3.28.59 \(\int \frac {x^3 (x^3 c_3-2 c_4) \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{(x^2+2 x^3 c_3+2 c_4) (-x^4+x^6 c_3{}^2+2 x^3 c_3 c_4+c_4{}^2)} \, dx\) [2759]

Optimal. Leaf size=259 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {1-c_0} \sqrt {-1+c_1} \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{-1+c_0}\right ) \sqrt {1-c_0}}{\sqrt {-1+c_1}}+\frac {\text {ArcTan}\left (\frac {\sqrt {-1-c_0} \sqrt {1+c_1} \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{1+c_0}\right ) \sqrt {-1-c_0}}{3 \sqrt {1+c_1}}-\frac {4 \text {ArcTan}\left (\frac {\sqrt {1-2 c_0} \sqrt {-1+2 c_1} \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{-1+2 c_0}\right ) \sqrt {1-2 c_0}}{3 \sqrt {-1+2 c_1}} \]

[Out]

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

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Rubi [F]
time = 21.51, 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^3 \left (x^3 c_3-2 c_4\right ) \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{\left (x^2+2 x^3 c_3+2 c_4\right ) \left (-x^4+x^6 c_3{}^2+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^3*(x^3*C[3] - 2*C[4])*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/((x^2 + 2*x^3*C[
3] + 2*C[4])*(-x^4 + x^6*C[3]^2 + 2*x^3*C[3]*C[4] + C[4]^2)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {x^3 \left (x^3 c_3-2 c_4\right ) \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{\left (x^2+2 x^3 c_3+2 c_4\right ) \left (-x^4+x^6 c_3{}^2+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx &=\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x^3 \left (x^3 c_3-2 c_4\right ) \sqrt {x^2 c_0+x^3 c_3+c_4}}{\sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right ) \left (-x^4+x^6 c_3{}^2+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx}{\sqrt {x^2 c_0+x^3 c_3+c_4}}\\ &=\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \left (\frac {x (-2+3 x c_3) \sqrt {x^2 c_0+x^3 c_3+c_4}}{6 \left (-x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}}+\frac {x (2+3 x c_3) \sqrt {x^2 c_0+x^3 c_3+c_4}}{2 \left (x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}}-\frac {4 x (1+3 x c_3) \sqrt {x^2 c_0+x^3 c_3+c_4}}{3 \sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right )}\right ) \, dx}{\sqrt {x^2 c_0+x^3 c_3+c_4}}\\ &=\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x (-2+3 x c_3) \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (-x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}} \, dx}{6 \sqrt {x^2 c_0+x^3 c_3+c_4}}+\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x (2+3 x c_3) \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}} \, dx}{2 \sqrt {x^2 c_0+x^3 c_3+c_4}}-\frac {\left (4 \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x (1+3 x c_3) \sqrt {x^2 c_0+x^3 c_3+c_4}}{\sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right )} \, dx}{3 \sqrt {x^2 c_0+x^3 c_3+c_4}}\\ &=\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \left (-\frac {2 x \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (-x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}}+\frac {3 x^2 c_3 \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (-x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}}\right ) \, dx}{6 \sqrt {x^2 c_0+x^3 c_3+c_4}}+\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \left (\frac {2 x \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}}+\frac {3 x^2 c_3 \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}}\right ) \, dx}{2 \sqrt {x^2 c_0+x^3 c_3+c_4}}-\frac {\left (4 \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \left (\frac {x \sqrt {x^2 c_0+x^3 c_3+c_4}}{\sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right )}+\frac {3 x^2 c_3 \sqrt {x^2 c_0+x^3 c_3+c_4}}{\sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right )}\right ) \, dx}{3 \sqrt {x^2 c_0+x^3 c_3+c_4}}\\ &=-\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (-x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}} \, dx}{3 \sqrt {x^2 c_0+x^3 c_3+c_4}}+\frac {\left (\sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}} \, dx}{\sqrt {x^2 c_0+x^3 c_3+c_4}}-\frac {\left (4 \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x \sqrt {x^2 c_0+x^3 c_3+c_4}}{\sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right )} \, dx}{3 \sqrt {x^2 c_0+x^3 c_3+c_4}}+\frac {\left (c_3 \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x^2 \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (-x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}} \, dx}{2 \sqrt {x^2 c_0+x^3 c_3+c_4}}+\frac {\left (3 c_3 \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x^2 \sqrt {x^2 c_0+x^3 c_3+c_4}}{\left (x^2+x^3 c_3+c_4\right ) \sqrt {x^2 c_1+x^3 c_3+c_4}} \, dx}{2 \sqrt {x^2 c_0+x^3 c_3+c_4}}-\frac {\left (4 c_3 \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}} \sqrt {x^2 c_1+x^3 c_3+c_4}\right ) \int \frac {x^2 \sqrt {x^2 c_0+x^3 c_3+c_4}}{\sqrt {x^2 c_1+x^3 c_3+c_4} \left (x^2+2 x^3 c_3+2 c_4\right )} \, dx}{\sqrt {x^2 c_0+x^3 c_3+c_4}}\\ \end {align*}

