3.12.83 \(\int \frac {x^2 (4+x^3)}{(1+x^3)^{3/4} (1+x^3+x^4)} \, dx\) [1183]

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

[Out]

-2^(1/2)*arctan((-1/2*x^2*2^(1/2)+1/2*(x^3+1)^(1/2)*2^(1/2))/x/(x^3+1)^(1/4))-2^(1/2)*arctanh(2^(1/2)*x*(x^3+1
)^(1/4)/(x^2+(x^3+1)^(1/2)))

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Rubi [F]
time = 0.70, 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 [A]
time = 2.00, size = 74, normalized size = 0.85 \begin {gather*} -\sqrt {2} \left (\text {ArcTan}\left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )+\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.64, size = 208, normalized size = 2.39

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{3}+1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )\)	\(208\)

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-RootOf(_Z^4+1)*x^3-2*(x^3+1)^(3/4)*x-RootOf(_Z^4+1))/(x^4+x^3+1))-RootOf(_Z^4+1)*ln((RootOf(_Z^4+1)^3*x^4
-RootOf(_Z^4+1)^3*x^3+2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*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*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (70) = 140\).
time = 0.38, 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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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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