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

Optimal. Leaf size=133 \[ \frac {3 \sqrt [3]{1+x^5}}{2 x}+\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2 \sqrt [3]{2}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\right )}{4 \sqrt [3]{2}} \]

[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)

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Rubi [F]
time = 0.56, 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 {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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 \, _2F_1\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 \, _2F_1\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*}

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Mathematica [A]
time = 1.64, size = 133, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+x^5}}{2 x}+\frac {\sqrt {3} \text {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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 49.39, size = 1015, normalized size = 7.63

method	result	size
trager	\(\text {Expression too large to display}\)	\(1015\)
risch	\(\text {Expression too large to display}\)	\(1655\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^5+1)^(1/3)/x+3/2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln((876387663*RootOf(RootOf(_Z^3-
4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3*x^5+76247796*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
36*_Z^2)*RootOf(_Z^3-4)^4*x^5-1752775326*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)
^3*x^3-152495592*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4*x^3-2337033768*RootOf(R
ootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^5-203327456*RootOf(_Z^3-4)^2*x^5+496072665*(x^5+
1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x+876387663*RootOf(RootOf(_Z^3-
4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3+76247796*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_
Z^2)*RootOf(_Z^3-4)^4-1460646105*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^3-12707
9660*RootOf(_Z^3-4)^2*x^3+3428378874*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*(x^5+1)^(1/3)*x^2+73
6754034*RootOf(_Z^3-4)*(x^5+1)^(1/3)*x^2-1142792958*(x^5+1)^(2/3)*x-2337033768*RootOf(RootOf(_Z^3-4)^2+6*_Z*Ro
otOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)-203327456*RootOf(_Z^3-4)^2)/(2*x^5-x^3+2))+1/4*RootOf(_Z^3-4)*ln(-(418900
887*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3*x^5-76247796*RootOf(RootOf(_Z^3-4)
^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4*x^5-837801774*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*
_Z^2)^2*RootOf(_Z^3-4)^3*x^3+152495592*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4*x
^3+1396336290*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^5-254159320*RootOf(_Z^3-4)
^2*x^5+496072665*(x^5+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x+4189008
87*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^3-76247796*RootOf(RootOf(_Z^3-4)^2+6*
_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^4+139633629*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*Roo
tOf(_Z^3-4)*x^3-25415932*RootOf(_Z^3-4)^2*x^3-4420524204*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*
(x^5+1)^(1/3)*x^2-571396479*RootOf(_Z^3-4)*(x^5+1)^(1/3)*x^2+1473508068*(x^5+1)^(2/3)*x+1396336290*RootOf(Root
Of(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)-254159320*RootOf(_Z^3-4)^2)/(2*x^5-x^3+2))

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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)^(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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (97) = 194\).
time = 38.23, size = 384, normalized size = 2.89 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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)^(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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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