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Mathematica [F]
time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3 \left (x^3 c_3-2 c_4\right ) \sqrt {\frac {x^2 c_0+x^3 c_3+c_4}{x^2 c_1+x^3 c_3+c_4}}}{\left (x^2+2 x^3 c_3+2 c_4\right ) \left (-x^4+x^6 c_3{}^2+2 x^3 c_3 c_4+c_4{}^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^3*(x^3*C[3] - 2*C[4])*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/((x^2 + 2*
x^3*C[3] + 2*C[4])*(-x^4 + x^6*C[3]^2 + 2*x^3*C[3]*C[4] + C[4]^2)),x]

[Out]

Integrate[(x^3*(x^3*C[3] - 2*C[4])*Sqrt[(x^2*C[0] + x^3*C[3] + C[4])/(x^2*C[1] + x^3*C[3] + C[4])])/((x^2 + 2*
x^3*C[3] + 2*C[4])*(-x^4 + x^6*C[3]^2 + 2*x^3*C[3]*C[4] + C[4]^2)), x]

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (\textit {\_C3} \,x^{3}-2 \textit {\_C4} \right ) \sqrt {\frac {\textit {\_C3} \,x^{3}+\textit {\_C0} \,x^{2}+\textit {\_C4}}{\textit {\_C3} \,x^{3}+\textit {\_C1} \,x^{2}+\textit {\_C4}}}}{\left (2 \textit {\_C3} \,x^{3}+x^{2}+2 \textit {\_C4} \right ) \left (\textit {\_C3}^{2} x^{6}+2 \textit {\_C3} \textit {\_C4} \,x^{3}-x^{4}+\textit {\_C4}^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(_C3*x^3-2*_C4)*((_C3*x^3+_C0*x^2+_C4)/(_C3*x^3+_C1*x^2+_C4))^(1/2)/(2*_C3*x^3+x^2+2*_C4)/(_C3^2*x^6+2
*_C3*_C4*x^3-x^4+_C4^2),x)

[Out]

int(x^3*(_C3*x^3-2*_C4)*((_C3*x^3+_C0*x^2+_C4)/(_C3*x^3+_C1*x^2+_C4))^(1/2)/(2*_C3*x^3+x^2+2*_C4)/(_C3^2*x^6+2
*_C3*_C4*x^3-x^4+_C4^2),x)

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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^3*(_C3*x^3-2*_C4)*((_C3*x^3+_C0*x^2+_C4)/(_C3*x^3+_C1*x^2+_C4))^(1/2)/(2*_C3*x^3+x^2+2*_C4)/(_C3^2
*x^6+2*_C3*_C4*x^3-x^4+_C4^2),x, algorithm="maxima")

[Out]

integrate((_C3*x^3 - 2*_C4)*x^3*sqrt((_C3*x^3 + _C0*x^2 + _C4)/(_C3*x^3 + _C1*x^2 + _C4))/((_C3^2*x^6 + 2*_C3*
_C4*x^3 - x^4 + _C4^2)*(2*_C3*x^3 + x^2 + 2*_C4)), x)

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Fricas [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^3*(_C3*x^3-2*_C4)*((_C3*x^3+_C0*x^2+_C4)/(_C3*x^3+_C1*x^2+_C4))^(1/2)/(2*_C3*x^3+x^2+2*_C4)/(_C3^2
*x^6+2*_C3*_C4*x^3-x^4+_C4^2),x, algorithm="fricas")

[Out]

Timed out

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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**3*(_C3*x**3-2*_C4)*((_C3*x**3+_C0*x**2+_C4)/(_C3*x**3+_C1*x**2+_C4))**(1/2)/(2*_C3*x**3+x**2+2*_C
4)/(_C3**2*x**6+2*_C3*_C4*x**3-x**4+_C4**2),x)

[Out]

Timed out

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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^3*(_C3*x^3-2*_C4)*((_C3*x^3+_C0*x^2+_C4)/(_C3*x^3+_C1*x^2+_C4))^(1/2)/(2*_C3*x^3+x^2+2*_C4)/(_C3^2
*x^6+2*_C3*_C4*x^3-x^4+_C4^2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x^3\,\sqrt {\frac {_{\mathrm {C3}}\,x^3+_{\mathrm {C0}}\,x^2+_{\mathrm {C4}}}{_{\mathrm {C3}}\,x^3+_{\mathrm {C1}}\,x^2+_{\mathrm {C4}}}}\,\left (2\,_{\mathrm {C4}}-_{\mathrm {C3}}\,x^3\right )}{\left (2\,_{\mathrm {C3}}\,x^3+x^2+2\,_{\mathrm {C4}}\right )\,\left ({_{\mathrm {C3}}}^2\,x^6+2\,_{\mathrm {C3}}\,_{\mathrm {C4}}\,x^3+{_{\mathrm {C4}}}^2-x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3*((_C4 + _C0*x^2 + _C3*x^3)/(_C4 + _C1*x^2 + _C3*x^3))^(1/2)*(2*_C4 - _C3*x^3))/((2*_C4 + 2*_C3*x^3 +
x^2)*(_C4^2 - x^4 + _C3^2*x^6 + 2*_C3*_C4*x^3)),x)

[Out]

int(-(x^3*((_C4 + _C0*x^2 + _C3*x^3)/(_C4 + _C1*x^2 + _C3*x^3))^(1/2)*(2*_C4 - _C3*x^3))/((2*_C4 + 2*_C3*x^3 +
x^2)*(_C4^2 - x^4 + _C3^2*x^6 + 2*_C3*_C4*x^3)), x)

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