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 + a*e^2 + 2*c*d*e*x))/(105*e*(c*d^2 - a*e^2)^
5*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (128*c*d*(3*c^2*d^4 + 14*a*c*d^2*e^2 + 7*a^2*e^4)*(c*d^2 +
a*e^2 + 2*c*d*e*x))/(105*(c*d^2 - a*e^2)^7*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 614
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]
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 )^{7/2}} \, dx &=\frac {2 x^2}{7 \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 )^{5/2}}+\frac {2 \int \frac {x \left (-2 a d e^2 \left (c d^2-a e^2\right )+4 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 )^{7/2}} \, dx}{7 d e \left (c d^2-a e^2\right )^2}\\ &=\frac {2 x^2}{7 \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 )^{5/2}}-\frac {8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}-\frac {\left (8 \left (3 c^2 d^4+14 a c d^2 e^2+7 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 )^{5/2}} \, dx}{35 e \left (c d^2-a e^2\right )^3}\\ &=\frac {2 x^2}{7 \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 )^{5/2}}-\frac {8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}+\frac {16 \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {\left (64 c d \left (3 c^2 d^4+14 a c d^2 e^2+7 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}{105 \left (c d^2-a e^2\right )^5}\\ &=\frac {2 x^2}{7 \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 )^{5/2}}-\frac {8 \left (2 a d e \left (c d^2+2 a e^2\right )+\left (2 c^2 d^4+a c d^2 e^2+3 a^2 e^4\right ) x\right )}{35 e \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}+\frac {16 \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 e \left (c d^2-a e^2\right )^5 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {128 c d \left (3 c^2 d^4+14 a c d^2 e^2+7 a^2 e^4\right ) \left (c d^2+a e^2+2 c d e x\right )}{105 \left (c d^2-a e^2\right )^7 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 433, normalized size = 1.27 \begin {gather*} -\frac {2 \sqrt {(d+e x) (a e+c d x)} \left (-a^6 e^{10} \left (8 d^2+28 d e x+35 e^2 x^2\right )+2 a^5 c d e^8 \left (112 d^3+382 d^2 e x+455 d e^2 x^2+140 e^3 x^3\right )+5 a^4 c^2 d^2 e^6 \left (336 d^4+1288 d^3 e x+1859 d^2 e^2 x^2+1288 d e^3 x^3+336 e^4 x^4\right )+20 a^3 c^3 d^3 e^4 \left (56 d^5+406 d^4 e x+1001 d^3 e^2 x^2+1084 d^2 e^3 x^3+560 d e^4 x^4+112 e^5 x^5\right )+a^2 c^4 d^4 e^2 \left (56 d^6+2996 d^5 e x+13195 d^4 e^2 x^2+24080 d^3 e^3 x^3+20320 d^2 e^4 x^4+7616 d e^5 x^5+896 e^6 x^6\right )+2 a c^5 d^6 e x \left (70 d^5+1295 d^4 e x+4060 d^3 e^2 x^2+5600 d^2 e^3 x^3+3616 d e^4 x^4+896 e^5 x^5\right )+3 c^6 d^8 x^2 \left (35 d^4+280 d^3 e x+560 d^2 e^2 x^2+448 d e^3 x^3+128 e^4 x^4\right )\right )}{105 (d+e x)^4 \left (c d^2-a e^2\right )^7 (a e+c d x)^3} \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)^(7/2)),x]
[Out]
(-2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(a^6*e^10*(8*d^2 + 28*d*e*x + 35*e^2*x^2)) + 2*a^5*c*d*e^8*(112*d^3 + 382*
d^2*e*x + 455*d*e^2*x^2 + 140*e^3*x^3) + 3*c^6*d^8*x^2*(35*d^4 + 280*d^3*e*x + 560*d^2*e^2*x^2 + 448*d*e^3*x^3
+ 128*e^4*x^4) + 5*a^4*c^2*d^2*e^6*(336*d^4 + 1288*d^3*e*x + 1859*d^2*e^2*x^2 + 1288*d*e^3*x^3 + 336*e^4*x^4)
+ 20*a^3*c^3*d^3*e^4*(56*d^5 + 406*d^4*e*x + 1001*d^3*e^2*x^2 + 1084*d^2*e^3*x^3 + 560*d*e^4*x^4 + 112*e^5*x^
5) + 2*a*c^5*d^6*e*x*(70*d^5 + 1295*d^4*e*x + 4060*d^3*e^2*x^2 + 5600*d^2*e^3*x^3 + 3616*d*e^4*x^4 + 896*e^5*x
^5) + a^2*c^4*d^4*e^2*(56*d^6 + 2996*d^5*e*x + 13195*d^4*e^2*x^2 + 24080*d^3*e^3*x^3 + 20320*d^2*e^4*x^4 + 761
