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

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]{1+x+x^3}}{x}-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x+x^3}}\right )-\log \left (-x+\sqrt [3]{1+x+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x+x^3}+\left (1+x+x^3\right )^{2/3}\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 = 6.98, size = 389, normalized size = 4.09

method	result	size
trager	\(-\frac {3 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}-\ln \left (\frac {18990 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{3}-37665 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x +3357 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+21648 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{3}-9495 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +2238 \left (x^{3}+x +1\right )^{\frac {2}{3}} x +22872 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-16670 x^{3}-9495 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2}+30321 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x +30321 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )-13336 x -13336}{1+x}\right )+\frac {3 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \ln \left (-\frac {7974 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x^{3}-75330 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {2}{3}} x +68616 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+1398 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x^{3}-3987 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2} x +45744 \left (x^{3}+x +1\right )^{\frac {2}{3}} x +4476 \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-46676 x^{3}-3987 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )^{2}-17976 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right ) x -17976 \RootOf \left (9 \textit {\_Z}^{2}-6 \textit {\_Z} +4\right )-20004 x -20004}{1+x}\right )}{2}\)	\(389\)
risch	\(-\frac {3 \left (x^{3}+x +1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}+4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+4 x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}} x^{2}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}-12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}+12 x^{2} \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}}+8 x^{4}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}+8 x^{3}+4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x -12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x +4 x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+8 x +4}{\left (1+x \right ) \left (x^{3}+x +1\right )}\right )+\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{6}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}-8 x^{6}+\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{3}+6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {2}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{3}-12 x^{4}-6 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x^{2}-12 x^{3}-4 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +12 \left (x^{6}+2 x^{4}+2 x^{3}+x^{2}+2 x +1\right )^{\frac {1}{3}} x -4 x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-8 x -4}{\left (1+x \right ) \left (x^{3}+x +1\right )}\right )}{2}\right ) \left (\left (x^{3}+x +1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{3}+x +1\right )^{\frac {2}{3}}}\)	\(753\)

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,method=_RETURNVERBOSE)

[Out]

-3*(x^3+x+1)^(1/3)/x-ln((18990*RootOf(9*_Z^2-6*_Z+4)^2*x^3-37665*RootOf(9*_Z^2-6*_Z+4)*(x^3+x+1)^(2/3)*x+3357*
RootOf(9*_Z^2-6*_Z+4)*(x^3+x+1)^(1/3)*x^2+21648*RootOf(9*_Z^2-6*_Z+4)*x^3-9495*RootOf(9*_Z^2-6*_Z+4)^2*x+2238*
(x^3+x+1)^(2/3)*x+22872*(x^3+x+1)^(1/3)*x^2-16670*x^3-9495*RootOf(9*_Z^2-6*_Z+4)^2+30321*RootOf(9*_Z^2-6*_Z+4)
*x+30321*RootOf(9*_Z^2-6*_Z+4)-13336*x-13336)/(1+x))+3/2*RootOf(9*_Z^2-6*_Z+4)*ln(-(7974*RootOf(9*_Z^2-6*_Z+4)
^2*x^3-75330*RootOf(9*_Z^2-6*_Z+4)*(x^3+x+1)^(2/3)*x+68616*RootOf(9*_Z^2-6*_Z+4)*(x^3+x+1)^(1/3)*x^2+1398*Root
Of(9*_Z^2-6*_Z+4)*x^3-3987*RootOf(9*_Z^2-6*_Z+4)^2*x+45744*(x^3+x+1)^(2/3)*x+4476*(x^3+x+1)^(1/3)*x^2-46676*x^
3-3987*RootOf(9*_Z^2-6*_Z+4)^2-17976*RootOf(9*_Z^2-6*_Z+4)*x-17976*RootOf(9*_Z^2-6*_Z+4)-20004*x-20004)/(1+x))

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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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