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

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

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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+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[x]*Sqrt[-1 + x^4]*Defer[Subst][Defer[Int][x^2/(Sqrt[-1 + x^8]*(a - x^4 - 2*a*x^8 + a*x^16)), x], x, Sq
rt[x]])/Sqrt[-x + x^5] + (6*Sqrt[x]*Sqrt[-1 + x^4]*Defer[Subst][Defer[Int][x^10/(Sqrt[-1 + x^8]*(a - x^4 - 2*a
*x^8 + a*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 (a-x^2-2 a x^4+a x^8\right )} \, dx &=\int \frac {x \left (1+3 x^4\right )}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a 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 (a-x^2-2 a x^4+a 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 (a-x^4-2 a x^8+a 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 (a-x^4-2 a x^8+a x^{16}\right )}+\frac {3 x^{10}}{\sqrt {-1+x^8} \left (a-x^4-2 a x^8+a 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 (a-x^4-2 a x^8+a 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 (a-x^4-2 a x^8+a x^{16}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+3 x^5}{\sqrt {-x+x^5} \left (a-x^2-2 a x^4+a x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.68, size = 52, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {-x+x^5}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {-x+x^5}}\right )}{\sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.73, size = 202, normalized size = 3.88 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {x^{5} - x}}{{\left (x^{4} - 1\right )} a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} - \frac {\log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} + 2 \, \sqrt {x^{5} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{4} - a}{a^{\frac {1}{4}}}\right )} + a + \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {a x^{8} - 2 \, a x^{4} + x^{2} - 2 \, \sqrt {x^{5} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{4} - a}{a^{\frac {1}{4}}}\right )} + a + \frac {2 \, {\left (a x^{5} - a x\right )}}{\sqrt {a}}}{a x^{8} - 2 \, a x^{4} - x^{2} + a}\right )}{4 \, a^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {3 x^{5}+x}{\sqrt {x^{5}-x}\, \left (a \,x^{8}-2 a \,x^{4}-x^{2}+a \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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mupad [B]  time = 3.56, size = 103, normalized size = 1.98 \begin {gather*} \frac {\ln \left (\frac {x-2\,a^{1/4}\,\sqrt {x^5-x}-\sqrt {a}+\sqrt {a}\,x^4}{x+\sqrt {a}-\sqrt {a}\,x^4}\right )}{2\,a^{1/4}}+\frac {\ln \left (\frac {x+\sqrt {a}-\sqrt {a}\,x^4+a^{1/4}\,\sqrt {x^5-x}\,2{}\mathrm {i}}{x-\sqrt {a}+\sqrt {a}\,x^4}\right )\,1{}\mathrm {i}}{2\,a^{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 - 2*a*x^4 + a*x^8 - x^2)),x)

[Out]

log((x - 2*a^(1/4)*(x^5 - x)^(1/2) - a^(1/2) + a^(1/2)*x^4)/(x + a^(1/2) - a^(1/2)*x^4))/(2*a^(1/4)) + (log((x
+ a^(1/4)*(x^5 - x)^(1/2)*2i + a^(1/2) - a^(1/2)*x^4)/(x - a^(1/2) + a^(1/2)*x^4))*1i)/(2*a^(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)/(a*x**8-2*a*x**4-x**2+a),x)

[Out]

Timed out

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