Optimal. Leaf size=223 \[ -\frac {\log \left (q^{2/3} x^2+2 q^{2/3} x+q^{2/3}+\left ((2 q+1) x^2+3 q x+q+x^3\right )^{2/3}+\left (\sqrt [3]{q} x+\sqrt [3]{q}\right ) \sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}\right )}{4 \sqrt [3]{q}}+\frac {\log \left (\sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}-\sqrt [3]{q} x-\sqrt [3]{q}\right )}{2 \sqrt [3]{q}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}}{\sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}+2 \sqrt [3]{q} x+2 \sqrt [3]{q}}\right )}{2 \sqrt [3]{q}} \]
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Rubi [F] time = 22.00, 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 {1}{x \sqrt [3]{(1+x) \left (q+2 q x+x^2\right )}} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {1}{x \sqrt [3]{(1+x) \left (q+2 q x+x^2\right )}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (-1-2 q)+x\right ) \sqrt [3]{\frac {2}{27} (1-q)^2 (1+8 q)-\frac {1}{3} (1-4 q) (1-q) x+x^3}} \, dx,x,\frac {1}{3} (1+2 q)+x\right )\\ &=\frac {\left (\sqrt [3]{1+2 q+\frac {1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {-(-1+q)^3 q}\right )^{2/3}}{\sqrt [3]{1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {-(-1+q)^3 q}}}+3 x} \sqrt [3]{-1+5 q-4 q^2+\frac {(1-4 q)^2 (1-q)^2}{\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}}+\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}+9 \left (\frac {1}{3} (1+2 q)+x\right )^2-\frac {\left (1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}\right ) (1+2 q+3 x)}{\sqrt [3]{1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{3} (-1-2 q)+x\right ) \sqrt [3]{\frac {1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}}{3 \sqrt [3]{1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}}}+x} \sqrt [3]{\frac {1}{9} \left (-1+5 q-4 q^2+\frac {(1-4 q)^2 (1-q)^2}{\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}}+\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}\right )-\frac {\left (1-5 q+4 q^2+\left (1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}\right )^{2/3}\right ) x}{3 \sqrt [3]{1+6 q-15 q^2+8 q^3-3 \sqrt {3} \sqrt {(1-q)^3 q}}}+x^2}} \, dx,x,\frac {1}{3} (1+2 q)+x\right )}{3 \sqrt [3]{q+3 q x+(1+2 q) x^2+x^3}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 55, normalized size = 0.25 \begin {gather*} -\frac {3 \left ((x+1) \left (2 q x+q+x^2\right )\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x^2+2 q x+q}{q (x+1)^2}\right )}{4 q (x+1)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.53, size = 223, normalized size = 1.00 \begin {gather*} -\frac {\log \left (q^{2/3} x^2+2 q^{2/3} x+q^{2/3}+\left ((2 q+1) x^2+3 q x+q+x^3\right )^{2/3}+\left (\sqrt [3]{q} x+\sqrt [3]{q}\right ) \sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}\right )}{4 \sqrt [3]{q}}+\frac {\log \left (\sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}-\sqrt [3]{q} x-\sqrt [3]{q}\right )}{2 \sqrt [3]{q}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}}{\sqrt [3]{(2 q+1) x^2+3 q x+q+x^3}+2 \sqrt [3]{q} x+2 \sqrt [3]{q}}\right )}{2 \sqrt [3]{q}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 46.43, size = 1383, normalized size = 6.20
result too large to display
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 {1}{\left ({\left (2 \, q x + x^{2} + q\right )} {\left (x + 1\right )}\right )^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (\left (1+x \right ) \left (2 q x +x^{2}+q \right )\right )^{\frac {1}{3}}}\, dx \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 {1}{\left ({\left (2 \, q x + x^{2} + q\right )} {\left (x + 1\right )}\right )^{\frac {1}{3}} x}\,{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 = -0.00 \begin {gather*} \int \frac {1}{x\,{\left (\left (x+1\right )\,\left (x^2+2\,q\,x+q\right )\right )}^{1/3}} \,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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