3.12.44 \(\int \frac {2+3 x^5}{(-1+x^2+x^5) \sqrt [3]{-x+x^6}} \, dx\) [1144]

Optimal. Leaf size=85 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x+x^6}}\right )-\log \left (x+\sqrt [3]{-x+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x+x^6}+\left (-x+x^6\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^6-x)^(1/3)))-ln(x+(x^6-x)^(1/3))+1/2*ln(x^2-x*(x^6-x)^(1/3)+(x^6-x)^(2/3))

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Rubi [F]
time = 0.76, 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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}\\ &=\frac {\left (9 \sqrt [3]{x} \sqrt [3]{1-x^5}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^{15}}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^6}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^5}\right ) \text {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 ) \text {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 ) \text {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 = 20.20, 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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 5.34, size = 539, normalized size = 6.34

method	result	size
trager	\(\text {Expression too large to display}\)	\(539\)

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]

RootOf(_Z^2-_Z+1)*ln((62685010802979296884*RootOf(_Z^2-_Z+1)^2*x^5-494135918029415995819*RootOf(_Z^2-_Z+1)*x^5
+64534903374029948502*x^5-485808833723089550851*RootOf(_Z^2-_Z+1)^2*x^2+433300799797487350553*(x^6-x)^(2/3)*Ro
otOf(_Z^2-_Z+1)-433300799797487350553*(x^6-x)^(1/3)*RootOf(_Z^2-_Z+1)*x+487658726294140202469*RootOf(_Z^2-_Z+1
)*x^2-62685010802979296884*RootOf(_Z^2-_Z+1)^2+427751122084335395699*(x^6-x)^(2/3)-427751122084335395699*x*(x^
6-x)^(1/3)-56207819067703503534*x^2+494135918029415995819*RootOf(_Z^2-_Z+1)-64534903374029948502)/(x^5+x^2-1))
-ln((62685010802979296884*RootOf(_Z^2-_Z+1)^2*x^5+368765896423457402051*RootOf(_Z^2-_Z+1)*x^5-3669160038524067
50433*x^5-485808833723089550851*RootOf(_Z^2-_Z+1)^2*x^2-433300799797487350553*(x^6-x)^(2/3)*RootOf(_Z^2-_Z+1)+
433300799797487350553*(x^6-x)^(1/3)*RootOf(_Z^2-_Z+1)*x+483958941152038899233*RootOf(_Z^2-_Z+1)*x^2-6268501080
2979296884*RootOf(_Z^2-_Z+1)^2+861051921881822746252*(x^6-x)^(2/3)-861051921881822746252*x*(x^6-x)^(1/3)-54357
926496652851916*x^2-368765896423457402051*RootOf(_Z^2-_Z+1)+366916003852406750433)/(x^5+x^2-1))*RootOf(_Z^2-_Z
+1)+ln((62685010802979296884*RootOf(_Z^2-_Z+1)^2*x^5+368765896423457402051*RootOf(_Z^2-_Z+1)*x^5-3669160038524
06750433*x^5-485808833723089550851*RootOf(_Z^2-_Z+1)^2*x^2-433300799797487350553*(x^6-x)^(2/3)*RootOf(_Z^2-_Z+
1)+433300799797487350553*(x^6-x)^(1/3)*RootOf(_Z^2-_Z+1)*x+483958941152038899233*RootOf(_Z^2-_Z+1)*x^2-6268501
0802979296884*RootOf(_Z^2-_Z+1)^2+861051921881822746252*(x^6-x)^(2/3)-861051921881822746252*x*(x^6-x)^(1/3)-54
357926496652851916*x^2-368765896423457402051*RootOf(_Z^2-_Z+1)+366916003852406750433)/(x^5+x^2-1))

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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((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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Fricas [A]
time = 1.31, size = 104, normalized size = 1.22 \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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{5} + 2}{\sqrt [3]{x \left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (x^{5} + x^{2} - 1\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 + 2)/((x*(x - 1)*(x**4 + x**3 + x**2 + x + 1))**(1/3)*(x**5 + x**2 - 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((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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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^6 - x)^(1/3)*(x^2 + x^5 - 1)),x)

[Out]

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

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