\(\int \frac {x (x^2 c_3-c_4)}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} (x+3 x^2 c_3+3 c_4) (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2)} \, dx\) [2728]

Optimal result

Integrand size = 90, antiderivative size = 250 \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right ) \sqrt {-1+c_1}}{2 \sqrt {1-c_0}}-\frac {\arctan \left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right ) \sqrt {1+c_1}}{4 \sqrt {-1-c_0}}+\frac {3 \arctan \left (\frac {\sqrt {1-3 c_0} \sqrt {-1+3 c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+3 c_0}\right ) \sqrt {-1+3 c_1}}{4 \sqrt {1-3 c_0}} \]

[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]

\[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx \]

[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*} \text {integral}& = \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] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(57413\) vs. \(2(250)=500\).

Time = 6.61 (sec) , antiderivative size = 57413, normalized size of antiderivative = 229.65 \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Result too large to show} \]

[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(942\) vs. \(2(181)=362\).

Time = 8.22 (sec) , antiderivative size = 943, normalized size of antiderivative = 3.77

method	result	size
default	\(\text {Expression too large to display}\)	\(943\)

[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,method=_RETURNVERBOSE)

[Out]

-1/8*(_C3*x^2+_C0*x+_C4)*(((-1+3*_C1)*(-1+3*_C0))^(1/2)*ln((x^2*_C0*_C3+x^2*_C1*_C3+2*x^2*_C3+2*x*_C0*_C1+2*((
1+_C1)*(1+_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)+x*_C0+x*_C1+_C0*_C4+_C1*_C4+2*_C4)*_C3/(
_C3*x^2+_C4-x))*((1+_C1)*(1+_C0))^(1/2)*_C0+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)*ln((x^2*_C0*_C3+x^2*_C1*_C3-2*x^2*
_C3+2*x*_C0*_C1+2*((-1+_C1)*(-1+_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C0-x*_C1+_C0*_C
4+_C1*_C4-2*_C4)*_C3/(_C3*x^2+_C4+x))*((-1+_C1)*(-1+_C0))^(1/2)*_C0-9*ln((3*x^2*_C0*_C3+3*x^2*_C1*_C3-2*x^2*_C
3+6*x*_C0*_C1+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C0-x*_C1+3*_C
0*_C4+3*_C1*_C4-2*_C4)*_C3/(3*_C3*x^2+3*_C4+x))*_C0^2*_C1-((-1+3*_C1)*(-1+3*_C0))^(1/2)*ln((x^2*_C0*_C3+x^2*_C
1*_C3+2*x^2*_C3+2*x*_C0*_C1+2*((1+_C1)*(1+_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)+x*_C0+x*
_C1+_C0*_C4+_C1*_C4+2*_C4)*_C3/(_C3*x^2+_C4-x))*((1+_C1)*(1+_C0))^(1/2)+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)*ln((x^
2*_C0*_C3+x^2*_C1*_C3-2*x^2*_C3+2*x*_C0*_C1+2*((-1+_C1)*(-1+_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C
4))^(1/2)-x*_C0-x*_C1+_C0*_C4+_C1*_C4-2*_C4)*_C3/(_C3*x^2+_C4+x))*((-1+_C1)*(-1+_C0))^(1/2)+3*ln((3*x^2*_C0*_C
3+3*x^2*_C1*_C3-2*x^2*_C3+6*x*_C0*_C1+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4)
)^(1/2)-x*_C0-x*_C1+3*_C0*_C4+3*_C1*_C4-2*_C4)*_C3/(3*_C3*x^2+3*_C4+x))*_C0^2+9*ln((3*x^2*_C0*_C3+3*x^2*_C1*_C
3-2*x^2*_C3+6*x*_C0*_C1+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C0-
x*_C1+3*_C0*_C4+3*_C1*_C4-2*_C4)*_C3/(3*_C3*x^2+3*_C4+x))*_C1-3*ln((3*x^2*_C0*_C3+3*x^2*_C1*_C3-2*x^2*_C3+6*x*
_C0*_C1+2*((-1+3*_C1)*(-1+3*_C0))^(1/2)*((_C3*x^2+_C1*x+_C4)*(_C3*x^2+_C0*x+_C4))^(1/2)-x*_C0-x*_C1+3*_C0*_C4+
3*_C1*_C4-2*_C4)*_C3/(3*_C3*x^2+3*_C4+x)))/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/((_C3*x^2+_C1*x+_C4
)*(_C3*x^2+_C0*x+_C4))^(1/2)/((-1+3*_C1)*(-1+3*_C0))^(1/2)/(_C0^2-1)

Fricas [A] (verification not implemented)

none

Time = 34.26 (sec) , antiderivative size = 6425, normalized size of antiderivative = 25.70 \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Too large to display} \]

[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]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \text {Failed to integrate} \]

[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)

Giac [F]

\[ \text {Failed to integrate} \]

[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(-1)]

Timed out. \[ \int \frac {x \left (x^2 c_3-c_4\right )}{\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (x+3 x^2 c_3+3 c_4\right ) \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=-\int \frac {x\,\left (_{\mathrm {C4}}-_{\mathrm {C3}}\,x^2\right )}{\sqrt {\frac {_{\mathrm {C3}}\,x^2+_{\mathrm {C0}}\,x+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^2+_{\mathrm {C1}}\,x+_{\mathrm {C4}}}}\,\left (3\,_{\mathrm {C3}}\,x^2+x+3\,_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^4+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^2+{_{\mathrm {C4}}}^2-x^2\right )} \,d x \]

[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)