\(\int \frac {(x^2 c_3-c_4) (x+3 x^2 c_3+3 c_4)}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2)} \, dx\) [2420]

Optimal result

Integrand size = 90, antiderivative size = 195 \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=6 \text {arctanh}\left (\sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}\right )-\frac {2 \arctan \left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{-1+c_0}\right ) \sqrt {-1+c_1}}{\sqrt {1-c_0}}-\frac {4 \arctan \left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}}}{1+c_0}\right ) \sqrt {1+c_1}}{\sqrt {-1-c_0}} \]

[Out]

6*arctanh(((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2))-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)-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)

Rubi [F]

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

[In]

Int[((x^2*C[3] - C[4])*(x + 3*x^2*C[3] + 3*C[4]))/(x*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[3] + C[4]
)]*(-x^2 + x^4*C[3]^2 + 2*x^2*C[3]*C[4] + C[4]^2)),x]

[Out]

(-3*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(x*Sqrt[x*C[0] + x^2*C[3] + C[4]]
), x])/(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]]) + (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])/(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]]) + (2*C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][Sqrt[x*C[1] + x^2*C[3] + C[4]]/(Sq
rt[x*C[0] + x^2*C[3] + C[4]]*(1 + 2*x*C[3] - Sqrt[1 - 4*C[3]*C[4]])), x])/(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]]) + (4*C[3]*Sqrt[x*C[0] + x^2*C[3] + C[4]]*Defer[Int][S
qrt[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])/(S
qrt[(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]]) + (2*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])/(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 {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \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 {\left (x^2 c_3-c_4\right ) \sqrt {x c_1+x^2 c_3+c_4} \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {x c_0+x^2 c_3+c_4} \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 {3 \sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}}+\frac {2 (-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}}+\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}}\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}{\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 (2 \sqrt {x c_0+x^2 c_3+c_4}\right ) \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}{\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 {\sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}} \, 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 {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}{\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 (2 \sqrt {x c_0+x^2 c_3+c_4}\right ) \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}{\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 {\sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}} \, 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 {\left (3 \sqrt {x c_0+x^2 c_3+c_4}\right ) \int \frac {\sqrt {x c_1+x^2 c_3+c_4}}{x \sqrt {x c_0+x^2 c_3+c_4}} \, 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 {\left (2 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}{\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 (2 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}{\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 (4 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}{\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 (4 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}{\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. \(43491\) vs. \(2(195)=390\).

Time = 6.51 (sec) , antiderivative size = 43491, normalized size of antiderivative = 223.03 \[ \int \frac {\left (x^2 c_3-c_4\right ) \left (x+3 x^2 c_3+3 c_4\right )}{x \sqrt {\frac {x c_0+x^2 c_3+c_4}{x c_1+x^2 c_3+c_4}} \left (-x^2+x^4 c_3{}^2+2 x^2 c_3 c_4+c_4{}^2\right )} \, dx=\text {Result too large to show} \]

[In]

Integrate[((x^2*C[3] - C[4])*(x + 3*x^2*C[3] + 3*C[4]))/(x*Sqrt[(x*C[0] + x^2*C[3] + C[4])/(x*C[1] + x^2*C[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] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 14.98 (sec) , antiderivative size = 35837869, normalized size of antiderivative = 183783.94

method	result	size
default	\(\text {Expression too large to display}\)	\(35837869\)

[In]

int((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3*_C4*x
^2+_C4^2-x^2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F(-1)]

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

[In]

integrate((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3
*_C4*x^2+_C4^2-x^2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

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

[In]

integrate((_C3*x**2-_C4)*(3*_C3*x**2+3*_C4+x)/x/((_C3*x**2+_C0*x+_C4)/(_C3*x**2+_C1*x+_C4))**(1/2)/(_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((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3
*_C4*x^2+_C4^2-x^2),x, algorithm="maxima")

[Out]

integrate((3*_C3*x^2 + 3*_C4 + x)*(_C3*x^2 - _C4)/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 - x^2)*x*sqrt((_C3*x^2 +
_C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))), x)

Giac [F]

\[ \text {Failed to integrate} \]

[In]

integrate((_C3*x^2-_C4)*(3*_C3*x^2+3*_C4+x)/x/((_C3*x^2+_C0*x+_C4)/(_C3*x^2+_C1*x+_C4))^(1/2)/(_C3^2*x^4+2*_C3
*_C4*x^2+_C4^2-x^2),x, algorithm="giac")

[Out]

integrate((3*_C3*x^2 + 3*_C4 + x)*(_C3*x^2 - _C4)/((_C3^2*x^4 + 2*_C3*_C4*x^2 + _C4^2 - x^2)*x*sqrt((_C3*x^2 +
_C0*x + _C4)/(_C3*x^2 + _C1*x + _C4))), x)

Mupad [F(-1)]

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

[In]

int(-((_C4 - _C3*x^2)*(3*_C4 + x + 3*_C3*x^2))/(x*((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2))^(1/2)*(_C4
^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)),x)

[Out]

int(-((_C4 - _C3*x^2)*(3*_C4 + x + 3*_C3*x^2))/(x*((_C4 + _C0*x + _C3*x^2)/(_C4 + _C1*x + _C3*x^2))^(1/2)*(_C4
^2 - x^2 + _C3^2*x^4 + 2*_C3*_C4*x^2)), x)