3.28.28 \(\int \frac {x (x^2 c_3-c_4)}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} (x+3 x^2 c_3+3 c_4) (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2)} \, dx\) [2728]
Optimal. Leaf size=250 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right ) \sqrt {-1+c_1}}{2 \sqrt {1-c_0}}-\frac {\text {ArcTan}\left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right ) \sqrt {1+c_1}}{4 \sqrt {-1-c_0}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {1-3 c_0} \sqrt {-1+3 c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+3 c_0}\right ) \sqrt {-1+3 c_1}}{4 \sqrt {1-3 c_0}} \]
[Out]
-1/2*arctan((1-_C0)^(1/2)*(-1+_C1)^(1/2)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(-1+_C0))*(-1+_C1)^(1
/2)/(1-_C0)^(1/2)-1/4*arctan((-1-_C0)^(1/2)*(1+_C1)^(1/2)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(1+_
C0))*(1+_C1)^(1/2)/(-1-_C0)^(1/2)+3/4*arctan((1-3*_C0)^(1/2)*(-1+3*_C1)^(1/2)*((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C
1*x+_C4))^(1/2)/(-1+3*_C0))*(-1+3*_C1)^(1/2)/(1-3*_C0)^(1/2)
________________________________________________________________________________________
Rubi [F]
time = 11.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(x*(x^2*C[3] - C[4]))/(Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*(x + 3*x^2*C[3] + 3*C[4
])*(-x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]
[Out]
(-9*C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(Sqrt[x*C[0] + x^2*C[3] + C[
4]]*(1 + 6*x*C[3] - Sqrt[1 - 36*C[3]*C[4]])), x])/(4*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]
)]*Sqrt[x*C[1] + x^2*C[3] + C[4]]) - (9*C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][Sqrt[x*C[1] + x^2*C[3]
+ C[4]]/(Sqrt[x*C[0] + x^2*C[3] + C[4]]*(1 + 6*x*C[3] + Sqrt[1 - 36*C[3]*C[4]])), x])/(4*Sqrt[(x*C[0] + x^2*C[
3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*Sqrt[x*C[1] + x^2*C[3] + C[4]]) + (C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*
Defer[Int][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(Sqrt[x*C[0] + x^2*C[3] + C[4]]*(-1 + 2*x*C[3] - Sqrt[1 - 4*C[3]*C[4
]])), x])/(4*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*Sqrt[x*C[1] + x^2*C[3] + C[4]]) + (C[
3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(Sqrt[x*C[0] + x^2*C[3] + C[4]]*(1
+ 2*x*C[3] - Sqrt[1 - 4*C[3]*C[4]])), x])/(2*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*Sqrt
[x*C[1] + x^2*C[3] + C[4]]) + (C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(
Sqrt[x*C[0] + x^2*C[3] + C[4]]*(-1 + 2*x*C[3] + Sqrt[1 - 4*C[3]*C[4]])), x])/(4*Sqrt[(x*C[0] + x^2*C[3] + C[4]
)/(x*C[1] + x^2*C[3] + C[4])]*Sqrt[x*C[1] + x^2*C[3] + C[4]]) + (C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int
][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(Sqrt[x*C[0] + x^2*C[3] + C[4]]*(1 + 2*x*C[3] + Sqrt[1 - 4*C[3]*C[4]])), x])/
(2*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*Sqrt[x*C[1] + x^2*C[3] + C[4]])
Rubi steps
\begin {align*} \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx &=\int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (x^4 c_3{}^2+c_4{}^2+x^2 (-1+2 c_3 c_4)\right )} \, dx\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \frac {x \left (x^2 c_3-c_4\right ) \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (x+3 x^2 c_3+3 c_4\right ) \left (x^4 c_3{}^2+c_4{}^2+x^2 (-1+2 c_3 c_4)\right )} \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \left (\frac {(-1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{8 \left (-x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}}+\frac {(1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{4 \left (x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}}-\frac {3 (1+6 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{8 \sqrt {x c_0+x^2 c_3+c_4} \left (x+3 x^2 c_3+3 c_4\right )}\right ) \, dx}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \frac {(-1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\left (-x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{8 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \frac {(1+2 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\left (x+x^2 c_3+c_4\right ) \sqrt {x c_0+x^2 c_3+c_4}} \, dx}{4 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}-\frac {\left (3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {(1+6 x c_3) \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (x+3 x^2 c_3+3 c_4\right )} \, dx}{8 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \left (\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )}+\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )}\right ) \, dx}{8 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\sqrt {x c_0+x^2 c_3+c_4} \int \left (\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )}+\frac {2 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )}\right ) \, dx}{4 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}-\frac {\left (3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \left (\frac {6 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+6 x c_3-\sqrt {1-36 c_3 c_4}\right )}+\frac {6 c_3 \sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+6 x c_3+\sqrt {1-36 c_3 c_4}\right )}\right ) \, dx}{8 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ &=\frac {\left (c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{4 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (-1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{4 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3-\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}+\frac {\left (c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+2 x c_3+\sqrt {1-4 c_3 c_4}\right )} \, dx}{2 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}-\frac {\left (9 c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+6 x c_3-\sqrt {1-36 c_3 c_4}\right )} \, dx}{4 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}-\frac {\left (9 c_3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{\sqrt {x c_0+x^2 c_3+c_4} \left (1+6 x c_3+\sqrt {1-36 c_3 c_4}\right )} \, dx}{4 \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \sqrt {x c_1+x^2 c_3+c_4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(57413\) vs. \(2(250)=500\).
