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

Optimal. Leaf size=52 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^5-x}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {x^5-x}}\right )}{a^{3/4}} \]

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Rubi [F]  time = 1.46, 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 = 0.50, 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 = 2.17, size = 52, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-x+x^5}}\right )}{a^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt {-x+x^5}}\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)*x)/Sqrt[-x + x^5]]/a^(3/4) - ArcTanh[(a^(1/4)*x)/Sqrt[-x + x^5]]/a^(3/4)

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fricas [B]  time = 0.72, size = 231, normalized size = 4.44 \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 = 0.03, 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 = 4.01, size = 134, normalized size = 2.58 \begin {gather*} \frac {\ln \left (\frac {a^2+2\,\sqrt {x^5-x}\,{\left (a^3\right )}^{3/4}-a^2\,x^4-a\,x\,\sqrt {a^3}}{a-a\,x^4+x\,\sqrt {a^3}}\right )}{2\,{\left (a^3\right )}^{1/4}}+\frac {\ln \left (\frac {a^2\,1{}\mathrm {i}-2\,\sqrt {x^5-x}\,{\left (a^3\right )}^{3/4}-a^2\,x^4\,1{}\mathrm {i}+a\,x\,\sqrt {a^3}\,1{}\mathrm {i}}{a\,x^4-a+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^5 - x)^(1/2)*(a*x^2 + 2*x^4 - x^8 - 1)),x)

[Out]

(log((a^2*1i - 2*(x^5 - x)^(1/2)*(a^3)^(3/4) - a^2*x^4*1i + a*x*(a^3)^(1/2)*1i)/(a*x^4 - a + x*(a^3)^(1/2)))*1
i)/(2*(a^3)^(1/4)) + log((a^2 + 2*(x^5 - x)^(1/2)*(a^3)^(3/4) - a^2*x^4 - a*x*(a^3)^(1/2))/(a - a*x^4 + x*(a^3
)^(1/2)))/(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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