3.13.36 \(\int \frac {\sqrt [3]{1+2 x^7} (-3+8 x^7)}{x^2 (1+x^3+2 x^7)} \, dx\)

Optimal. Leaf size=100 \[ \frac {3 \sqrt [3]{2 x^7+1}}{x}-\log \left (\sqrt [3]{2 x^7+1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2 x^7+1}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{2 x^7+1} x+\left (2 x^7+1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.87, 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+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{1+2 x^7}}{x^2}+\frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{1+2 x^7}}{x^2} \, dx\right )+\int \frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{7};\frac {6}{7};-2 x^7\right )}{x}+\int \left (\frac {3 x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}+\frac {14 x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{7};\frac {6}{7};-2 x^7\right )}{x}+3 \int \frac {x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx+14 \int \frac {x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]

[Out]

Integrate[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)), x]

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IntegrateAlgebraic [A]  time = 17.82, size = 100, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{2 x^7+1}}{x}-\log \left (\sqrt [3]{2 x^7+1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2 x^7+1}-x}\right )+\frac {1}{2} \log \left (-\sqrt [3]{2 x^7+1} x+\left (2 x^7+1\right )^{2/3}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + 2*x^7)^(1/3)*(-3 + 8*x^7))/(x^2*(1 + x^3 + 2*x^7)),x]

[Out]

(3*(1 + 2*x^7)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + 2*x^7)^(1/3))] - Log[x + (1 + 2*x^7)^(1/3)]
+ Log[x^2 - x*(1 + 2*x^7)^(1/3) + (1 + 2*x^7)^(2/3)]/2

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fricas [A]  time = 13.78, size = 141, normalized size = 1.41 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {8377128467638 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 15171948325814 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2102123379894 \, x^{7} + 4448471619035 \, x^{3} + 1051061689947\right )}}{60468559237154 \, x^{7} - 5089335571601 \, x^{3} + 30234279618577}\right ) - x \log \left (\frac {2 \, x^{7} + x^{3} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{7} + x^{3} + 1}\right ) + 6 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x*arctan((8377128467638*sqrt(3)*(2*x^7 + 1)^(1/3)*x^2 + 15171948325814*sqrt(3)*(2*x^7 + 1)^(2/3
)*x + sqrt(3)*(2102123379894*x^7 + 4448471619035*x^3 + 1051061689947))/(60468559237154*x^7 - 5089335571601*x^3
+ 30234279618577)) - x*log((2*x^7 + x^3 + 3*(2*x^7 + 1)^(1/3)*x^2 + 3*(2*x^7 + 1)^(2/3)*x + 1)/(2*x^7 + x^3 +
1)) + 6*(2*x^7 + 1)^(1/3))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x, algorithm="giac")

[Out]

integrate((8*x^7 - 3)*(2*x^7 + 1)^(1/3)/((2*x^7 + x^3 + 1)*x^2), x)

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maple [C]  time = 6.00, size = 778, normalized size = 7.78

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x)

[Out]

3*(2*x^7+1)^(1/3)/x+(RootOf(_Z^2-_Z+1)*ln((-8*x^14*RootOf(_Z^2-_Z+1)+4*x^14+4*RootOf(_Z^2-_Z+1)^2*x^10-2*x^10*
RootOf(_Z^2-_Z+1)+6*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(1/3)*x^8-6*(4*x^14+4*x^7+1)^(1/3)*x^8-8*x^7*RootOf(_Z^
2-_Z+1)+4*x^7+2*RootOf(_Z^2-_Z+1)^2*x^3-3*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(2/3)*x^2-RootOf(_Z^2-_Z+1)*x^3+3
*(4*x^14+4*x^7+1)^(2/3)*x^2+3*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(1/3)*x-3*(4*x^14+4*x^7+1)^(1/3)*x-2*RootOf(_
Z^2-_Z+1)+1)/(2*x^7+1)/(2*x^7+x^3+1))-ln((8*x^14*RootOf(_Z^2-_Z+1)-4*x^14+4*RootOf(_Z^2-_Z+1)^2*x^10-6*x^10*Ro
otOf(_Z^2-_Z+1)-6*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(1/3)*x^8+2*x^10+8*x^7*RootOf(_Z^2-_Z+1)-4*x^7+2*RootOf(_
Z^2-_Z+1)^2*x^3+3*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(2/3)*x^2-3*RootOf(_Z^2-_Z+1)*x^3-3*RootOf(_Z^2-_Z+1)*(4*
x^14+4*x^7+1)^(1/3)*x+x^3+2*RootOf(_Z^2-_Z+1)-1)/(2*x^7+1)/(2*x^7+x^3+1))*RootOf(_Z^2-_Z+1)+ln((8*x^14*RootOf(
_Z^2-_Z+1)-4*x^14+4*RootOf(_Z^2-_Z+1)^2*x^10-6*x^10*RootOf(_Z^2-_Z+1)-6*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(1/
3)*x^8+2*x^10+8*x^7*RootOf(_Z^2-_Z+1)-4*x^7+2*RootOf(_Z^2-_Z+1)^2*x^3+3*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(2/
3)*x^2-3*RootOf(_Z^2-_Z+1)*x^3-3*RootOf(_Z^2-_Z+1)*(4*x^14+4*x^7+1)^(1/3)*x+x^3+2*RootOf(_Z^2-_Z+1)-1)/(2*x^7+
1)/(2*x^7+x^3+1)))/(2*x^7+1)^(2/3)*((2*x^7+1)^2)^(1/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+1)^(1/3)*(8*x^7-3)/x^2/(2*x^7+x^3+1),x, algorithm="maxima")

[Out]

integrate((8*x^7 - 3)*(2*x^7 + 1)^(1/3)/((2*x^7 + x^3 + 1)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (2\,x^7+1\right )}^{1/3}\,\left (8\,x^7-3\right )}{x^2\,\left (2\,x^7+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^7 + 1)^(1/3)*(8*x^7 - 3))/(x^2*(x^3 + 2*x^7 + 1)),x)

[Out]

int(((2*x^7 + 1)^(1/3)*(8*x^7 - 3))/(x^2*(x^3 + 2*x^7 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{2 x^{7} + 1} \left (8 x^{7} - 3\right )}{x^{2} \left (2 x^{7} + x^{3} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**7+1)**(1/3)*(8*x**7-3)/x**2/(2*x**7+x**3+1),x)

[Out]

Integral((2*x**7 + 1)**(1/3)*(8*x**7 - 3)/(x**2*(2*x**7 + x**3 + 1)), x)

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