time = 6.50, size = 57413, normalized size = 229.65 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(x*(x^2*C[3] - C[4]))/(Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4])]*(x + 3*x^2*C[3] +
3*C[4])*(-x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]
[Out]
Result too large to show
________________________________________________________________________________________
Maple [F(-1)] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 180.00, size = 0, normalized size = 0.00 hanged
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x
^2+_C4^2-x^2),x)
[Out]
int(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3*_C4*x
^2+_C4^2-x^2),x)
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3
*_C4*x^2+_C4^2-x^2),x, algorithm="maxima")
[Out]
integrate((_C3*x^2 - _C4)*x/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 - x^2)*(3*_C3*x^2 + 3*_C4 + x)*sqrt((_C3*x^2 +
_C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))), x)
________________________________________________________________________________________
Fricas [A]
time = 34.32, size = 6425, normalized size = 25.70 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3
*_C4*x^2+_C4^2-x^2),x, algorithm="fricas")
[Out]
[3/16*sqrt((3*C1 - 1)/(3*C0 - 1))*log(-((9*C0^2 + 6*(9*C0 - 4)*C1 + 9*C1^2 - 24*C0 + 8)*C3^2*x^4 + 2*(9*(4*C0
- 1)*C1^2 - 9*C0^2 + 2*(18*C0^2 - 15*C0 + 2)*C1 + 4*C0)*C3*x^3 + (9*C0^2 + 6*(9*C0 - 4)*C1 + 9*C1^2 - 24*C0 +
8)*C4^2 + 2*(9*(4*C0 - 1)*C1^2 - 9*C0^2 + 2*(18*C0^2 - 15*C0 + 2)*C1 + 4*C0)*C4*x + ((72*C0^2 - 24*C0 + 1)*C1^
2 + 2*(9*C0^2 + 6*(9*C0 - 4)*C1 + 9*C1^2 - 24*C0 + 8)*C3*C4 + C0^2 - 6*(4*C0^2 - C0)*C1)*x^2 + 4*((9*C0^2 + 3*
(3*C0 - 1)*C1 - 9*C0 + 2)*C3^2*x^4 + (3*(3*C0 - 1)*C1^2 - 3*C0^2 + 3*(9*C0^2 - 6*C0 + 1)*C1 + C0)*C3*x^3 + (9*
C0^2 + 3*(3*C0 - 1)*C1 - 9*C0 + 2)*C4^2 + (3*(3*C0 - 1)*C1^2 - 3*C0^2 + 3*(9*C0^2 - 6*C0 + 1)*C1 + C0)*C4*x +
((18*C0^2 - 9*C0 + 1)*C1^2 + 2*(9*C0^2 + 3*(3*C0 - 1)*C1 - 9*C0 + 2)*C3*C4 - (3*C0^2 - C0)*C1)*x^2)*sqrt((C3*x
^2 + C0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt((3*C1 - 1)/(3*C0 - 1)))/(9*C3^2*x^4 + 6*C3*x^3 + (18*C3*C4 + 1)*x^2
+ 9*C4^2 + 6*C4*x)) + 1/8*sqrt((C1 - 1)/(C0 - 1))*log(-((C0^2 + 2*(3*C0 - 4)*C1 + C1^2 - 8*C0 + 8)*C3^2*x^4 +
2*((4*C0 - 3)*C1^2 - 3*C0^2 + 2*(2*C0^2 - 5*C0 + 2)*C1 + 4*C0)*C3*x^3 + (C0^2 + 2*(3*C0 - 4)*C1 + C1^2 - 8*C0
+ 8)*C4^2 + 2*((4*C0 - 3)*C1^2 - 3*C0^2 + 2*(2*C0^2 - 5*C0 + 2)*C1 + 4*C0)*C4*x + ((8*C0^2 - 8*C0 + 1)*C1^2 +
2*(C0^2 + 2*(3*C0 - 4)*C1 + C1^2 - 8*C0 + 8)*C3*C4 + C0^2 - 2*(4*C0^2 - 3*C0)*C1)*x^2 - 4*((C0^2 + (C0 - 1)*C
