Optimal. Leaf size=87 \[ 2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x^4+2 x^2+x}}{2 x^3-2 x-1}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-2 x^4+2 x^2+x}}{2 x^3-2 x-1}\right ) \]
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Rubi [F] time = 3.38, 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 {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(3+4 x) \sqrt {x+2 x^2-2 x^4}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx &=\frac {\sqrt {x+2 x^2-2 x^4} \int \frac {\sqrt {x} (3+4 x) \sqrt {1+2 x-2 x^3}}{(1+2 x) \left (1+2 x+x^3\right )} \, dx}{\sqrt {x} \sqrt {1+2 x-2 x^3}}\\ &=\frac {\left (2 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3+4 x^2\right ) \sqrt {1+2 x^2-2 x^6}}{\left (1+2 x^2\right ) \left (1+2 x^2+x^6\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}\\ &=\frac {\left (2 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {4 \sqrt {1+2 x^2-2 x^6}}{1+2 x^2}+\frac {\left (-4+3 x^2-2 x^4\right ) \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}\\ &=\frac {\left (2 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-4+3 x^2-2 x^4\right ) \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}+\frac {\left (8 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^2-2 x^6}}{1+2 x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}\\ &=\frac {\left (2 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6}+\frac {3 x^2 \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6}-\frac {2 x^4 \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}+\frac {\left (8 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {1+2 x^2-2 x^6}}{2 \left (i-\sqrt {2} x\right )}+\frac {i \sqrt {1+2 x^2-2 x^6}}{2 \left (i+\sqrt {2} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}\\ &=\frac {\left (4 i \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^2-2 x^6}}{i-\sqrt {2} x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}+\frac {\left (4 i \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^2-2 x^6}}{i+\sqrt {2} x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}-\frac {\left (4 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}+\frac {\left (6 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}-\frac {\left (8 \sqrt {x+2 x^2-2 x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^2-2 x^6}}{1+2 x^2+x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x-2 x^3}}\\ \end {align*}
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Mathematica [C] time = 6.70, size = 18077, normalized size = 207.78 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.42, size = 87, normalized size = 1.00 \begin {gather*} 2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x^4+2 x^2+x}}{2 x^3-2 x-1}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-2 x^4+2 x^2+x}}{2 x^3-2 x-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 168, normalized size = 1.93 \begin {gather*} \frac {2}{5} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x^{3} - 4 \, x^{2} - x + 1\right )}}{16 \, x^{5} - 16 \, x^{4} - 12 \, x^{3} + 8 \, x^{2} + 4 \, x - 1}\right ) - \frac {1}{5} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x^{2} + 5 \, x + 2\right )}}{32 \, x^{5} + 80 \, x^{4} + 84 \, x^{3} + 40 \, x^{2} + 6 \, x - 1}\right ) - \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-2 \, x^{4} + 2 \, x^{2} + x} x}{5 \, x^{3} - 2 \, x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x + 3\right )}}{{\left (x^{3} + 2 \, x + 1\right )} {\left (2 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.61, size = 6278, normalized size = 72.16 \begin {gather*} \text {output too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-2 \, x^{4} + 2 \, x^{2} + x} {\left (4 \, x + 3\right )}}{{\left (x^{3} + 2 \, x + 1\right )} {\left (2 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (4\,x+3\right )\,\sqrt {-2\,x^4+2\,x^2+x}}{\left (2\,x+1\right )\,\left (x^3+2\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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