3.12.72 \(\int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx\)

Optimal. Leaf size=95 \[ -\frac {3 \sqrt [3]{x^3+x+1}}{x}-\log \left (\sqrt [3]{x^3+x+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x+1}+x}\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3+x+1} x+\left (x^3+x+1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 1.47, 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 {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*2^(2/3)*(1 + x + x^3)^(1/3)*Defer[Int][(((2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3))/6^(2/3)
+ x)^(1/3)*((6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3))/18 - (((6/(-9 + Sqrt
[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x)/3^(2/3) + x^2)^(1/3))/x^2, x])/((6^(1/3)*(2*(3/(-9 + Sqrt[93]))^(
1/3) - (2*(-9 + Sqrt[93]))^(1/3)) + 6*x)^(1/3)*(6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqr
t[93]))^(2/3) - 6*3^(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2)^(1/3)) - (3*2^(2
/3)*(1 + x + x^3)^(1/3)*Defer[Int][(((2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3))/6^(2/3) + x)^(1
/3)*((6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3))/18 - (((6/(-9 + Sqrt[93]))^
(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x)/3^(2/3) + x^2)^(1/3))/x, x])/((6^(1/3)*(2*(3/(-9 + Sqrt[93]))^(1/3) - (2
*(-9 + Sqrt[93]))^(1/3)) + 6*x)^(1/3)*(6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(
2/3) - 6*3^(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2)^(1/3)) + (3*2^(2/3)*(1 +
x + x^3)^(1/3)*Defer[Int][(((2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-9 + Sqrt[93]))^(1/3))/6^(2/3) + x)^(1/3)*((6 +
6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3))/18 - (((6/(-9 + Sqrt[93]))^(1/3) - (
(-9 + Sqrt[93])/2)^(1/3))*x)/3^(2/3) + x^2)^(1/3))/(1 + x), x])/((6^(1/3)*(2*(3/(-9 + Sqrt[93]))^(1/3) - (2*(-
9 + Sqrt[93]))^(1/3)) + 6*x)^(1/3)*(6 + 6*3^(1/3)*(2/(-9 + Sqrt[93]))^(2/3) + 2^(1/3)*(3*(-9 + Sqrt[93]))^(2/3
) - 6*3^(1/3)*((6/(-9 + Sqrt[93]))^(1/3) - ((-9 + Sqrt[93])/2)^(1/3))*x + 18*x^2)^(1/3))

