\(\int \frac {\sqrt [3]{1+x^5} (-3+2 x^5)}{x^2 (2-x^3+2 x^5)} \, dx\) [1917]

Optimal result

Integrand size = 34, antiderivative size = 133 \[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\frac {3 \sqrt [3]{1+x^5}}{2 x}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx \]

[In]

Int[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)),x]

[Out]

(3*Hypergeometric2F1[-1/3, -1/5, 4/5, -x^5])/(2*x) - (3*Defer[Int][(x*(1 + x^5)^(1/3))/(2 - x^3 + 2*x^5), x])/
2 + 5*Defer[Int][(x^3*(1 + x^5)^(1/3))/(2 - x^3 + 2*x^5), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{1+x^5}}{2 x^2}+\frac {x \left (-3+10 x^2\right ) \sqrt [3]{1+x^5}}{2 \left (2-x^3+2 x^5\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {x \left (-3+10 x^2\right ) \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^5}}{x^2} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{5},\frac {4}{5},-x^5\right )}{2 x}+\frac {1}{2} \int \left (-\frac {3 x \sqrt [3]{1+x^5}}{2-x^3+2 x^5}+\frac {10 x^3 \sqrt [3]{1+x^5}}{2-x^3+2 x^5}\right ) \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{5},\frac {4}{5},-x^5\right )}{2 x}-\frac {3}{2} \int \frac {x \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx+5 \int \frac {x^3 \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.86 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\frac {3 \sqrt [3]{1+x^5}}{2 x}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]

[In]

Integrate[((1 + x^5)^(1/3)*(-3 + 2*x^5))/(x^2*(2 - x^3 + 2*x^5)),x]

[Out]

(3*(1 + x^5)^(1/3))/(2*x) + (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^5)^(1/3))])/(2*2^(1/3)) + Log[-x
+ 2^(1/3)*(1 + x^5)^(1/3)]/(2*2^(1/3)) - Log[x^2 + 2^(1/3)*x*(1 + x^5)^(1/3) + 2^(2/3)*(1 + x^5)^(2/3)]/(4*2^
(1/3))

Maple [A] (verified)

Time = 54.50 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.26

method	result	size
pseudoelliptic	\(\frac {-2 \,2^{\frac {2}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{5}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x +2 \,2^{\frac {2}{3}} x \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-2^{\frac {2}{3}} x \ln \left (\frac {2^{\frac {2}{3}} {\left (\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} x^{2}+2 {\left (\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2^{\frac {2}{3}} x \ln \left (2\right )+12 \left (x^{5}+1\right )^{\frac {1}{3}}}{8 x}\)	\(168\)
trager	\(\text {Expression too large to display}\)	\(1015\)
risch	\(\text {Expression too large to display}\)	\(1061\)

[In]

int((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x,method=_RETURNVERBOSE)

[Out]

1/8*(-2*2^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(x+2*2^(1/3)*(x^5+1)^(1/3))/x)*x+2*2^(2/3)*x*ln((-2^(2/3)*x+2*((1+x
)*(x^4-x^3+x^2-x+1))^(1/3))/x)-2^(2/3)*x*ln((2^(2/3)*((1+x)*(x^4-x^3+x^2-x+1))^(1/3)*x+2^(1/3)*x^2+2*((1+x)*(x
^4-x^3+x^2-x+1))^(2/3))/x^2)-2^(2/3)*x*ln(2)+12*(x^5+1)^(1/3))/x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (97) = 194\).

Time = 62.47 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.89 \[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\frac {2 \, \sqrt {3} 2^{\frac {2}{3}} x \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (24 \, \sqrt {2} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 2^{\frac {5}{6}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \cdot 2^{\frac {1}{6}} {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) + 2 \cdot 2^{\frac {2}{3}} x \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )}}{2 \, x^{5} - x^{3} + 2}\right ) - 2^{\frac {2}{3}} x \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{24 \, x} \]

[In]

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

[Out]

1/24*(2*sqrt(3)*2^(2/3)*x*arctan(1/6*sqrt(3)*2^(1/6)*(24*sqrt(2)*(2*x^11 + x^9 - x^7 + 4*x^6 + x^4 + 2*x)*(x^5
+ 1)^(2/3) + 2^(5/6)*(8*x^15 + 60*x^13 + 24*x^11 + 24*x^10 - x^9 + 120*x^8 + 24*x^6 + 24*x^5 + 60*x^3 + 8) +
12*2^(1/6)*(4*x^12 + 14*x^10 + x^8 + 8*x^7 + 14*x^5 + 4*x^2)*(x^5 + 1)^(1/3))/(8*x^15 - 12*x^13 - 48*x^11 + 24
*x^10 - x^9 - 24*x^8 - 48*x^6 + 24*x^5 - 12*x^3 + 8)) + 2*2^(2/3)*x*log((3*2^(2/3)*(x^5 + 1)^(1/3)*x^2 - 6*(x^
5 + 1)^(2/3)*x + 2^(1/3)*(2*x^5 - x^3 + 2))/(2*x^5 - x^3 + 2)) - 2^(2/3)*x*log((12*2^(1/3)*(x^6 + x^4 + x)*(x^
5 + 1)^(2/3) + 2^(2/3)*(4*x^10 + 14*x^8 + x^6 + 8*x^5 + 14*x^3 + 4) + 6*(4*x^7 + x^5 + 4*x^2)*(x^5 + 1)^(1/3))
/(4*x^10 - 4*x^8 + x^6 + 8*x^5 - 4*x^3 + 4)) + 36*(x^5 + 1)^(1/3))/x

Sympy [F]

\[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\int \frac {\sqrt [3]{\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (2 x^{5} - 3\right )}{x^{2} \cdot \left (2 x^{5} - x^{3} + 2\right )}\, dx \]

[In]

integrate((x**5+1)**(1/3)*(2*x**5-3)/x**2/(2*x**5-x**3+2),x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{5} - x^{3} + 2\right )} x^{2}} \,d x } \]

[In]

integrate((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)*(x^5 + 1)^(1/3)/((2*x^5 - x^3 + 2)*x^2), x)

Giac [F]

\[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{5} - x^{3} + 2\right )} x^{2}} \,d x } \]

[In]

integrate((x^5+1)^(1/3)*(2*x^5-3)/x^2/(2*x^5-x^3+2),x, algorithm="giac")

[Out]

integrate((2*x^5 - 3)*(x^5 + 1)^(1/3)/((2*x^5 - x^3 + 2)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx=\int \frac {{\left (x^5+1\right )}^{1/3}\,\left (2\,x^5-3\right )}{x^2\,\left (2\,x^5-x^3+2\right )} \,d x \]

[In]

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

[Out]

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