\(\int \frac {(4+x^5) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx\) [807]

Optimal result

Integrand size = 26, antiderivative size = 61 \[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\frac {\left (-1+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{2 x^8}-2 \log (x)+\frac {1}{2} \log \left (-1+x^5+\sqrt {1-2 x^5+x^8+x^{10}}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx \]

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.95 \[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\frac {1}{2} \left (\frac {\left (-1+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^8}-4 \log (x)+\log \left (-1+x^5+\sqrt {1-2 x^5+x^8+x^{10}}\right )\right ) \]

[In]

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

Maple [A] (verified)

Time = 1.96 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {\operatorname {arcsinh}\left (\frac {x^{5}-1}{x^{4}}\right ) x^{6}+\sqrt {\frac {x^{10}+x^{8}-2 x^{5}+1}{x^{4}}}\, x^{5}-\sqrt {\frac {x^{10}+x^{8}-2 x^{5}+1}{x^{4}}}}{2 x^{6}}\)	\(65\)
trager	\(\frac {\left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \sqrt {x^{10}+x^{8}-2 x^{5}+1}}{2 x^{8}}-\frac {\ln \left (-\frac {-x^{5}+\sqrt {x^{10}+x^{8}-2 x^{5}+1}+1}{x^{4}}\right )}{2}\)	\(67\)
risch	\(\frac {x^{15}+x^{13}-3 x^{10}-x^{8}+3 x^{5}-1}{2 x^{8} \sqrt {x^{10}+x^{8}-2 x^{5}+1}}+\frac {\ln \left (-\frac {-1+x^{5}+\sqrt {x^{10}+x^{8}-2 x^{5}+1}}{x^{4}}\right )}{2}\)	\(73\)

[In]

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

[Out]

1/2*(arcsinh((x^5-1)/x^4)*x^6+(1/x^4*(x^10+x^8-2*x^5+1))^(1/2)*x^5-(1/x^4*(x^10+x^8-2*x^5+1))^(1/2))/x^6

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\int \frac {\left (x^{5} + 4\right ) \sqrt {x^{10} + x^{8} - 2 x^{5} + 1}}{x^{9}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\int { \frac {\sqrt {x^{10} + x^{8} - 2 \, x^{5} + 1} {\left (x^{5} + 4\right )}}{x^{9}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\int { \frac {\sqrt {x^{10} + x^{8} - 2 \, x^{5} + 1} {\left (x^{5} + 4\right )}}{x^{9}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (4+x^5\right ) \sqrt {1-2 x^5+x^8+x^{10}}}{x^9} \, dx=\int \frac {\left (x^5+4\right )\,\sqrt {x^{10}+x^8-2\,x^5+1}}{x^9} \,d x \]

[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)