6*d*e^5*x^5 + 896*e^6*x^6)))/(105*(c*d^2 - a*e^2)^7*(a*e + c*d*x)^3*(d + e*x)^4)
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 180.03, 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)^(7/2)),x]
[Out]
$Aborted
________________________________________________________________________________________
fricas [B] time = 167.97, size = 1540, normalized size = 4.52
result too large to display
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)^(7/2),x, algorithm="fricas")
[Out]
-2/105*(56*a^2*c^4*d^10*e^2 + 1120*a^3*c^3*d^8*e^4 + 1680*a^4*c^2*d^6*e^6 + 224*a^5*c*d^4*e^8 - 8*a^6*d^2*e^10
+ 128*(3*c^6*d^8*e^4 + 14*a*c^5*d^6*e^6 + 7*a^2*c^4*d^4*e^8)*x^6 + 64*(21*c^6*d^9*e^3 + 113*a*c^5*d^7*e^5 + 1
19*a^2*c^4*d^5*e^7 + 35*a^3*c^3*d^3*e^9)*x^5 + 80*(21*c^6*d^10*e^2 + 140*a*c^5*d^8*e^4 + 254*a^2*c^4*d^6*e^6 +
140*a^3*c^3*d^4*e^8 + 21*a^4*c^2*d^2*e^10)*x^4 + 40*(21*c^6*d^11*e + 203*a*c^5*d^9*e^3 + 602*a^2*c^4*d^7*e^5
+ 542*a^3*c^3*d^5*e^7 + 161*a^4*c^2*d^3*e^9 + 7*a^5*c*d*e^11)*x^3 + 5*(21*c^6*d^12 + 518*a*c^5*d^10*e^2 + 2639
*a^2*c^4*d^8*e^4 + 4004*a^3*c^3*d^6*e^6 + 1859*a^4*c^2*d^4*e^8 + 182*a^5*c*d^2*e^10 - 7*a^6*e^12)*x^2 + 4*(35*
a*c^5*d^11*e + 749*a^2*c^4*d^9*e^3 + 2030*a^3*c^3*d^7*e^5 + 1610*a^4*c^2*d^5*e^7 + 191*a^5*c*d^3*e^9 - 7*a^6*d
*e^11)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a^3*c^7*d^18*e^3 - 7*a^4*c^6*d^16*e^5 + 21*a^5*c^5*d^14
*e^7 - 35*a^6*c^4*d^12*e^9 + 35*a^7*c^3*d^10*e^11 - 21*a^8*c^2*d^8*e^13 + 7*a^9*c*d^6*e^15 - a^10*d^4*e^17 + (
c^10*d^17*e^4 - 7*a*c^9*d^15*e^6 + 21*a^2*c^8*d^13*e^8 - 35*a^3*c^7*d^11*e^10 + 35*a^4*c^6*d^9*e^12 - 21*a^5*c
^5*d^7*e^14 + 7*a^6*c^4*d^5*e^16 - a^7*c^3*d^3*e^18)*x^7 + (4*c^10*d^18*e^3 - 25*a*c^9*d^16*e^5 + 63*a^2*c^8*d
^14*e^7 - 77*a^3*c^7*d^12*e^9 + 35*a^4*c^6*d^10*e^11 + 21*a^5*c^5*d^8*e^13 - 35*a^6*c^4*d^6*e^15 + 17*a^7*c^3*
d^4*e^17 - 3*a^8*c^2*d^2*e^19)*x^6 + 3*(2*c^10*d^19*e^2 - 10*a*c^9*d^17*e^4 + 15*a^2*c^8*d^15*e^6 + 7*a^3*c^7*
d^13*e^8 - 49*a^4*c^6*d^11*e^10 + 63*a^5*c^5*d^9*e^12 - 35*a^6*c^4*d^7*e^14 + 5*a^7*c^3*d^5*e^16 + 3*a^8*c^2*d
^3*e^18 - a^9*c*d*e^20)*x^5 + (4*c^10*d^20*e - 10*a*c^9*d^18*e^3 - 30*a^2*c^8*d^16*e^5 + 155*a^3*c^7*d^14*e^7
- 245*a^4*c^6*d^12*e^9 + 147*a^5*c^5*d^10*e^11 + 35*a^6*c^4*d^8*e^13 - 95*a^7*c^3*d^6*e^15 + 45*a^8*c^2*d^4*e^
17 - 5*a^9*c*d^2*e^19 - a^10*e^21)*x^4 + (c^10*d^21 + 5*a*c^9*d^19*e^2 - 45*a^2*c^8*d^17*e^4 + 95*a^3*c^7*d^15
*e^6 - 35*a^4*c^6*d^13*e^8 - 147*a^5*c^5*d^11*e^10 + 245*a^6*c^4*d^9*e^12 - 155*a^7*c^3*d^7*e^14 + 30*a^8*c^2*
d^5*e^16 + 10*a^9*c*d^3*e^18 - 4*a^10*d*e^20)*x^3 + 3*(a*c^9*d^20*e - 3*a^2*c^8*d^18*e^3 - 5*a^3*c^7*d^16*e^5
+ 35*a^4*c^6*d^14*e^7 - 63*a^5*c^5*d^12*e^9 + 49*a^6*c^4*d^10*e^11 - 7*a^7*c^3*d^8*e^13 - 15*a^8*c^2*d^6*e^15
+ 10*a^9*c*d^4*e^17 - 2*a^10*d^2*e^19)*x^2 + (3*a^2*c^8*d^19*e^2 - 17*a^3*c^7*d^17*e^4 + 35*a^4*c^6*d^15*e^6 -
21*a^5*c^5*d^13*e^8 - 35*a^6*c^4*d^11*e^10 + 77*a^7*c^3*d^9*e^12 - 63*a^8*c^2*d^7*e^14 + 25*a^9*c*d^5*e^16 -
4*a^10*d^3*e^18)*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)^(7/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.6Unable to transpose Error: Bad Argument Value
