3.25.63 \(\int \frac {x (3+7 x^4)}{\sqrt [3]{1+x^4} (-4+x^3+x^7)} \, dx\)

Optimal. Leaf size=200 \[ \sqrt [3]{2} \tanh ^{-1}\left (1-\sqrt [3]{2} x \sqrt [3]{x^4+1}\right )-\frac {\sqrt {3} \tan ^{-1}\left (\frac {3 \sqrt {3} x \sqrt [3]{x^4+1}-6 x^2 \sqrt [3]{x^4+1}}{-3 \sqrt [3]{x^4+1} x+2 \sqrt {3} \sqrt [3]{x^4+1} x^2+4\ 2^{2/3} \sqrt {3} x-6\ 2^{2/3}}\right )}{2^{2/3}}-\frac {\tanh ^{-1}\left (\frac {2^{2/3} \sqrt [3]{x^4+1} x+2 \sqrt [3]{2}}{2^{2/3} \sqrt [3]{x^4+1} x+2 \left (x^4+1\right )^{2/3} x^2+2 \sqrt [3]{2}}\right )}{2^{2/3}} \]

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Rubi [F]  time = 0.49, 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 {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)),x]

[Out]

3*Defer[Int][x/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)), x] + 7*Defer[Int][x^5/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)), x
]

Rubi steps

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

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (3+7 x^4\right )}{\sqrt [3]{1+x^4} \left (-4+x^3+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)),x]

[Out]

Integrate[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)), x]

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IntegrateAlgebraic [A]  time = 19.20, size = 200, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {3 \sqrt {3} x \sqrt [3]{1+x^4}-6 x^2 \sqrt [3]{1+x^4}}{-6 2^{2/3}+4\ 2^{2/3} \sqrt {3} x-3 x \sqrt [3]{1+x^4}+2 \sqrt {3} x^2 \sqrt [3]{1+x^4}}\right )}{2^{2/3}}+\sqrt [3]{2} \tanh ^{-1}\left (1-\sqrt [3]{2} x \sqrt [3]{1+x^4}\right )-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [3]{2}+2^{2/3} x \sqrt [3]{1+x^4}}{2 \sqrt [3]{2}+2^{2/3} x \sqrt [3]{1+x^4}+2 x^2 \left (1+x^4\right )^{2/3}}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x*(3 + 7*x^4))/((1 + x^4)^(1/3)*(-4 + x^3 + x^7)),x]

[Out]

-((Sqrt[3]*ArcTan[(3*Sqrt[3]*x*(1 + x^4)^(1/3) - 6*x^2*(1 + x^4)^(1/3))/(-6*2^(2/3) + 4*2^(2/3)*Sqrt[3]*x - 3*
x*(1 + x^4)^(1/3) + 2*Sqrt[3]*x^2*(1 + x^4)^(1/3))])/2^(2/3)) + 2^(1/3)*ArcTanh[1 - 2^(1/3)*x*(1 + x^4)^(1/3)]
- ArcTanh[(2*2^(1/3) + 2^(2/3)*x*(1 + x^4)^(1/3))/(2*2^(1/3) + 2^(2/3)*x*(1 + x^4)^(1/3) + 2*x^2*(1 + x^4)^(2
/3))]/2^(2/3)

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fricas [B]  time = 76.58, size = 344, normalized size = 1.72 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} {\left (x^{16} + 2 \, x^{12} + 4 \, x^{9} + x^{8} + 4 \, x^{5} - 32 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{21} + 3 \, x^{17} + 60 \, x^{14} + 3 \, x^{13} + 120 \, x^{10} + x^{9} + 192 \, x^{7} + 60 \, x^{6} + 192 \, x^{3} - 64\right )} + 24 \, {\left (x^{15} + 2 \, x^{11} + 28 \, x^{8} + x^{7} + 28 \, x^{4} + 16 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{21} + 3 \, x^{17} - 12 \, x^{14} + 3 \, x^{13} - 24 \, x^{10} + x^{9} - 384 \, x^{7} - 12 \, x^{6} - 384 \, x^{3} - 64\right )}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {12 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x^{2} - 4^{\frac {2}{3}} {\left (x^{7} + x^{3} - 4\right )} - 12 \cdot 4^{\frac {1}{3}} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x}{x^{7} + x^{3} - 4}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{9} + x^{5} + 8 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{14} + 2 \, x^{10} + 28 \, x^{7} + x^{6} + 28 \, x^{3} + 16\right )} + 24 \, {\left (x^{8} + x^{4} + 2 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{x^{14} + 2 \, x^{10} - 8 \, x^{7} + x^{6} - 8 \, x^{3} + 16}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x, algorithm="fricas")

