3.7.16 \(\int \frac {x-4 x^6}{\sqrt {x+x^6} (1-a x^2+2 x^5+x^{10})} \, dx\)

Optimal. Leaf size=48 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^6+x}}\right )}{a^{3/4}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^6+x}}\right )}{a^{3/4}} \]

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Rubi [F]  time = 1.51, 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-4 x^6}{\sqrt {x+x^6} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)),x]

[Out]

(2*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^2/(Sqrt[1 + x^10]*(1 - a*x^4 + 2*x^10 + x^20)), x], x, Sqrt
[x]])/Sqrt[x + x^6] - (8*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^12/(Sqrt[1 + x^10]*(1 - a*x^4 + 2*x^1
0 + x^20)), x], x, Sqrt[x]])/Sqrt[x + x^6]

Rubi steps

\begin {align*} \int \frac {x-4 x^6}{\sqrt {x+x^6} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx &=\int \frac {x \left (1-4 x^5\right )}{\sqrt {x+x^6} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {\sqrt {x} \left (1-4 x^5\right )}{\sqrt {1+x^5} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1-4 x^{10}\right )}{\sqrt {1+x^{10}} \left (1-a x^4+2 x^{10}+x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {1+x^{10}} \left (1-a x^4+2 x^{10}+x^{20}\right )}-\frac {4 x^{12}}{\sqrt {1+x^{10}} \left (1-a x^4+2 x^{10}+x^{20}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^{10}} \left (1-a x^4+2 x^{10}+x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}-\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt {1+x^{10}} \left (1-a x^4+2 x^{10}+x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x-4 x^6}{\sqrt {x+x^6} \left (1-a x^2+2 x^5+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)),x]

[Out]

Integrate[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)), x]

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IntegrateAlgebraic [A]  time = 2.71, size = 48, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x+x^6}}\right )}{a^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x+x^6}}\right )}{a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x - 4*x^6)/(Sqrt[x + x^6]*(1 - a*x^2 + 2*x^5 + x^10)),x]

[Out]

-(ArcTan[(a^(1/4)*x)/Sqrt[x + x^6]]/a^(3/4)) + ArcTanh[(a^(1/4)*x)/Sqrt[x + x^6]]/a^(3/4)

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fricas [B]  time = 1.16, size = 321, normalized size = 6.69 \begin {gather*} -\frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} {\left (a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (a x^{5} + a\right )} \frac {1}{a^{3}}^{\frac {1}{4}}\right )} + {\left (2 \, {\left (a^{3} x^{6} + a^{3} x\right )} \frac {1}{a^{3}}^{\frac {3}{4}} + {\left (a x^{10} + 2 \, a x^{5} + a^{2} x^{2} + a\right )} \frac {1}{a^{3}}^{\frac {1}{4}}\right )} \sqrt {a \sqrt {\frac {1}{a^{3}}}}}{x^{10} + 2 \, x^{5} - a x^{2} + 1}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (-\frac {{\left (a^{2} x^{10} + 2 \, a^{2} x^{5} + a^{3} x^{2} + a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} + 2 \, \sqrt {x^{6} + x} {\left (x^{5} + a^{2} \sqrt {\frac {1}{a^{3}}} x + 1\right )} + 2 \, {\left (a x^{6} + a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )}}\right ) - \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {{\left (a^{2} x^{10} + 2 \, a^{2} x^{5} + a^{3} x^{2} + a^{2}\right )} \frac {1}{a^{3}}^{\frac {3}{4}} - 2 \, \sqrt {x^{6} + x} {\left (x^{5} + a^{2} \sqrt {\frac {1}{a^{3}}} x + 1\right )} + 2 \, {\left (a x^{6} + a x\right )} \frac {1}{a^{3}}^{\frac {1}{4}}}{2 \, {\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^6+x)/(x^6+x)^(1/2)/(x^10+2*x^5-a*x^2+1),x, algorithm="fricas")

[Out]