________________________________________________________________________________________
maple [B] time = 0.02, size = 663, normalized size = 1.94 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-896 a^{2} c^{4} d^{4} e^{8} x^{6}-1792 a \,c^{5} d^{6} e^{6} x^{6}-384 c^{6} d^{8} e^{4} x^{6}-2240 a^{3} c^{3} d^{3} e^{9} x^{5}-7616 a^{2} c^{4} d^{5} e^{7} x^{5}-7232 a \,c^{5} d^{7} e^{5} x^{5}-1344 c^{6} d^{9} e^{3} x^{5}-1680 a^{4} c^{2} d^{2} e^{10} x^{4}-11200 a^{3} c^{3} d^{4} e^{8} x^{4}-20320 a^{2} c^{4} d^{6} e^{6} x^{4}-11200 a \,c^{5} d^{8} e^{4} x^{4}-1680 c^{6} d^{10} e^{2} x^{4}-280 a^{5} c d \,e^{11} x^{3}-6440 a^{4} c^{2} d^{3} e^{9} x^{3}-21680 a^{3} c^{3} d^{5} e^{7} x^{3}-24080 a^{2} c^{4} d^{7} e^{5} x^{3}-8120 a \,c^{5} d^{9} e^{3} x^{3}-840 c^{6} d^{11} e \,x^{3}+35 a^{6} e^{12} x^{2}-910 a^{5} c \,d^{2} e^{10} x^{2}-9295 a^{4} c^{2} d^{4} e^{8} x^{2}-20020 a^{3} c^{3} d^{6} e^{6} x^{2}-13195 a^{2} c^{4} d^{8} e^{4} x^{2}-2590 a \,c^{5} d^{10} e^{2} x^{2}-105 c^{6} d^{12} x^{2}+28 a^{6} d \,e^{11} x -764 a^{5} c \,d^{3} e^{9} x -6440 a^{4} c^{2} d^{5} e^{7} x -8120 a^{3} c^{3} d^{7} e^{5} x -2996 a^{2} c^{4} d^{9} e^{3} x -140 a \,c^{5} d^{11} e x +8 a^{6} d^{2} e^{10}-224 a^{5} c \,d^{4} e^{8}-1680 a^{4} c^{2} d^{6} e^{6}-1120 a^{3} c^{3} d^{8} e^{4}-56 a^{2} c^{4} d^{10} e^{2}\right )}{105 \left (a^{7} e^{14}-7 a^{6} c \,d^{2} e^{12}+21 a^{5} c^{2} d^{4} e^{10}-35 a^{4} c^{3} d^{6} e^{8}+35 a^{3} c^{4} d^{8} e^{6}-21 a^{2} c^{5} d^{10} e^{4}+7 a \,c^{6} d^{12} e^{2}-c^{7} d^{14}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {7}{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)^(7/2),x)
[Out]
-2/105*(c*d*x+a*e)*(-896*a^2*c^4*d^4*e^8*x^6-1792*a*c^5*d^6*e^6*x^6-384*c^6*d^8*e^4*x^6-2240*a^3*c^3*d^3*e^9*x
^5-7616*a^2*c^4*d^5*e^7*x^5-7232*a*c^5*d^7*e^5*x^5-1344*c^6*d^9*e^3*x^5-1680*a^4*c^2*d^2*e^10*x^4-11200*a^3*c^
3*d^4*e^8*x^4-20320*a^2*c^4*d^6*e^6*x^4-11200*a*c^5*d^8*e^4*x^4-1680*c^6*d^10*e^2*x^4-280*a^5*c*d*e^11*x^3-644
0*a^4*c^2*d^3*e^9*x^3-21680*a^3*c^3*d^5*e^7*x^3-24080*a^2*c^4*d^7*e^5*x^3-8120*a*c^5*d^9*e^3*x^3-840*c^6*d^11*
e*x^3+35*a^6*e^12*x^2-910*a^5*c*d^2*e^10*x^2-9295*a^4*c^2*d^4*e^8*x^2-20020*a^3*c^3*d^6*e^6*x^2-13195*a^2*c^4*
d^8*e^4*x^2-2590*a*c^5*d^10*e^2*x^2-105*c^6*d^12*x^2+28*a^6*d*e^11*x-764*a^5*c*d^3*e^9*x-6440*a^4*c^2*d^5*e^7*
x-8120*a^3*c^3*d^7*e^5*x-2996*a^2*c^4*d^9*e^3*x-140*a*c^5*d^11*e*x+8*a^6*d^2*e^10-224*a^5*c*d^4*e^8-1680*a^4*c
^2*d^6*e^6-1120*a^3*c^3*d^8*e^4-56*a^2*c^4*d^10*e^2)/(a^7*e^14-7*a^6*c*d^2*e^12+21*a^5*c^2*d^4*e^10-35*a^4*c^3
*d^6*e^8+35*a^3*c^4*d^8*e^6-21*a^2*c^5*d^10*e^4+7*a*c^6*d^12*e^2-c^7*d^14)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(
7/2)
________________________________________________________________________________________
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)^(7/2),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 = 7.72, size = 11469, normalized size = 33.63
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)^(7/2)),x)
[Out]
((6*c^3*d^5 + 36*a*c^2*d^3*e^2 - 10*a^2*c*d*e^4)/(105*(a*e^2 - c*d^2)^6) - x*((16*c^2*d^2*e)/(105*(a*e^2 - c*d
^2)^5) - (8*c^2*d^2*e*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6)) + (8*a*c^2*d^3*e^2)/(105*(a*e^2 - c*d^2)^6))/(
x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) + (x*((a*((64*c^5*d^5*e^4*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6
*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(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)*((64*c^5*d^5*e^4*(a*e^2 +
c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*(5*a*e^2 - 3*c*d
^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^4*d^4*e^3*(7*c^2*d^
4 - 9*a^2*e^4 + 18*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (128*a*
c^5*d^6*e^5)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^4*d^4*e^3*(a*e^2 + c*
d^2)*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(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*(60*c^4*d^7 - 204*a*c^3*d^5*e^2 - 156*a^2*c^2*d^3*e^4 + 44*a^3*c*d*e^6))/(105*(a*e^2 - c*d^2)^6*(c^