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(6*4^(2/3)*(x^16 + 2*x^12 + 4*x^9 + x^8 + 4*x^5 - 32*x^2)*(x^4
+ 1)^(2/3) + 4^(1/3)*(x^21 + 3*x^17 + 60*x^14 + 3*x^13 + 120*x^10 + x^9 + 192*x^7 + 60*x^6 + 192*x^3 - 64) +
24*(x^15 + 2*x^11 + 28*x^8 + x^7 + 28*x^4 + 16*x)*(x^4 + 1)^(1/3))/(x^21 + 3*x^17 - 12*x^14 + 3*x^13 - 24*x^10
+ x^9 - 384*x^7 - 12*x^6 - 384*x^3 - 64)) + 1/12*4^(2/3)*log(-(12*(x^4 + 1)^(2/3)*x^2 - 4^(2/3)*(x^7 + x^3 -
4) - 12*4^(1/3)*(x^4 + 1)^(1/3)*x)/(x^7 + x^3 - 4)) - 1/24*4^(2/3)*log((3*4^(2/3)*(x^9 + x^5 + 8*x^2)*(x^4 + 1
)^(2/3) + 4^(1/3)*(x^14 + 2*x^10 + 28*x^7 + x^6 + 28*x^3 + 16) + 24*(x^8 + x^4 + 2*x)*(x^4 + 1)^(1/3))/(x^14 +
2*x^10 - 8*x^7 + x^6 - 8*x^3 + 16))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (7 \, x^{4} + 3\right )} x}{{\left (x^{7} + x^{3} - 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x, algorithm="giac")

[Out]

integrate((7*x^4 + 3)*x/((x^7 + x^3 - 4)*(x^4 + 1)^(1/3)), x)

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maple [C]  time = 63.42, size = 937, normalized size = 4.68

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (\frac {-\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{7}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{7}-\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{7}+2 x^{7} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )+6 \left (x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}-\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+6 x^{2} \left (x^{4}+1\right )^{\frac {2}{3}}-12 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x -\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-2\right )+8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{x^{7}+x^{3}-4}\right )}{2}-\frac {\ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{7}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{7}+6 \left (x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+6 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{3}-2\right )-16 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{x^{7}+x^{3}-4}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{2}-\ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{7}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{7}+6 \left (x^{4}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+6 \left (x^{4}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -4 \RootOf \left (\textit {\_Z}^{3}-2\right )-16 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{x^{7}+x^{3}-4}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )\)	\(937\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^3-2)*ln((-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^7+2*RootOf(Root
Of(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^7-RootOf(_Z^3-2)*x^7+2*x^7*RootOf(RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+6*(x^4+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*
_Z^2)*x^2-RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+2*RootOf(RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+6*x^2*(x^4+1)^(2/3)-12*(x^4+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2
+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-RootOf(_Z^3-2)*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)
+4*_Z^2)*x^3-4*RootOf(_Z^3-2)+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(x^7+x^3-4))-1/2*ln((Root
Of(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^7+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^
3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^7+6*(x^4+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-
2)+4*_Z^2)*x^2+RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+4*RootOf(RootOf(_Z^3-2
)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+6*(x^4+1)^(1/3)*RootOf(_Z^3-2)^2*x-4*RootOf(_Z^3-2)-16*
RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(x^7+x^3-4))*RootOf(_Z^3-2)-ln((RootOf(RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^7+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf
(_Z^3-2)^2*x^7+6*(x^4+1)^(2/3)*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2+RootOf
(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-
2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+6*(x^4+1)^(1/3)*RootOf(_Z^3-2)^2*x-4*RootOf(_Z^3-2)-16*RootOf(RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(x^7+x^3-4))*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (7 \, x^{4} + 3\right )} x}{{\left (x^{7} + x^{3} - 4\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(7*x^4+3)/(x^4+1)^(1/3)/(x^7+x^3-4),x, algorithm="maxima")

[Out]

integrate((7*x^4 + 3)*x/((x^7 + x^3 - 4)*(x^4 + 1)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x\,\left (7\,x^4+3\right )}{{\left (x^4+1\right )}^{1/3}\,\left (x^7+x^3-4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(7*x^4 + 3))/((x^4 + 1)^(1/3)*(x^3 + x^7 - 4)),x)

[Out]

int((x*(7*x^4 + 3))/((x^4 + 1)^(1/3)*(x^3 + x^7 - 4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (7 x^{4} + 3\right )}{\sqrt [3]{x^{4} + 1} \left (x^{7} + x^{3} - 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(7*x**4+3)/(x**4+1)**(1/3)/(x**7+x**3-4),x)

[Out]

Integral(x*(7*x**4 + 3)/((x**4 + 1)**(1/3)*(x**7 + x**3 - 4)), x)

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