-(a^(-3))^(1/4)*arctan((2*sqrt(x^6 + x)*(a^3*(a^(-3))^(3/4)*x + (a*x^5 + a)*(a^(-3))^(1/4)) + (2*(a^3*x^6 + a^
3*x)*(a^(-3))^(3/4) + (a*x^10 + 2*a*x^5 + a^2*x^2 + a)*(a^(-3))^(1/4))*sqrt(a*sqrt(a^(-3))))/(x^10 + 2*x^5 - a
*x^2 + 1)) + 1/4*(a^(-3))^(1/4)*log(-1/2*((a^2*x^10 + 2*a^2*x^5 + a^3*x^2 + a^2)*(a^(-3))^(3/4) + 2*sqrt(x^6 +
x)*(x^5 + a^2*sqrt(a^(-3))*x + 1) + 2*(a*x^6 + a*x)*(a^(-3))^(1/4))/(x^10 + 2*x^5 - a*x^2 + 1)) - 1/4*(a^(-3)
)^(1/4)*log(1/2*((a^2*x^10 + 2*a^2*x^5 + a^3*x^2 + a^2)*(a^(-3))^(3/4) - 2*sqrt(x^6 + x)*(x^5 + a^2*sqrt(a^(-3
))*x + 1) + 2*(a*x^6 + a*x)*(a^(-3))^(1/4))/(x^10 + 2*x^5 - a*x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, x^{6} - x}{{\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )} \sqrt {x^{6} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^6+x)/(x^6+x)^(1/2)/(x^10+2*x^5-a*x^2+1),x, algorithm="giac")

[Out]

integrate(-(4*x^6 - x)/((x^10 + 2*x^5 - a*x^2 + 1)*sqrt(x^6 + x)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-4 x^{6}+x}{\sqrt {x^{6}+x}\, \left (x^{10}+2 x^{5}-a \,x^{2}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*x^6+x)/(x^6+x)^(1/2)/(x^10+2*x^5-a*x^2+1),x)

[Out]

int((-4*x^6+x)/(x^6+x)^(1/2)/(x^10+2*x^5-a*x^2+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {4 \, x^{6} - x}{{\left (x^{10} + 2 \, x^{5} - a x^{2} + 1\right )} \sqrt {x^{6} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x^6+x)/(x^6+x)^(1/2)/(x^10+2*x^5-a*x^2+1),x, algorithm="maxima")

[Out]

-integrate((4*x^6 - x)/((x^10 + 2*x^5 - a*x^2 + 1)*sqrt(x^6 + x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x-4\,x^6}{\sqrt {x^6+x}\,\left (x^{10}+2\,x^5-a\,x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 4*x^6)/((x + x^6)^(1/2)*(2*x^5 - a*x^2 + x^10 + 1)),x)

[Out]

int((x - 4*x^6)/((x + x^6)^(1/2)*(2*x^5 - a*x^2 + x^10 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {x}{- a x^{2} \sqrt {x^{6} + x} + x^{10} \sqrt {x^{6} + x} + 2 x^{5} \sqrt {x^{6} + x} + \sqrt {x^{6} + x}}\right )\, dx - \int \frac {4 x^{6}}{- a x^{2} \sqrt {x^{6} + x} + x^{10} \sqrt {x^{6} + x} + 2 x^{5} \sqrt {x^{6} + x} + \sqrt {x^{6} + x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*x**6+x)/(x**6+x)**(1/2)/(x**10+2*x**5-a*x**2+1),x)

[Out]

-Integral(-x/(-a*x**2*sqrt(x**6 + x) + x**10*sqrt(x**6 + x) + 2*x**5*sqrt(x**6 + x) + sqrt(x**6 + x)), x) - In
tegral(4*x**6/(-a*x**2*sqrt(x**6 + x) + x**10*sqrt(x**6 + x) + 2*x**5*sqrt(x**6 + x) + sqrt(x**6 + x)), x)

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