Rubi steps

\begin {align*} \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx &=\int \left (\frac {3 \sqrt [3]{1+x+x^3}}{x^2}-\frac {\sqrt [3]{1+x+x^3}}{x}+\frac {\sqrt [3]{1+x+x^3}}{1+x}\right ) \, dx\\ &=3 \int \frac {\sqrt [3]{1+x+x^3}}{x^2} \, dx-\int \frac {\sqrt [3]{1+x+x^3}}{x} \, dx+\int \frac {\sqrt [3]{1+x+x^3}}{1+x} \, dx\\ &=-\frac {\sqrt [3]{1+x+x^3} \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{x} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}+\frac {\sqrt [3]{1+x+x^3} \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{1+x} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}+\frac {\left (3 \sqrt [3]{1+x+x^3}\right ) \int \frac {\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}{x^2} \, dx}{\sqrt [3]{\frac {2 \sqrt [3]{\frac {3}{-9+\sqrt {93}}}-\sqrt [3]{2 \left (-9+\sqrt {93}\right )}}{6^{2/3}}+x} \sqrt [3]{\frac {1}{18} \left (6+6 \sqrt [3]{3} \left (\frac {2}{-9+\sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (-9+\sqrt {93}\right )\right )^{2/3}\right )-\frac {\left (\sqrt [3]{\frac {6}{-9+\sqrt {93}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {93}\right )}\right ) x}{3^{2/3}}+x^2}}\\ \end {align*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+2 x) \sqrt [3]{1+x+x^3}}{x^2 (1+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.26, size = 95, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{x^3+x+1}}{x}-\log \left (\sqrt [3]{x^3+x+1}-x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x+1}+x}\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3+x+1} x+\left (x^3+x+1\right )^{2/3}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.11, size = 1096, normalized size = 11.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(x^3+x+1)^(1/3)/x+(1/2*RootOf(_Z^2-2*_Z+4)*ln((RootOf(_Z^2-2*_Z+4)^2*x^6+RootOf(_Z^2-2*_Z+4)*x^6+RootOf(_Z^
2-2*_Z+4)^2*x^4+3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x^4-2*x^6+RootOf(_Z^2-2*_Z+4)^2*x^3+3*
RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(2/3)*x^2+3*RootOf(_Z^2-2*_Z+4)*x^4-6*(x^6+2*x^4+2*x^3+x^2+2*x
+1)^(1/3)*x^4+3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x^2+3*RootOf(_Z^2-2*_Z+4)*x^3-6*x^2*(x^6
+2*x^4+2*x^3+x^2+2*x+1)^(2/3)-4*x^4+3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x+2*RootOf(_Z^2-2*
_Z+4)*x^2-6*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x^2-4*x^3+4*RootOf(_Z^2-2*_Z+4)*x-6*(x^6+2*x^4+2*x^3+x^2+2*x+1)^
(1/3)*x-2*x^2+2*RootOf(_Z^2-2*_Z+4)-4*x-2)/(x^3+x+1)/(1+x))+ln((RootOf(_Z^2-2*_Z+4)^2*x^6-5*RootOf(_Z^2-2*_Z+4
)*x^6+RootOf(_Z^2-2*_Z+4)^2*x^4-3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x^4+4*x^6+RootOf(_Z^2-
2*_Z+4)^2*x^3-3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(2/3)*x^2-7*RootOf(_Z^2-2*_Z+4)*x^4-3*RootOf(_
Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x^2-7*RootOf(_Z^2-2*_Z+4)*x^3+6*x^4-3*RootOf(_Z^2-2*_Z+4)*(x^6+2
*x^4+2*x^3+x^2+2*x+1)^(1/3)*x-2*RootOf(_Z^2-2*_Z+4)*x^2+6*x^3-4*RootOf(_Z^2-2*_Z+4)*x+2*x^2-2*RootOf(_Z^2-2*_Z
+4)+4*x+2)/(x^3+x+1)/(1+x))-1/2*ln((RootOf(_Z^2-2*_Z+4)^2*x^6-5*RootOf(_Z^2-2*_Z+4)*x^6+RootOf(_Z^2-2*_Z+4)^2*
x^4-3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*x^4+4*x^6+RootOf(_Z^2-2*_Z+4)^2*x^3-3*RootOf(_Z^2-
2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(2/3)*x^2-7*RootOf(_Z^2-2*_Z+4)*x^4-3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3
+x^2+2*x+1)^(1/3)*x^2-7*RootOf(_Z^2-2*_Z+4)*x^3+6*x^4-3*RootOf(_Z^2-2*_Z+4)*(x^6+2*x^4+2*x^3+x^2+2*x+1)^(1/3)*
x-2*RootOf(_Z^2-2*_Z+4)*x^2+6*x^3-4*RootOf(_Z^2-2*_Z+4)*x+2*x^2-2*RootOf(_Z^2-2*_Z+4)+4*x+2)/(x^3+x+1)/(1+x))*
RootOf(_Z^2-2*_Z+4))/(x^3+x+1)^(2/3)*((x^3+x+1)^2)^(1/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^3 + x + 1)^(1/3)*(2*x + 3)/((x + 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+3\right )\,{\left (x^3+x+1\right )}^{1/3}}{x^2\,\left (x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((2*x + 3)*(x**3 + x + 1)**(1/3)/(x**2*(x + 1)), x)

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