3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^3*d^3*e^2*(a*e^2 + c*d^2)*(7*c^2*d^4 - 9*a^2*e^4 + 18*a*c*d^
2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) - (a*(((a*e^2 + c*d^2)*((64*c^5*d
^5*e^4*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (64*c^5*d^5*e^4*
(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^4*d
^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 18*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d
*e^5)) - (128*a*c^5*d^6*e^5)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^4*d^4
*e^3*(a*e^2 + c*d^2)*(5*a*e^2 - 3*c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))
)/c + (c*d*e*(a*e^2 + c*d^2)*(60*c^4*d^7 - 204*a*c^3*d^5*e^2 - 156*a^2*c^2*d^3*e^4 + 44*a^3*c*d*e^6))/(105*(a*
e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2) + (
x*((a*((8*c^3*d^3*e^2*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6) - (8*c^3*d^3*e^2*(3*a*e^2 - c*d^2))/(105*(a*e^2
- c*d^2)^6)))/c + (36*c^4*d^7*e - 76*a*c^3*d^5*e^3 - 36*a^2*c^2*d^3*e^5 + 12*a^3*c*d*e^7)/(105*e*(a*e^2 - c*d
^2)^6) + ((a*e^2 + c*d^2)*((8*a*c^3*d^4*e^3)/(105*(a*e^2 - c*d^2)^6) - (((8*c^3*d^3*e^2*(a*e^2 + c*d^2))/(105*
(a*e^2 - c*d^2)^6) - (8*c^3*d^3*e^2*(3*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6))*(a*e^2 + c*d^2))/(c*d*e) + (2*
c^2*d^2*e*(11*c^2*d^4 - 13*a^2*e^4 + 14*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6)))/(c*d*e)) + (30*a^4*e^8 - 22*c^
4*d^8 + 20*a*c^3*d^6*e^2 - 132*a^3*c*d^2*e^6 + 72*a^2*c^2*d^4*e^4)/(105*e*(a*e^2 - c*d^2)^6) + (a*((8*a*c^3*d^
4*e^3)/(105*(a*e^2 - c*d^2)^6) - (((8*c^3*d^3*e^2*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6) - (8*c^3*d^3*e^2*(3
*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6))*(a*e^2 + c*d^2))/(c*d*e) + (2*c^2*d^2*e*(11*c^2*d^4 - 13*a^2*e^4 + 1
4*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6)))/c)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2) - (((d*((e*(2*a*e^4
- 2*c*d^2*e^2))/(7*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (4*c*d^2*e^3)/(7*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5
*c*d^2*e))))/e + (e*(2*a*d*e^3 + 2*c*d^3*e))/(7*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) +
a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((e*(10*a*e^3 - 14*c*d^2*e))/(35*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^
2*e)) - (4*c*d^2*e^2)/(7*(a*e^2 - c*d^2)^4*(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*((a*((4*c^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*
e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(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^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(
c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5
*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*d^2*e^2))/(35*(
a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^5*d^6*e^5)/(35*(a*e^2 - c*d^2)^4*(c^3*d
^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^4*d^4*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)
^4*(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*(14*c^4*d^7 - 56*a*c^3*d^5*e^2 - 12
*a^2*c^2*d^3*e^4 + 10*a^3*c*d*e^6))/(35*(a*e^2 - c*d^2)^4*(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)*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(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*(((a*e^2 + c*d^2)*((4*c^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e
^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*
d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*
d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^5*d^6*e^5)/(35*(a*e^2 -