1 - 3*C0 + 2)*C3^2*x^4 + ((C0 - 1)*C1^2 - C0^2 + 3*(C0^2 - 2*C0 + 1)*C1 + C0)*C3*x^3 + (C0^2 + (C0 - 1)*C1 - 3
*C0 + 2)*C4^2 + ((C0 - 1)*C1^2 - C0^2 + 3*(C0^2 - 2*C0 + 1)*C1 + C0)*C4*x + ((2*C0^2 - 3*C0 + 1)*C1^2 + 2*(C0^
2 + (C0 - 1)*C1 - 3*C0 + 2)*C3*C4 - (C0^2 - C0)*C1)*x^2)*sqrt((C3*x^2 + C0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt(
(C1 - 1)/(C0 - 1)))/(C3^2*x^4 + 2*C3*x^3 + (2*C3*C4 + 1)*x^2 + C4^2 + 2*C4*x)) + 1/16*sqrt((C1 + 1)/(C0 + 1))*
log(-((C0^2 + 2*(3*C0 + 4)*C1 + C1^2 + 8*C0 + 8)*C3^2*x^4 + 2*((4*C0 + 3)*C1^2 + 3*C0^2 + 2*(2*C0^2 + 5*C0 + 2
)*C1 + 4*C0)*C3*x^3 + (C0^2 + 2*(3*C0 + 4)*C1 + C1^2 + 8*C0 + 8)*C4^2 + 2*((4*C0 + 3)*C1^2 + 3*C0^2 + 2*(2*C0^
2 + 5*C0 + 2)*C1 + 4*C0)*C4*x + ((8*C0^2 + 8*C0 + 1)*C1^2 + 2*(C0^2 + 2*(3*C0 + 4)*C1 + C1^2 + 8*C0 + 8)*C3*C4
+ C0^2 + 2*(4*C0^2 + 3*C0)*C1)*x^2 - 4*((C0^2 + (C0 + 1)*C1 + 3*C0 + 2)*C3^2*x^4 + ((C0 + 1)*C1^2 + C0^2 + 3*
(C0^2 + 2*C0 + 1)*C1 + C0)*C3*x^3 + (C0^2 + (C0 + 1)*C1 + 3*C0 + 2)*C4^2 + ((C0 + 1)*C1^2 + C0^2 + 3*(C0^2 + 2
*C0 + 1)*C1 + C0)*C4*x + ((2*C0^2 + 3*C0 + 1)*C1^2 + 2*(C0^2 + (C0 + 1)*C1 + 3*C0 + 2)*C3*C4 + (C0^2 + C0)*C1)
*x^2)*sqrt((C3*x^2 + C0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt((C1 + 1)/(C0 + 1)))/(C3^2*x^4 - 2*C3*x^3 + (2*C3*C4
+ 1)*x^2 + C4^2 - 2*C4*x)), 1/4*sqrt(-(C1 - 1)/(C0 - 1))*arctan(2*((C0 - 1)*C3*x^2 + (C0 - 1)*C1*x + (C0 - 1)
*C4)*sqrt((C3*x^2 + C0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt(-(C1 - 1)/(C0 - 1))/((C0 + C1 - 2)*C3*x^2 + (C0 + C1
- 2)*C4 + ((2*C0 - 1)*C1 - C0)*x)) + 3/16*sqrt((3*C1 - 1)/(3*C0 - 1))*log(-((9*C0^2 + 6*(9*C0 - 4)*C1 + 9*C1^
2 - 24*C0 + 8)*C3^2*x^4 + 2*(9*(4*C0 - 1)*C1^2 - 9*C0^2 + 2*(18*C0^2 - 15*C0 + 2)*C1 + 4*C0)*C3*x^3 + (9*C0^2
+ 6*(9*C0 - 4)*C1 + 9*C1^2 - 24*C0 + 8)*C4^2 + 2*(9*(4*C0 - 1)*C1^2 - 9*C0^2 + 2*(18*C0^2 - 15*C0 + 2)*C1 + 4*
C0)*C4*x + ((72*C0^2 - 24*C0 + 1)*C1^2 + 2*(9*C0^2 + 6*(9*C0 - 4)*C1 + 9*C1^2 - 24*C0 + 8)*C3*C4 + C0^2 - 6*(4
*C0^2 - C0)*C1)*x^2 + 4*((9*C0^2 + 3*(3*C0 - 1)*C1 - 9*C0 + 2)*C3^2*x^4 + (3*(3*C0 - 1)*C1^2 - 3*C0^2 + 3*(9*C
0^2 - 6*C0 + 1)*C1 + C0)*C3*x^3 + (9*C0^2 + 3*(3*C0 - 1)*C1 - 9*C0 + 2)*C4^2 + (3*(3*C0 - 1)*C1^2 - 3*C0^2 + 3
*(9*C0^2 - 6*C0 + 1)*C1 + C0)*C4*x + ((18*C0^2 - 9*C0 + 1)*C1^2 + 2*(9*C0^2 + 3*(3*C0 - 1)*C1 - 9*C0 + 2)*C3*C
4 - (3*C0^2 - C0)*C1)*x^2)*sqrt((C3*x^2 + C0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt((3*C1 - 1)/(3*C0 - 1)))/(9*C3^
2*x^4 + 6*C3*x^3 + (18*C3*C4 + 1)*x^2 + 9*C4^2 + 6*C4*x)) + 1/16*sqrt((C1 + 1)/(C0 + 1))*log(-((C0^2 + 2*(3*C0
+ 4)*C1 + C1^2 + 8*C0 + 8)*C3^2*x^4 + 2*((4*C0 + 3)*C1^2 + 3*C0^2 + 2*(2*C0^2 + 5*C0 + 2)*C1 + 4*C0)*C3*x^3 +
