3.21.16 \(\int \frac {(-3+2 x^4) (1+2 x^4)^{2/3}}{x^3 (2-x^3+4 x^4)} \, dx\) [2016]

Optimal. Leaf size=143 \[ \frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+2 x^4}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+2 x^4}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+2 x^4}+2^{2/3} \left (1+2 x^4\right )^{2/3}\right )}{4\ 2^{2/3}} \]

[Out]

3/4*(2*x^4+1)^(2/3)/x^2-1/4*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(1/3)*(2*x^4+1)^(1/3)))*2^(1/3)+1/4*ln(-x+2^(1/3)*
(2*x^4+1)^(1/3))*2^(1/3)-1/8*ln(x^2+2^(1/3)*x*(2*x^4+1)^(1/3)+2^(2/3)*(2*x^4+1)^(2/3))*2^(1/3)

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Rubi [F]
time = 0.79, 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 {\left (-3+2 x^4\right ) \left (1+2 x^4\right )^{2/3}}{x^3 \left (2-x^3+4 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(1 + 2*x^4)^(2/3))/(4*x^2) + (6*x^2)/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3)) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 -
(1 + 2*x^4)^(1/3))*Sqrt[(1 + (1 + 2*x^4)^(1/3) + (1 + 2*x^4)^(2/3))/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3))^2]*Ellip
ticE[ArcSin[(1 + Sqrt[3] - (1 + 2*x^4)^(1/3))/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3))], -7 + 4*Sqrt[3]])/(2*x^2*Sqrt
[-((1 - (1 + 2*x^4)^(1/3))/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3))^2)]) + (Sqrt[2]*3^(3/4)*(1 - (1 + 2*x^4)^(1/3))*S
qrt[(1 + (1 + 2*x^4)^(1/3) + (1 + 2*x^4)^(2/3))/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqr
t[3] - (1 + 2*x^4)^(1/3))/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3))], -7 + 4*Sqrt[3]])/(x^2*Sqrt[-((1 - (1 + 2*x^4)^(1
/3))/(1 - Sqrt[3] - (1 + 2*x^4)^(1/3))^2)]) - (3*Defer[Int][(1 + 2*x^4)^(2/3)/(2 - x^3 + 4*x^4), x])/2 + 8*Def
er[Int][(x*(1 + 2*x^4)^(2/3))/(2 - x^3 + 4*x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (-3+2 x^4\right ) \left (1+2 x^4\right )^{2/3}}{x^3 \left (2-x^3+4 x^4\right )} \, dx &=\int \left (-\frac {3 \left (1+2 x^4\right )^{2/3}}{2 x^3}+\frac {(-3+16 x) \left (1+2 x^4\right )^{2/3}}{2 \left (2-x^3+4 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {(-3+16 x) \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3 \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4}+\frac {16 x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4}\right ) \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {\left (1+2 x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx-2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+2 x^2}} \, dx,x,x^2\right )+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx-\frac {\left (3 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+2 x^4}\right )}{\sqrt {2} x^2}\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+2 x^4}\right )}{\sqrt {2} x^2}-\frac {\left (3 \sqrt {2+\sqrt {3}} \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+2 x^4}\right )}{x^2}\\ &=\frac {3 \left (1+2 x^4\right )^{2/3}}{4 x^2}+\frac {6 x^2}{1-\sqrt {3}-\sqrt [3]{1+2 x^4}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1+2 x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+2 x^4}+\left (1+2 x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+2 x^4}}{1-\sqrt {3}-\sqrt [3]{1+2 x^4}}\right )|-7+4 \sqrt {3}\right )}{2 x^2 \sqrt {-\frac {1-\sqrt [3]{1+2 x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1+2 x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+2 x^4}+\left (1+2 x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1+2 x^4}}{1-\sqrt {3}-\sqrt [3]{1+2 x^4}}\right )|-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+2 x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+2 x^4}\right )^2}}}-\frac {3}{2} \int \frac {\left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx+8 \int \frac {x \left (1+2 x^4\right )^{2/3}}{2-x^3+4 x^4} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.97, size = 117, normalized size = 0.82 \begin {gather*} \frac {1}{8} \left (\frac {6 \left (1+2 x^4\right )^{2/3}}{x^2}-2 \sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+4 x^4}}\right )+2 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2+4 x^4}\right )-\sqrt [3]{2} \log \left (x^2+x \sqrt [3]{2+4 x^4}+\left (2+4 x^4\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*(1 + 2*x^4)^(2/3))/x^2 - 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + 4*x^4)^(1/3))] + 2*2^(1/3)*Log[-
x + (2 + 4*x^4)^(1/3)] - 2^(1/3)*Log[x^2 + x*(2 + 4*x^4)^(1/3) + (2 + 4*x^4)^(2/3)])/8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 85.70, size = 813, normalized size = 5.69