c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^4*d^4*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(35*
(a*e^2 - c*d^2)^4*(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^5*d^5*e^4*(a*e
^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*
d^2))/(35*(a*e^2 - c*d^2)^4*(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^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c
^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (
4*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^
2*c*d*e^5)) - (8*a*c^5*d^6*e^5)/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^4*d^
4*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(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*(14*c^4*d^7 - 56*a*c^3*d^5*e^2 - 12*a^2*c^2*d^3*e^4 + 10*a^3*c*d*e^6))/(35*(a*e^2 - c
*d^2)^4*(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)*(7*c^2*d^4 - 9*a^2*e^4 +
4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(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
*(16*a^4*e^7 - 64*a^3*c*d^2*e^5))/(35*(a*e^2 - c*d^2)^4*(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)*(14*c^4*d^7 - 56*a*c^3*d^5*e^2 - 12*a^2*c^2*d^3*e^4 + 10*a^3*c*d*e^6))/(35*(a*e^2 - c*d^2)^4*(
c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (32*a^4*c^2*d^3*e^8)/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (c*d*e*(16*a^4*e^7 - 64*a^3*c*d^2*e^5)*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2
)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) - (a*((a*(((a*e^2 + c*d^2)*((4*c^5*d^5*e^4*(a*e^2 + c*d^2))/
(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*
e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (4*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4
+ 4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*a*c^5*d^6*e^5)/(35*(
a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (2*c^4*d^4*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^
2))/(35*(a*e^2 - c*d^2)^4*(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^5*d^5*
e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (4*c^5*d^5*e^4*(7*a*
e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(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^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)
) - (4*c^5*d^5*e^4*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*
d*e) - (4*c^4*d^4*e^3*(7*c^2*d^4 - 9*a^2*e^4 + 4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*
e^3 + a^2*c*d*e^5)) - (8*a*c^5*d^6*e^5)/(35*(a*e^2 - c*d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (
2*c^4*d^4*e^3*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(35*(a*e^2 - c*d^2)^4*(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*(14*c^4*d^7 - 56*a*c^3*d^5*e^2 - 12*a^2*c^2*d^3*e^4 + 10*a^3*c*d*e^6))/(35*(a
*e^2 - c*d^2)^4*(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)*(7*c^2*d^4 - 9*a
^2*e^4 + 4*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^4*(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*(16*a^4*e^7 - 64*a^3*c*d^2*e^5))/(35*(a*e^2 - c*d^2)^4*(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)*(14*c^4*d^7 - 56*a*c^3*d^5*e^2 - 12*a^2*c^2*d^3*e^4 + 10*a^3*c*d*e^6))/(35*(a*e^2 - c*
d^2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (16*a^4*c*d^2*e^7*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^
2)^4*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*
x)^3*(d + e*x)^3) - ((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*(x*((a*((16*c^5*d^5*e^4*(a*e^2 + c*d^2))/(3
5*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^5*d^5*e^4*(5*a*e^2 - 3*c*d^2))/(35*(a
