3.12.83 \(\int \frac {x^2 (4+x^3)}{(1+x^3)^{3/4} (1+x^3+x^4)} \, dx\)

Optimal. Leaf size=87 \[ -\sqrt {2} \tan ^{-1}\left (\frac {\frac {\sqrt {x^3+1}}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{x^3+1}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+1}}{\sqrt {x^3+1}+x^2}\right ) \]

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Rubi [F]  time = 1.06, 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^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]

[Out]

-(x*Hypergeometric2F1[1/3, 3/4, 4/3, -x^3]) + (x^2*Hypergeometric2F1[2/3, 3/4, 5/3, -x^3])/2 + Defer[Int][1/((
1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] - Defer[Int][x/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] + 4*Defer[Int][x^2/((
1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] + Defer[Int][x^3/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x]

Rubi steps

\begin {align*} \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx &=\int \left (-\frac {1}{\left (1+x^3\right )^{3/4}}+\frac {x}{\left (1+x^3\right )^{3/4}}+\frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-\int \frac {1}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx\\ &=-x \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-x^3\right )+\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+\int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}-\frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-x \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-x^3\right )+\frac {1}{2} x^2 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};-x^3\right )+4 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\int \frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-\int \frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]

[Out]

Integrate[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x]

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IntegrateAlgebraic [A]  time = 2.55, size = 87, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tan ^{-1}\left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]

[Out]

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

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fricas [B]  time = 0.54, size = 189, normalized size = 2.17 \begin {gather*} 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} + \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}}{x^{2}}} - x - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}}{x^{2}}} + x - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} + \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}\right )}}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} - \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} + 1}\right )}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x, algorithm="fricas")

[Out]

2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 + sqrt(2)*(x^3 + 1)^(1/4)*x + sqrt(x^3 + 1))/x^2) - x - sqrt(2)*(x^3 + 1
)^(1/4))/x) + 2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 - sqrt(2)*(x^3 + 1)^(1/4)*x + sqrt(x^3 + 1))/x^2) + x - sq
rt(2)*(x^3 + 1)^(1/4))/x) - 1/2*sqrt(2)*log(4*(x^2 + sqrt(2)*(x^3 + 1)^(1/4)*x + sqrt(x^3 + 1))/x^2) + 1/2*sqr
t(2)*log(4*(x^2 - sqrt(2)*(x^3 + 1)^(1/4)*x + sqrt(x^3 + 1))/x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 4\right )} x^{2}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x, algorithm="giac")

[Out]

integrate((x^3 + 4)*x^2/((x^4 + x^3 + 1)*(x^3 + 1)^(3/4)), x)

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maple [C]  time = 1.67, size = 205, normalized size = 2.36

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-2 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^4+1)^3*ln((-2*(x^3+1)^(1/2)*RootOf(_Z^4+1)^3*x^2-2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*x^3-RootOf(_Z^4+1)
*x^4+2*(x^3+1)^(3/4)*x+RootOf(_Z^4+1)*x^3+RootOf(_Z^4+1))/(x^4+x^3+1))-RootOf(_Z^4+1)*ln((RootOf(_Z^4+1)^3*x^4
+2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*x^3-RootOf(_Z^4+1)^3*x^3+2*(x^3+1)^(1/2)*RootOf(_Z^4+1)*x^2+2*(x^3+1)^(3/4)*
x-RootOf(_Z^4+1)^3)/(x^4+x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 4\right )} x^{2}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x, algorithm="maxima")

[Out]

integrate((x^3 + 4)*x^2/((x^4 + x^3 + 1)*(x^3 + 1)^(3/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (x^3+4\right )}{{\left (x^3+1\right )}^{3/4}\,\left (x^4+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(x^3 + 4))/((x^3 + 1)^(3/4)*(x^3 + x^4 + 1)),x)

[Out]

int((x^2*(x^3 + 4))/((x^3 + 1)^(3/4)*(x^3 + x^4 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**3+4)/(x**3+1)**(3/4)/(x**4+x**3+1),x)

[Out]

Timed out

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