method	result	size
risch	\(\text {Expression too large to display}\)	\(813\)
trager	\(\text {Expression too large to display}\)	\(1001\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(2*x^4+1)^(2/3)/x^2-1/2*ln(-(2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+
2*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(2*x^4+1)^(2/3)*x-(2*x^4+1)^(1/3)*RootO
f(_Z^3-2)^2*x^2-4*RootOf(_Z^3-2)*(2*x^4+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2+8*Roo
tOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^4+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3
+4*(2*x^4+1)^(2/3)*x+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(4*x^4-x^3+2))*RootOf(RootOf(_Z^3-
2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-1/4*ln(-(2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-
2)^2*x^3+2*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(2*x^4+1)^(2/3)*x-(2*x^4+1)^(1
/3)*RootOf(_Z^3-2)^2*x^2-4*RootOf(_Z^3-2)*(2*x^4+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*
x^2+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^4+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*
_Z^2)*x^3+4*(2*x^4+1)^(2/3)*x+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(4*x^4-x^3+2))*RootOf(_Z^
3-2)+1/4*RootOf(_Z^3-2)*ln((RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+RootOf(
_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*(2*x^4+1)^(2/3)*x+(2*x^4+1)^(1/3)*RootOf(_Z^3-2)
^2*x^2+RootOf(_Z^3-2)*(2*x^4+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-4*RootOf(RootOf(
_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^4-(2*x^4+1)^(2/3)*x-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z
^2))/(4*x^4-x^3+2))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (107) = 214\).
time = 39.24, size = 404, normalized size = 2.83 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (8 \, x^{9} + 2 \, x^{8} - x^{7} + 8 \, x^{5} + x^{4} + 2 \, x\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (64 \, x^{12} + 240 \, x^{11} + 48 \, x^{10} - x^{9} + 96 \, x^{8} + 240 \, x^{7} + 24 \, x^{6} + 48 \, x^{4} + 60 \, x^{3} + 8\right )} + 12 \, {\left (16 \, x^{10} + 28 \, x^{9} + x^{8} + 16 \, x^{6} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (64 \, x^{12} - 48 \, x^{11} - 96 \, x^{10} - x^{9} + 96 \, x^{8} - 48 \, x^{7} - 48 \, x^{6} + 48 \, x^{4} - 12 \, x^{3} + 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (4 \, x^{4} - x^{3} + 2\right )} - 12 \, {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}} x}{4 \, x^{4} - x^{3} + 2}\right ) + 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{5} + x^{4} + x\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (16 \, x^{8} + 28 \, x^{7} + x^{6} + 16 \, x^{4} + 14 \, x^{3} + 4\right )} + 6 \, {\left (8 \, x^{6} + x^{5} + 4 \, x^{2}\right )} {\left (2 \, x^{4} + 1\right )}^{\frac {1}{3}}}{16 \, x^{8} - 8 \, x^{7} + x^{6} + 16 \, x^{4} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (2 \, x^{4} + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(4*4^(1/6)*sqrt(3)*x^2*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(8*x^9 + 2*x^8 - x^7 + 8*x^5 + x^4 + 2*x)*
(2*x^4 + 1)^(2/3) + 4^(1/3)*(64*x^12 + 240*x^11 + 48*x^10 - x^9 + 96*x^8 + 240*x^7 + 24*x^6 + 48*x^4 + 60*x^3
+ 8) + 12*(16*x^10 + 28*x^9 + x^8 + 16*x^6 + 14*x^5 + 4*x^2)*(2*x^4 + 1)^(1/3))/(64*x^12 - 48*x^11 - 96*x^10 -
x^9 + 96*x^8 - 48*x^7 - 48*x^6 + 48*x^4 - 12*x^3 + 8)) - 2*4^(2/3)*x^2*log((6*4^(1/3)*(2*x^4 + 1)^(1/3)*x^2 +
4^(2/3)*(4*x^4 - x^3 + 2) - 12*(2*x^4 + 1)^(2/3)*x)/(4*x^4 - x^3 + 2)) + 4^(2/3)*x^2*log((6*4^(2/3)*(2*x^5 +
x^4 + x)*(2*x^4 + 1)^(2/3) + 4^(1/3)*(16*x^8 + 28*x^7 + x^6 + 16*x^4 + 14*x^3 + 4) + 6*(8*x^6 + x^5 + 4*x^2)*(
2*x^4 + 1)^(1/3))/(16*x^8 - 8*x^7 + x^6 + 16*x^4 - 4*x^3 + 4)) - 36*(2*x^4 + 1)^(2/3))/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((2*x**4-3)*(2*x**4+1)**(2/3)/x**3/(4*x**4-x**3+2),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((2*x^4-3)*(2*x^4+1)^(2/3)/x^3/(4*x^4-x^3+2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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