(C0^2 + 2*(3*C0 + 4)*C1 + C1^2 + 8*C0 + 8)*C4^2 + 2*((4*C0 + 3)*C1^2 + 3*C0^2 + 2*(2*C0^2 + 5*C0 + 2)*C1 + 4*
C0)*C4*x + ((8*C0^2 + 8*C0 + 1)*C1^2 + 2*(C0^2 + 2*(3*C0 + 4)*C1 + C1^2 + 8*C0 + 8)*C3*C4 + C0^2 + 2*(4*C0^2 +
3*C0)*C1)*x^2 - 4*((C0^2 + (C0 + 1)*C1 + 3*C0 + 2)*C3^2*x^4 + ((C0 + 1)*C1^2 + C0^2 + 3*(C0^2 + 2*C0 + 1)*C1
+ C0)*C3*x^3 + (C0^2 + (C0 + 1)*C1 + 3*C0 + 2)*C4^2 + ((C0 + 1)*C1^2 + C0^2 + 3*(C0^2 + 2*C0 + 1)*C1 + C0)*C4*
x + ((2*C0^2 + 3*C0 + 1)*C1^2 + 2*(C0^2 + (C0 + 1)*C1 + 3*C0 + 2)*C3*C4 + (C0^2 + C0)*C1)*x^2)*sqrt((C3*x^2 +
C0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt((C1 + 1)/(C0 + 1)))/(C3^2*x^4 - 2*C3*x^3 + (2*C3*C4 + 1)*x^2 + C4^2 - 2*
C4*x)), 1/8*sqrt(-(C1 + 1)/(C0 + 1))*arctan(2*((C0 + 1)*C3*x^2 + (C0 + 1)*C1*x + (C0 + 1)*C4)*sqrt((C3*x^2 + C
0*x + C4)/(C3*x^2 + C1*x + C4))*sqrt(-(C1 + 1)/(C0 + 1))/((C0 + C1 + 2)*C3*x^2 + (C0 + C1 + 2)*C4 + ((2*C0 + 1
)*C1 + C0)*x)) + 3/16*sqrt((3*C1 - 1)/(3*C0 - 1))*log(-((9*C0^2 + 6*(9*C0 - 4)*C1 + 9*C1^2 - 24*C0 + 8)*C3^2*x
^4 + 2*(9*(4*C0 - 1)*C1^2 - 9*C0^2 + 2*(18*C0^2 - 15*C0 + 2)*C1 + 4*C0)*C3*x^3 + (9*C0^2 + 6*(9*C0 - 4)*C1 + 9
*C1^2 - 24*C0 + 8)*C4^2 + 2*(9*(4*C0 - 1)*C1^2 ...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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*(_C3*x**2-_C4)/((_C3*x**2+_C0*x+_C4)/(_C3*x**2+_C1*x+_C4))**(1/2)/(3*_C3*x**2+3*_C4+x)/(_C3**2*x**
4+2*_C3*_C4*x**2+_C4**2-x**2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*(_C3*x^2-_C4)/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(3*_C3*x^2+3*_C4+x)/(_C3^2*x^4+2*_C3
*_C4*x^2+_C4^2-x^2),x, algorithm="giac")
[Out]
integrate((_C3*x^2 - _C4)*x/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 - x^2)*(3*_C3*x^2 + 3*_C4 + x)*sqrt((_C3*x^2 +
_C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))), x)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x\,\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )}{\sqrt {\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}}\,\left (3\,_{\mathrm {C3}}\,x^2+x+3\,_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2-x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(x*(_C4 - _C3*x^2))/(((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2))^(1/2)*(3*_C4 + x + 3*_C3*x^2)*(_C4
^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)),x)
[Out]
-int((x*(_C4 - _C3*x^2))/(((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2))^(1/2)*(3*_C4 + x + 3*_C3*x^2)*(_C4
^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)), x)
________________________________________________________________________________________