3.10.93 \(\int \frac {2-3 x^5}{(1-x^2+x^5) \sqrt [3]{x+x^6}} \, dx\)

Optimal. Leaf size=75 \[ -\log \left (\sqrt [3]{x^6+x}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+x}+x}\right )+\frac {1}{2} \log \left (\sqrt [3]{x^6+x} x+\left (x^6+x\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 1.24, 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 {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 - 3*x^5)/((1 - x^2 + x^5)*(x + x^6)^(1/3)),x]

[Out]

(-9*x*(1 + x^5)^(1/3)*Hypergeometric2F1[2/15, 1/3, 17/15, -x^5])/(2*(x + x^6)^(1/3)) + (15*x^(1/3)*(1 + x^5)^(
1/3)*Defer[Subst][Defer[Int][x/((1 + x^15)^(1/3)*(1 - x^6 + x^15)), x], x, x^(1/3)])/(x + x^6)^(1/3) - (9*x^(1
/3)*(1 + x^5)^(1/3)*Defer[Subst][Defer[Int][x^7/((1 + x^15)^(1/3)*(1 - x^6 + x^15)), x], x, x^(1/3)])/(x + x^6
)^(1/3)

Rubi steps

\begin {align*} \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \int \frac {2-3 x^5}{\sqrt [3]{x} \sqrt [3]{1+x^5} \left (1-x^2+x^5\right )} \, dx}{\sqrt [3]{x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (2-3 x^{15}\right )}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x}{\sqrt [3]{1+x^{15}}}+\frac {x \left (5-3 x^6\right )}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x \left (5-3 x^6\right )}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=-\frac {9 x \sqrt [3]{1+x^5} \, _2F_1\left (\frac {2}{15},\frac {1}{3};\frac {17}{15};-x^5\right )}{2 \sqrt [3]{x+x^6}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {5 x}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )}-\frac {3 x^7}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ &=-\frac {9 x \sqrt [3]{1+x^5} \, _2F_1\left (\frac {2}{15},\frac {1}{3};\frac {17}{15};-x^5\right )}{2 \sqrt [3]{x+x^6}}-\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}+\frac {\left (15 \sqrt [3]{x} \sqrt [3]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1+x^{15}} \left (1-x^6+x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2-3 x^5}{\left (1-x^2+x^5\right ) \sqrt [3]{x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2 - 3*x^5)/((1 - x^2 + x^5)*(x + x^6)^(1/3)),x]

[Out]

Integrate[(2 - 3*x^5)/((1 - x^2 + x^5)*(x + x^6)^(1/3)), x]

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IntegrateAlgebraic [A]  time = 2.75, size = 75, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^6}}\right )-\log \left (-x+\sqrt [3]{x+x^6}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+x^6}+\left (x+x^6\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 - 3*x^5)/((1 - x^2 + x^5)*(x + x^6)^(1/3)),x]

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x + x^6)^(1/3))] - Log[-x + (x + x^6)^(1/3)] + Log[x^2 + x*(x + x^6)^(1/3)
+ (x + x^6)^(2/3)]/2

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fricas [A]  time = 2.28, size = 100, normalized size = 1.33 \begin {gather*} \sqrt {3} \arctan \left (-\frac {4 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + 1\right )} - 2 \, \sqrt {3} {\left (x^{6} + x\right )}^{\frac {2}{3}}}{x^{5} + 8 \, x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} - x^{2} + 3 \, {\left (x^{6} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{6} + x\right )}^{\frac {2}{3}} + 1}{x^{5} - x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+2)/(x^5-x^2+1)/(x^6+x)^(1/3),x, algorithm="fricas")

[Out]

sqrt(3)*arctan(-(4*sqrt(3)*(x^6 + x)^(1/3)*x + sqrt(3)*(x^5 + 1) - 2*sqrt(3)*(x^6 + x)^(2/3))/(x^5 + 8*x^2 + 1
)) - 1/2*log((x^5 - x^2 + 3*(x^6 + x)^(1/3)*x - 3*(x^6 + x)^(2/3) + 1)/(x^5 - x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, x^{5} - 2}{{\left (x^{6} + x\right )}^{\frac {1}{3}} {\left (x^{5} - x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+2)/(x^5-x^2+1)/(x^6+x)^(1/3),x, algorithm="giac")

