3.13.43 \(\int \frac {(1+x^5)^{2/3} (-3+2 x^5)}{x^3 (1-x^3+x^5)} \, dx\) [1243]

Optimal. Leaf size=90 \[ \frac {3 \left (1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^5}}\right )+\log \left (-x+\sqrt [3]{1+x^5}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right ) \]

[Out]

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

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Rubi [F]
time = 0.49, 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 (1+x^5\right )^{2/3} \left (-3+2 x^5\right )}{x^3 \left (1-x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right )}{x^3 \left (1-x^3+x^5\right )} \, dx &=\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{x^3}+\frac {\left (-3+5 x^2\right ) \left (1+x^5\right )^{2/3}}{1-x^3+x^5}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^5\right )^{2/3}}{x^3} \, dx\right )+\int \frac {\left (-3+5 x^2\right ) \left (1+x^5\right )^{2/3}}{1-x^3+x^5} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}+\int \left (-\frac {3 \left (1+x^5\right )^{2/3}}{1-x^3+x^5}+\frac {5 x^2 \left (1+x^5\right )^{2/3}}{1-x^3+x^5}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {2}{3},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2}-3 \int \frac {\left (1+x^5\right )^{2/3}}{1-x^3+x^5} \, dx+5 \int \frac {x^2 \left (1+x^5\right )^{2/3}}{1-x^3+x^5} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.87, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^5}}\right )+\log \left (-x+\sqrt [3]{1+x^5}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 6.32, size = 290, normalized size = 3.22

method	result	size
risch	\(\frac {3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-x^{5}-3 \left (x^{5}+1\right )^{\frac {2}{3}} x -3 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{5}-x^{3}+1}\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{5}+1\right )^{\frac {2}{3}} x -3 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{5}-x^{3}+1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{5}+1\right )^{\frac {2}{3}} x -3 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{5}-x^{3}+1}\right )\)	\(290\)
trager	\(\frac {3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}-3 \ln \left (-\frac {525713499 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-1051426998 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}[phone] x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-471795651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}[phone] x^{3}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )[phone]}{x^{5}-x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {525713499 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-498754575 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-1051426998 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}[phone] x^{5}-552672423 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x -552672423 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-229155681 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}[phone] x^{3}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-498754575 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )[phone]}{x^{5}-x^{3}+1}\right )-\ln \left (-\frac {525713499 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}-1051426998 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}[phone] x^{5}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-471795651 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}[phone] x^{3}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}[phone] \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )[phone]}{x^{5}-x^{3}+1}\right )\)	\(609\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^5+1)^(2/3)/x^2+RootOf(_Z^2+_Z+1)*ln((RootOf(_Z^2+_Z+1)*x^5+RootOf(_Z^2+_Z+1)^2*x^3-x^5-3*(x^5+1)^(2/3)*
x-3*(x^5+1)^(1/3)*x^2-x^3+RootOf(_Z^2+_Z+1)-1)/(x^5-x^3+1))-ln((-RootOf(_Z^2+_Z+1)*x^5+RootOf(_Z^2+_Z+1)^2*x^3
-2*x^5+2*RootOf(_Z^2+_Z+1)*x^3-3*(x^5+1)^(2/3)*x-3*(x^5+1)^(1/3)*x^2-RootOf(_Z^2+_Z+1)-2)/(x^5-x^3+1))*RootOf(
_Z^2+_Z+1)-ln((-RootOf(_Z^2+_Z+1)*x^5+RootOf(_Z^2+_Z+1)^2*x^3-2*x^5+2*RootOf(_Z^2+_Z+1)*x^3-3*(x^5+1)^(2/3)*x-
3*(x^5+1)^(1/3)*x^2-RootOf(_Z^2+_Z+1)-2)/(x^5-x^3+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.03, size = 135, normalized size = 1.50 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {67616276 \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 10249526 \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (1423013 \, x^{5} + 37509888 \, x^{3} + 1423013\right )}}{300763 \, x^{5} - 86350888 \, x^{3} + 300763}\right ) - x^{2} \log \left (\frac {x^{5} - x^{3} + 3 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + 1}{x^{5} - x^{3} + 1}\right ) - 3 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(3)*x^2*arctan((67616276*sqrt(3)*(x^5 + 1)^(1/3)*x^2 + 10249526*sqrt(3)*(x^5 + 1)^(2/3)*x + sqrt(3
)*(1423013*x^5 + 37509888*x^3 + 1423013))/(300763*x^5 - 86350888*x^3 + 300763)) - x^2*log((x^5 - x^3 + 3*(x^5
+ 1)^(1/3)*x^2 - 3*(x^5 + 1)^(2/3)*x + 1)/(x^5 - x^3 + 1)) - 3*(x^5 + 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^5+1\right )}^{2/3}\,\left (2\,x^5-3\right )}{x^3\,\left (x^5-x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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