Optimal. Leaf size=118 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{q} (-1+x)}{\sqrt {3} \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}}\right )}{2 \sqrt [3]{q}}+\frac {\log (1-x)}{4 \sqrt [3]{q}}+\frac {\log (x)}{2 \sqrt [3]{q}}-\frac {3 \log \left (-\sqrt [3]{q} (-1+x)+\sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}\right )}{4 \sqrt [3]{q}} \]
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Rubi [F]
time = 10.22, 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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Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \, dx &=\text {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 ) \text {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 [A]
time = 2.23, size = 189, normalized size = 1.60 \begin {gather*} \frac {\sqrt [3]{-1+x} \sqrt [3]{q-2 q x+x^2} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{q} (-1+x)^{2/3}}{\sqrt [3]{q} (-1+x)^{2/3}+2 \sqrt [3]{q-2 q x+x^2}}\right )-2 \log \left (-\sqrt [3]{q} (-1+x)^{2/3}+\sqrt [3]{q-2 q x+x^2}\right )+\log \left (q^{2/3} (-1+x)^{4/3}+\sqrt [3]{q} (-1+x)^{2/3} \sqrt [3]{q-2 q x+x^2}+\left (q-2 q x+x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{q} \sqrt [3]{(-1+x) \left (q-2 q x+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (\left (-1+x \right ) \left (-2 q x +x^{2}+q \right )\right )^{\frac {1}{3}}}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 661 vs.
\(2 (100) = 200\).
time = 10.68, size = 1496, normalized size = 12.68 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt [3]{\left (x - 1\right ) \left (- 2 q x + q + x^{2}\right )}}\, dx \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*} \text {could not integrate} \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 {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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