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

Optimal. Leaf size=52 \[ \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^6-x}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {x^6-x}}\right )}{\sqrt [4]{a}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.55, 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 (a-x^2-2 a x^5+a x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[x]*Sqrt[-1 + x^5]*Defer[Subst][Defer[Int][x^2/(Sqrt[-1 + x^10]*(a - x^4 - 2*a*x^10 + a*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]*(a - x^4 -
2*a*x^10 + a*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 (a-x^2-2 a x^5+a x^{10}\right )} \, dx &=\int \frac {x \left (1+4 x^5\right )}{\sqrt {-x+x^6} \left (a-x^2-2 a x^5+a 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 (a-x^2-2 a x^5+a 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 (a-x^4-2 a x^{10}+a 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 (a-x^4-2 a x^{10}+a x^{20}\right )}+\frac {4 x^{12}}{\sqrt {-1+x^{10}} \left (a-x^4-2 a x^{10}+a 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 (a-x^4-2 a x^{10}+a 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 (a-x^4-2 a x^{10}+a x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+4 x^6}{\sqrt {-x+x^6} \left (a-x^2-2 a x^5+a x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 2.72, size = 52, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {-x+x^6}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt [4]{a} \sqrt {-x+x^6}}\right )}{\sqrt [4]{a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.37, size = 280, normalized size = 5.38 \begin {gather*} -\frac {\arctan \left (-\frac {2 \, \sqrt {x^{6} - x} {\left (a^{\frac {1}{4}} x + \frac {a x^{5} - a}{a^{\frac {1}{4}}}\right )} - {\left (\frac {a^{2} x^{10} - 2 \, a^{2} x^{5} + a x^{2} + a^{2}}{a^{\frac {1}{4}}} + \frac {2 \, {\left (a^{2} x^{6} - a^{2} x\right )}}{a^{\frac {3}{4}}}\right )} \sqrt {\frac {1}{a^{\frac {3}{2}}}}}{a x^{10} - 2 \, a x^{5} - x^{2} + a}\right )}{a^{\frac {1}{4}}} - \frac {\log \left (\frac {2 \, \sqrt {x^{6} - x} {\left (x^{5} + \frac {x}{\sqrt {a}} - 1\right )} + \frac {2 \, {\left (x^{6} - x\right )}}{a^{\frac {1}{4}}} + \frac {a x^{10} - 2 \, a x^{5} + x^{2} + a}{a^{\frac {3}{4}}}}{2 \, {\left (a x^{10} - 2 \, a x^{5} - x^{2} + a\right )}}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\log \left (\frac {2 \, \sqrt {x^{6} - x} {\left (x^{5} + \frac {x}{\sqrt {a}} - 1\right )} - \frac {2 \, {\left (x^{6} - x\right )}}{a^{\frac {1}{4}}} - \frac {a x^{10} - 2 \, a x^{5} + x^{2} + a}{a^{\frac {3}{4}}}}{2 \, {\left (a x^{10} - 2 \, a x^{5} - x^{2} + a\right )}}\right )}{4 \, a^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {4 x^{6}+x}{\sqrt {x^{6}-x}\, \left (a \,x^{10}-2 a \,x^{5}-x^{2}+a \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (4 x^{5} + 1\right )}{\sqrt {x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (a x^{10} - 2 a x^{5} + a - x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(4*x**5 + 1)/(sqrt(x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))*(a*x**10 - 2*a*x**5 + a - x**2)), x)

________________________________________________________________________________________