*e^2 - c*d^2)^6*(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)*((16*c^5
*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^5*d^5*e^4
*(5*a*e^2 - 3*c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (16*c^4*d
^4*e^3*(c^2*d^4 - 7*a^2*e^4 + 14*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^
5)) - (32*a*c^5*d^6*e^5)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^4*d^4*e^3*(
a*e^2 + c*d^2)*(5*a*e^2 - 3*c*d^2))/(35*(a*e^2 - c*d^2)^6*(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*(484*c^4*d^7 + 1228*a*c^3*d^5*e^2 - 1092*a^2*c^2*d^3*e^4 - 812*a^3*c*d*e^6))/(105*(a*e^2 -
c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (8*c^3*d^3*e^2*(a*e^2 + c*d^2)*(c^2*d^4 - 7*a^2*e^4 +
14*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) - (a*(((a*e^2 + c*d^2)*(
(16*c^5*d^5*e^4*(a*e^2 + c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^5*
d^5*e^4*(5*a*e^2 - 3*c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (1
6*c^4*d^4*e^3*(c^2*d^4 - 7*a^2*e^4 + 14*a*c*d^2*e^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2
*c*d*e^5)) - (32*a*c^5*d^6*e^5)/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^4*d^
4*e^3*(a*e^2 + c*d^2)*(5*a*e^2 - 3*c*d^2))/(35*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))
)/c + (c*d*e*(a*e^2 + c*d^2)*(484*c^4*d^7 + 1228*a*c^3*d^5*e^2 - 1092*a^2*c^2*d^3*e^4 - 812*a^3*c*d*e^6))/(105
*(a*e^2 - c*d^2)^6*(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*((16*c^6*d^6*e^5*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e
- 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^5*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(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)*((16*c^6*d^6*e^5*(a*e^2 + c*d^2))/(105*(
a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^5*(7*a*e^2 - c*d^2))/(105*(a*e^2
- c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^5*d^5*e^4*(2*c^2*d^4 - 6*a^2*e^4 +
5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*a*c^6*d^7*e^6)/(105*
(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^4*(a*e^2 + c*d^2)*(7*a*e^2 - c*d
^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (8*c^4*d^4*e^3*(25*a^3*e^
6 + 5*c^3*d^6 - 23*a*c^2*d^4*e^2 - 51*a^2*c*d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^
2*c*d*e^5)) - (16*c^4*d^4*e^3*(a*e^2 + c*d^2)*(2*c^2*d^4 - 6*a^2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*
(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*(((a*e^2 + c*d^2)*((16*c^6*d^6*e^5*(a*
e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^5*(7*a*e^2 -
c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^5*d^5*e^4*(2*c^
2*d^4 - 6*a^2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*
a*c^6*d^7*e^6)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^4*(a*e^2 + c
*d^2)*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(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*((16*c^6*d^6*e^5*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)
) - (16*c^6*d^6*e^5*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(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)*((16*c^6*d^6*e^5*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*
a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^5*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2
*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^5*d^5*e^4*(2*c^2*d^4 - 6*a^2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*
d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*a*c^6*d^7*e^6)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2
*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^4*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3