[Out]

integrate(-(3*x^5 - 2)/((x^6 + x)^(1/3)*(x^5 - x^2 + 1)), x)

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maple [C]  time = 5.81, size = 348, normalized size = 4.64

method	result	size
trager	\(-\ln \left (-\frac {540903516171 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-2129063429050 x^{5}-1081807032342 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-4781755056454 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x [phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}[phone] \left (x^{6}+x \right )^{\frac {2}{3}}[phone] x \left (x^{6}+x \right )^{\frac {1}{3}}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-3193595143575 x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2129063429050}{x^{5}-x^{2}+1}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-25445006 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{5}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}-103219569 x^{5}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {2}{3}}-163071098 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}+x \right )^{\frac {1}{3}} x -43368040 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}[phone] \left (x^{6}+x \right )^{\frac {2}{3}}[phone] x \left (x^{6}+x \right )^{\frac {1}{3}}-25445006 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-34406523 x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-103219569}{x^{5}-x^{2}+1}\right )\)	\(348\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^5+2)/(x^5-x^2+1)/(x^6+x)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-ln(-(540903516171*RootOf(_Z^2-_Z+1)^2*x^5+17275317817*RootOf(_Z^2-_Z+1)*x^5-2129063429050*x^5-1081807032342*R
ootOf(_Z^2-_Z+1)^2*x^2+1553609277245*RootOf(_Z^2-_Z+1)*(x^6+x)^(2/3)-4781755056454*RootOf(_Z^2-_Z+1)*(x^6+x)^(
1/3)*x+3751773977563*RootOf(_Z^2-_Z+1)*x^2+3228145779209*(x^6+x)^(2/3)[phone]*x*(x^6+x)^(1/3)[phone]
171*RootOf(_Z^2-_Z+1)^2-3193595143575*x^2+17275317817*RootOf(_Z^2-_Z+1)-2129063429050)/(x^5-x^2+1))+RootOf(_Z^
2-_Z+1)*ln(-(-25445006*RootOf(_Z^2-_Z+1)^2*x^5+110741541*RootOf(_Z^2-_Z+1)*x^5-103219569*x^5+50890012*RootOf(_
Z^2-_Z+1)^2*x^2+70252591*RootOf(_Z^2-_Z+1)*(x^6+x)^(2/3)-163071098*RootOf(_Z^2-_Z+1)*(x^6+x)^(1/3)*x-43368040*
RootOf(_Z^2-_Z+1)*x^2+92818507*(x^6+x)^(2/3)[phone]*x*(x^6+x)^(1/3)-25445006*RootOf(_Z^2-_Z+1)^2-34406523*x^
2+110741541*RootOf(_Z^2-_Z+1)-103219569)/(x^5-x^2+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {3 \, x^{5} - 2}{{\left (x^{6} + x\right )}^{\frac {1}{3}} {\left (x^{5} - x^{2} + 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^5+2)/(x^5-x^2+1)/(x^6+x)^(1/3),x, algorithm="maxima")

[Out]

-integrate((3*x^5 - 2)/((x^6 + x)^(1/3)*(x^5 - x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {3\,x^5-2}{{\left (x^6+x\right )}^{1/3}\,\left (x^5-x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x^5 - 2)/((x + x^6)^(1/3)*(x^5 - x^2 + 1)),x)

[Out]

-int((3*x^5 - 2)/((x + x^6)^(1/3)*(x^5 - x^2 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {3 x^{5}}{x^{5} \sqrt [3]{x^{6} + x} - x^{2} \sqrt [3]{x^{6} + x} + \sqrt [3]{x^{6} + x}}\, dx - \int \left (- \frac {2}{x^{5} \sqrt [3]{x^{6} + x} - x^{2} \sqrt [3]{x^{6} + x} + \sqrt [3]{x^{6} + x}}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**5+2)/(x**5-x**2+1)/(x**6+x)**(1/3),x)

[Out]

-Integral(3*x**5/(x**5*(x**6 + x)**(1/3) - x**2*(x**6 + x)**(1/3) + (x**6 + x)**(1/3)), x) - Integral(-2/(x**5
*(x**6 + x)**(1/3) - x**2*(x**6 + x)**(1/3) + (x**6 + x)**(1/3)), x)

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