3.8.88 \(\int \frac {x-3 x^5}{\sqrt {x+x^5} (1-a x^2+2 x^4+x^8)} \, dx\)

Optimal. Leaf size=60 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x^5+x}}{x^4+1}\right )}{a^{3/4}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x^5+x}}{x^4+1}\right )}{a^{3/4}} \]

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Rubi [F]  time = 1.44, 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 x^5}{\sqrt {x+x^5} \left (1-a x^2+2 x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x - 3*x^5)/(Sqrt[x + x^5]*(1 - a*x^2 + 2*x^4 + x^8)),x]

[Out]

(2*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^2/(Sqrt[1 + x^8]*(1 - a*x^4 + 2*x^8 + x^16)), x], x, Sqrt[x
]])/Sqrt[x + x^5] - (6*Sqrt[x]*Sqrt[1 + x^4]*Defer[Subst][Defer[Int][x^10/(Sqrt[1 + x^8]*(1 - a*x^4 + 2*x^8 +
x^16)), x], x, Sqrt[x]])/Sqrt[x + x^5]

Rubi steps

\begin {align*} \int \frac {x-3 x^5}{\sqrt {x+x^5} \left (1-a x^2+2 x^4+x^8\right )} \, dx &=\int \frac {x \left (1-3 x^4\right )}{\sqrt {x+x^5} \left (1-a x^2+2 x^4+x^8\right )} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {\sqrt {x} \left (1-3 x^4\right )}{\sqrt {1+x^4} \left (1-a x^2+2 x^4+x^8\right )} \, dx}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1-3 x^8\right )}{\sqrt {1+x^8} \left (1-a x^4+2 x^8+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\sqrt {1+x^8} \left (1-a x^4+2 x^8+x^{16}\right )}-\frac {3 x^{10}}{\sqrt {1+x^8} \left (1-a x^4+2 x^8+x^{16}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^8} \left (1-a x^4+2 x^8+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (6 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt {1+x^8} \left (1-a x^4+2 x^8+x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x-3 x^5}{\sqrt {x+x^5} \left (1-a x^2+2 x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x - 3*x^5)/(Sqrt[x + x^5]*(1 - a*x^2 + 2*x^4 + x^8)),x]

[Out]

Integrate[(x - 3*x^5)/(Sqrt[x + x^5]*(1 - a*x^2 + 2*x^4 + x^8)), x]

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IntegrateAlgebraic [A]  time = 1.90, size = 60, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x+x^5}}{1+x^4}\right )}{a^{3/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x+x^5}}{1+x^4}\right )}{a^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x - 3*x^5)/(Sqrt[x + x^5]*(1 - a*x^2 + 2*x^4 + x^8)),x]

[Out]

-(ArcTan[(a^(1/4)*Sqrt[x + x^5])/(1 + x^4)]/a^(3/4)) + ArcTanh[(a^(1/4)*Sqrt[x + x^5])/(1 + x^4)]/a^(3/4)

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fricas [B]  time = 0.57, size = 220, normalized size = 3.67 \begin {gather*} -\frac {1}{a^{3}}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {x^{5} + x} a \frac {1}{a^{3}}^{\frac {1}{4}}}{x^{4} + 1}\right ) + \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {x^{8} + 2 \, x^{4} + a x^{2} + 2 \, \sqrt {x^{5} + x} {\left (a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (a x^{4} + a\right )} \frac {1}{a^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a^{2} x^{5} + a^{2} x\right )} \sqrt {\frac {1}{a^{3}}} + 1}{x^{8} + 2 \, x^{4} - a x^{2} + 1}\right ) - \frac {1}{4} \, \frac {1}{a^{3}}^{\frac {1}{4}} \log \left (\frac {x^{8} + 2 \, x^{4} + a x^{2} - 2 \, \sqrt {x^{5} + x} {\left (a^{3} \frac {1}{a^{3}}^{\frac {3}{4}} x + {\left (a x^{4} + a\right )} \frac {1}{a^{3}}^{\frac {1}{4}}\right )} + 2 \, {\left (a^{2} x^{5} + a^{2} x\right )} \sqrt {\frac {1}{a^{3}}} + 1}{x^{8} + 2 \, x^{4} - a x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+x)/(x^5+x)^(1/2)/(x^8+2*x^4-a*x^2+1),x, algorithm="fricas")

[Out]