*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (8*c^4*d^4*e^3*(25*a^3*e^6 + 5*c^3*d^6 - 23*a*c^2*d^4*e^2
- 51*a^2*c*d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^4*d^4*e^3*(a
*e^2 + c*d^2)*(2*c^2*d^4 - 6*a^2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a
^2*c*d*e^5))))/(c*d*e) - (32*c^3*d^3*e^2*(7*a^4*e^8 - 3*c^4*d^8 - 28*a^3*c*d^2*e^6 + 12*a^2*c^2*d^4*e^4))/(105
*(a*e^2 - c*d^2)^6*(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)*(25*a^3*e^6 +
5*c^3*d^6 - 23*a*c^2*d^4*e^2 - 51*a^2*c*d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(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*(88*c^5*d^10 - 152*a^5*e^10 + 80*a*c^4*d^8*e^2 + 272*a^4*c*d^2*e^8 - 520*a^2
*c^3*d^6*e^4 + 296*a^3*c^2*d^4*e^6))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16
*c^2*d^2*e*(a*e^2 + c*d^2)*(7*a^4*e^8 - 3*c^4*d^8 - 28*a^3*c*d^2*e^6 + 12*a^2*c^2*d^4*e^4))/(105*(a*e^2 - c*d^
2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))) - (a*((a*(((a*e^2 + c*d^2)*((16*c^6*d^6*e^5*(a*e^2 + c*d^2)
)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^5*(7*a*e^2 - c*d^2))/(10
5*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (32*c^5*d^5*e^4*(2*c^2*d^4 - 6*a^
2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*a*c^6*d^7*e^
6)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^4*(a*e^2 + c*d^2)*(7*a*e
^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(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*((
16*c^6*d^6*e^5*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^6*
d^6*e^5*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(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)*((16*c^6*d^6*e^5*(a*e^2 + c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^
3 + a^2*c*d*e^5)) - (16*c^6*d^6*e^5*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a
^2*c*d*e^5))))/(c*d*e) - (32*c^5*d^5*e^4*(2*c^2*d^4 - 6*a^2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*
d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (32*a*c^6*d^7*e^6)/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e
^3 + a^2*c*d*e^5)) + (8*c^5*d^5*e^4*(a*e^2 + c*d^2)*(7*a*e^2 - c*d^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a
*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (8*c^4*d^4*e^3*(25*a^3*e^6 + 5*c^3*d^6 - 23*a*c^2*d^4*e^2 - 51*a^2*c*
d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^4*d^4*e^3*(a*e^2 + c*d^2
)*(2*c^2*d^4 - 6*a^2*e^4 + 5*a*c*d^2*e^2))/(105*(a*e^2 - c*d^2)^6*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))
))/(c*d*e) - (32*c^3*d^3*e^2*(7*a^4*e^8 - 3*c^4*d^8 - 28*a^3*c*d^2*e^6 + 12*a^2*c^2*d^4*e^4))/(105*(a*e^2 - c*
d^2)^6*(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)*(25*a^3*e^6 + 5*c^3*d^6 -
23*a*c^2*d^4*e^2 - 51*a^2*c*d^2*e^4))/(105*(a*e^2 - c*d^2)^6*(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)*(88*c^5*d^10 - 152*a^5*e^10 + 80*a*c^4*d^8*e^2 + 272*a^4*c*d^2*e^8 - 520*a^2*c^3*d^6*e
^4 + 296*a^3*c^2*d^4*e^6))/(105*(a*e^2 - c*d^2)^6*(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^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((d + e*x)^4*(7*a^4*e^9 + 7*c^4*d
^8*e - 28*a*c^3*d^6*e^3 - 28*a^3*c*d^2*e^7 + 42*a^2*c^2*d^4*e^5)) + (8*c*d*e*(5*a*e^2 + c*d^2)*(x*(a*e^2 + c*d
^2) + a*d*e + c*d*e*x^2)^(1/2))/(105*(a*e^2 - c*d^2)^6*(d + e*x))
________________________________________________________________________________________
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)**(7/2),x)
[Out]
Timed out
________________________________________________________________________________________