3.9.7 \(\int \frac {(4+x^5) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx\)

Optimal. Leaf size=61 \[ \frac {\sqrt {x^{10}+x^8-2 x^5+1} \left (x^5-1\right )}{2 x^8}+\frac {1}{2} \log \left (x^5+\sqrt {x^{10}+x^8-2 x^5+1}-1\right )-2 \log (x) \]

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Rubi [F]  time = 0.28, 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 {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/x^9,x]

[Out]

4*Defer[Int][Sqrt[1 - 2*x^5 + x^8 + x^10]/x^9, x] + Defer[Int][Sqrt[1 - 2*x^5 + x^8 + x^10]/x^4, x]

Rubi steps

\begin {align*} \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx &=\int \left (\frac {4 \sqrt {1-2 x^5+x^8+x^{10}}}{x^9}+\frac {\sqrt {1-2 x^5+x^8+x^{10}}}{x^4}\right ) \, dx\\ &=4 \int \frac {\sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx+\int \frac {\sqrt {1-2 x^5+x^8+x^{10}}}{x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/x^9,x]

[Out]

Integrate[((4 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/x^9, x]

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IntegrateAlgebraic [A]  time = 0.33, size = 61, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{2 x^8}-2 \log (x)+\frac {1}{2} \log \left (-1+x^5+\sqrt {1-2 x^5+x^8+x^{10}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((4 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/x^9,x]

[Out]

((-1 + x^5)*Sqrt[1 - 2*x^5 + x^8 + x^10])/(2*x^8) - 2*Log[x] + Log[-1 + x^5 + Sqrt[1 - 2*x^5 + x^8 + x^10]]/2

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fricas [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((x^5+4)*(x^10+x^8-2*x^5+1)^(1/2)/x^9,x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{10} + x^{8} - 2 \, x^{5} + 1} {\left (x^{5} + 4\right )}}{x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(x^10+x^8-2*x^5+1)^(1/2)/x^9,x, algorithm="giac")

[Out]

integrate(sqrt(x^10 + x^8 - 2*x^5 + 1)*(x^5 + 4)/x^9, x)

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maple [A]  time = 0.77, size = 67, normalized size = 1.10

method	result	size
trager	\(\frac {\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \sqrt {x^{10}+x^{8}-2 x^{5}+1}}{2 x^{8}}-\frac {\ln \left (-\frac {-x^{5}+\sqrt {x^{10}+x^{8}-2 x^{5}+1}+1}{x^{4}}\right )}{2}\)	\(67\)
risch	\(\frac {x^{15}+x^{13}-3 x^{10}-x^{8}+3 x^{5}-1}{2 x^{8} \sqrt {x^{10}+x^{8}-2 x^{5}+1}}+\frac {\ln \left (-\frac {-1+x^{5}+\sqrt {x^{10}+x^{8}-2 x^{5}+1}}{x^{4}}\right )}{2}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+4)*(x^10+x^8-2*x^5+1)^(1/2)/x^9,x,method=_RETURNVERBOSE)

[Out]

1/2*(-1+x)*(x^4+x^3+x^2+x+1)/x^8*(x^10+x^8-2*x^5+1)^(1/2)-1/2*ln(-(-x^5+(x^10+x^8-2*x^5+1)^(1/2)+1)/x^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{10} + x^{8} - 2 \, x^{5} + 1} {\left (x^{5} + 4\right )}}{x^{9}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(x^10+x^8-2*x^5+1)^(1/2)/x^9,x, algorithm="maxima")

[Out]

integrate(sqrt(x^10 + x^8 - 2*x^5 + 1)*(x^5 + 4)/x^9, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^5+4\right )\,\sqrt {x^{10}+x^8-2\,x^5+1}}{x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 4)*(x^8 - 2*x^5 + x^10 + 1)^(1/2))/x^9,x)

[Out]

int(((x^5 + 4)*(x^8 - 2*x^5 + x^10 + 1)^(1/2))/x^9, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{5} + 4\right ) \sqrt {x^{10} + x^{8} - 2 x^{5} + 1}}{x^{9}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+4)*(x**10+x**8-2*x**5+1)**(1/2)/x**9,x)

[Out]

Integral((x**5 + 4)*sqrt(x**10 + x**8 - 2*x**5 + 1)/x**9, x)

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