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

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 = 17.79, size = 100, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\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 = 12.40, 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 = 35.09, size = 632, normalized size = 6.32

method	result	size
trager	\(\frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \ln \left (\frac {-88928146808112798 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{7}-55447617558383298 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{7}+2558542431350156 x^{7}+133392220212169197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+66431161712710197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (2 x^{7}+1\right )^{\frac {2}{3}} x -66431161712710197 \left (2 x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}-20578078271889984 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-20401446009640383 \left (2 x^{7}+1\right )^{\frac {2}{3}} x +20401446009640383 \left (2 x^{7}+1\right )^{\frac {1}{3}} x^{2}[phone] x^{3}-44464073404056399 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-27723808779191649 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+1279271215675078}{2 x^{7}+x^{3}+1}\right )-3 \ln \left (-\frac {88928146808112798 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{7}-114733048763791830 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{7}+25804901955679032 x^{7}-133392220212169197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+66431161712710197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (2 x^{7}+1\right )^{\frac {2}{3}} x -66431161712710197 \left (2 x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+68350068536222814 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-1742274561263016 \left (2 x^{7}+1\right )^{\frac {2}{3}} x +1742274561263016 \left (2 x^{7}+1\right )^{\frac {1}{3}} x^{2}-8601633985226344 x^{3}+44464073404056399 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-57366524381895915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+12902450977839516}{2 x^{7}+x^{3}+1}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+\ln \left (-\frac {88928146808112798 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{7}-114733048763791830 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{7}+25804901955679032 x^{7}-133392220212169197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2} x^{3}+66431161712710197 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (2 x^{7}+1\right )^{\frac {2}{3}} x -66431161712710197 \left (2 x^{7}+1\right )^{\frac {1}{3}} \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{2}+68350068536222814 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x^{3}-1742274561263016 \left (2 x^{7}+1\right )^{\frac {2}{3}} x +1742274561263016 \left (2 x^{7}+1\right )^{\frac {1}{3}} x^{2}-8601633985226344 x^{3}+44464073404056399 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-57366524381895915 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )+12902450977839516}{2 x^{7}+x^{3}+1}\right )\)	\(632\)
risch	\(\frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+6 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )-\ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+2 x^{10}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+2 x^{10}+8 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +x^{3}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )\right ) \left (\left (2 x^{7}+1\right )^{2}\right )^{\frac {1}{3}}}{\left (2 x^{7}+1\right )^{\frac {2}{3}}}\)	\(778\)

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,method=_RETURNVERBOSE)

[Out]

3*(2*x^7+1)^(1/3)/x+3*RootOf(9*_Z^2-3*_Z+1)*ln((-88928146808112798*RootOf(9*_Z^2-3*_Z+1)^2*x^7-554476175583832
98*RootOf(9*_Z^2-3*_Z+1)*x^7+2558542431350156*x^7+133392220212169197*RootOf(9*_Z^2-3*_Z+1)^2*x^3+6643116171271
0197*RootOf(9*_Z^2-3*_Z+1)*(2*x^7+1)^(2/3)*x-66431161712710197*(2*x^7+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2-20578
078271889984*RootOf(9*_Z^2-3*_Z+1)*x^3-20401446009640383*(2*x^7+1)^(2/3)*x+20401446009640383*(2*x^7+1)^(1/3)*x
^2+639635607837539*x^3-44464073404056399*RootOf(9*_Z^2-3*_Z+1)^2-27723808779191649*RootOf(9*_Z^2-3*_Z+1)+12792
71215675078)/(2*x^7+x^3+1))-3*ln(-(88928146808112798*RootOf(9*_Z^2-3*_Z+1)^2*x^7-114733048763791830*RootOf(9*_
Z^2-3*_Z+1)*x^7+25804901955679032*x^7-133392220212169197*RootOf(9*_Z^2-3*_Z+1)^2*x^3+66431161712710197*RootOf(
9*_Z^2-3*_Z+1)*(2*x^7+1)^(2/3)*x-66431161712710197*(2*x^7+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+68350068536222814
*RootOf(9*_Z^2-3*_Z+1)*x^3-1742274561263016*(2*x^7+1)^(2/3)*x+1742274561263016*(2*x^7+1)^(1/3)*x^2-86016339852
26344*x^3+44464073404056399*RootOf(9*_Z^2-3*_Z+1)^2-57366524381895915*RootOf(9*_Z^2-3*_Z+1)+12902450977839516)
/(2*x^7+x^3+1))*RootOf(9*_Z^2-3*_Z+1)+ln(-(88928146808112798*RootOf(9*_Z^2-3*_Z+1)^2*x^7-114733048763791830*Ro
otOf(9*_Z^2-3*_Z+1)*x^7+25804901955679032*x^7-133392220212169197*RootOf(9*_Z^2-3*_Z+1)^2*x^3+66431161712710197
*RootOf(9*_Z^2-3*_Z+1)*(2*x^7+1)^(2/3)*x-66431161712710197*(2*x^7+1)^(1/3)*RootOf(9*_Z^2-3*_Z+1)*x^2+683500685
36222814*RootOf(9*_Z^2-3*_Z+1)*x^3-1742274561263016*(2*x^7+1)^(2/3)*x+1742274561263016*(2*x^7+1)^(1/3)*x^2-860
1633985226344*x^3+44464073404056399*RootOf(9*_Z^2-3*_Z+1)^2-57366524381895915*RootOf(9*_Z^2-3*_Z+1)[phone]
7839516)/(2*x^7+x^3+1))

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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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