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

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

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Rubi [F]  time = 1.60, 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, S
qrt[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*}

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Mathematica [F]  time = 0.49, 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]

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IntegrateAlgebraic [A]  time = 2.71, size = 48, 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)

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fricas [B]  time = 1.22, size = 264, normalized size = 5.50 \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)

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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)

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

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)

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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)

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

[Out]

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

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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)

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