-(a^(-3))^(1/4)*arctan(sqrt(x^5 + x)*a*(a^(-3))^(1/4)/(x^4 + 1)) + 1/4*(a^(-3))^(1/4)*log((x^8 + 2*x^4 + a*x^2
+ 2*sqrt(x^5 + x)*(a^3*(a^(-3))^(3/4)*x + (a*x^4 + a)*(a^(-3))^(1/4)) + 2*(a^2*x^5 + a^2*x)*sqrt(a^(-3)) + 1)
/(x^8 + 2*x^4 - a*x^2 + 1)) - 1/4*(a^(-3))^(1/4)*log((x^8 + 2*x^4 + a*x^2 - 2*sqrt(x^5 + x)*(a^3*(a^(-3))^(3/4
)*x + (a*x^4 + a)*(a^(-3))^(1/4)) + 2*(a^2*x^5 + a^2*x)*sqrt(a^(-3)) + 1)/(x^8 + 2*x^4 - a*x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{5} - x}{{\left (x^{8} + 2 \, x^{4} - a x^{2} + 1\right )} \sqrt {x^{5} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+x)/(x^5+x)^(1/2)/(x^8+2*x^4-a*x^2+1),x, algorithm="giac")

[Out]

integrate(-(3*x^5 - x)/((x^8 + 2*x^4 - a*x^2 + 1)*sqrt(x^5 + x)), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {-3 x^{5}+x}{\sqrt {x^{5}+x}\, \left (x^{8}+2 x^{4}-a \,x^{2}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^5+x)/(x^5+x)^(1/2)/(x^8+2*x^4-a*x^2+1),x)

[Out]

int((-3*x^5+x)/(x^5+x)^(1/2)/(x^8+2*x^4-a*x^2+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {3 \, x^{5} - x}{{\left (x^{8} + 2 \, x^{4} - a x^{2} + 1\right )} \sqrt {x^{5} + x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+x)/(x^5+x)^(1/2)/(x^8+2*x^4-a*x^2+1),x, algorithm="maxima")

[Out]

-integrate((3*x^5 - x)/((x^8 + 2*x^4 - a*x^2 + 1)*sqrt(x^5 + x)), x)

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mupad [B]  time = 4.76, size = 217, normalized size = 3.62 \begin {gather*} \frac {\ln \left (\frac {512\,\sqrt {x^5+x}\,{\left (a^3\right )}^{7/4}+256\,a^5-27\,a^7+256\,a^5\,x^4-27\,x\,{\left (a^3\right )}^{5/2}-27\,a^7\,x^4-54\,a^5\,\sqrt {x^5+x}\,{\left (a^3\right )}^{3/4}+256\,a^4\,x\,\sqrt {a^3}}{a+a\,x^4-x\,\sqrt {a^3}}\right )}{2\,{\left (a^3\right )}^{1/4}}+\frac {\ln \left (\frac {54\,a^6\,\sqrt {x^5+x}\,{\left (a^3\right )}^{3/4}-512\,a\,\sqrt {x^5+x}\,{\left (a^3\right )}^{7/4}+a^6\,256{}\mathrm {i}-a^8\,27{}\mathrm {i}+a^6\,x^4\,256{}\mathrm {i}-a^8\,x^4\,27{}\mathrm {i}-a^5\,x\,\sqrt {a^3}\,256{}\mathrm {i}+a^7\,x\,\sqrt {a^3}\,27{}\mathrm {i}}{a+a\,x^4+x\,\sqrt {a^3}}\right )\,1{}\mathrm {i}}{2\,{\left (a^3\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 3*x^5)/((x + x^5)^(1/2)*(2*x^4 - a*x^2 + x^8 + 1)),x)

[Out]

log((512*(x + x^5)^(1/2)*(a^3)^(7/4) + 256*a^5 - 27*a^7 + 256*a^5*x^4 - 27*x*(a^3)^(5/2) - 27*a^7*x^4 - 54*a^5
*(x + x^5)^(1/2)*(a^3)^(3/4) + 256*a^4*x*(a^3)^(1/2))/(a + a*x^4 - x*(a^3)^(1/2)))/(2*(a^3)^(1/4)) + (log((a^6
*256i - a^8*27i + a^6*x^4*256i - a^8*x^4*27i + 54*a^6*(x + x^5)^(1/2)*(a^3)^(3/4) - a^5*x*(a^3)^(1/2)*256i + a
^7*x*(a^3)^(1/2)*27i - 512*a*(x + x^5)^(1/2)*(a^3)^(7/4))/(a + a*x^4 + x*(a^3)^(1/2)))*1i)/(2*(a^3)^(1/4))

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sympy [F(-1)]  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((-3*x**5+x)/(x**5+x)**(1/2)/(x**8+2*x**4-a*x**2+1),x)

[Out]

Timed out

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