Optimal. Leaf size=230 \[ -\frac {\log \left (\left (k x^3+(-k-1) x^2+x\right )^{2/3} \left (b^{2/3} x-b^{2/3} k x^2\right )+b k^2 x^4-2 b k x^3+\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{4/3}+b x^2\right )}{2 \sqrt [3]{b}}+\frac {\log \left (\sqrt {b} k x^2+\sqrt [6]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}-\sqrt {b} x\right )}{\sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}}{-2 \sqrt [3]{b} k x^2+2 \sqrt [3]{b} x+\left (k x^3+(-k-1) x^2+x\right )^{2/3}}\right )}{\sqrt [3]{b}} \]
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Rubi [F] time = 4.18, 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+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {-1+2 k x+(1-2 k) x^2}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-x} (-1+(-1+2 k) x)}{x^{2/3} (1-k x)^{2/3} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \left (\frac {\left (-1-\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k\right ) \sqrt [3]{1-x}}{x^{2/3} (1-k x)^{2/3} \left (-2-b-\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right )}+\frac {\left (-1+\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k\right ) \sqrt [3]{1-x}}{x^{2/3} (1-k x)^{2/3} \left (-2-b+\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right )}\right ) \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (\left (-1-\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-x}}{x^{2/3} (1-k x)^{2/3} \left (-2-b-\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right )} \, dx}{((1-x) x (1-k x))^{2/3}}+\frac {\left (\left (-1+\frac {\sqrt {4+b-4 k}}{\sqrt {b}}+2 k\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{1-x}}{x^{2/3} (1-k x)^{2/3} \left (-2-b+\sqrt {b} \sqrt {4+b-4 k}+2 (1+b k) x\right )} \, dx}{((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 8.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left (1-(2+b) x+(1+b k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.87, size = 230, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{2 \sqrt [3]{b} x-2 \sqrt [3]{b} k x^2+\left (x+(-1-k) x^2+k x^3\right )^{2/3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (-\sqrt {b} x+\sqrt {b} k x^2+\sqrt [6]{b} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{\sqrt [3]{b}}-\frac {\log \left (b x^2-2 b k x^3+b k^2 x^4+\left (b^{2/3} x-b^{2/3} k x^2\right ) \left (x+(-1-k) x^2+k x^3\right )^{2/3}+\sqrt [3]{b} \left (x+(-1-k) x^2+k x^3\right )^{4/3}\right )}{2 \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, k - 1\right )} x^{2} - 2 \, k x + 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (b k + 1\right )} x^{2} - {\left (b + 2\right )} x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {-1+2 k x +\left (1-2 k \right ) x^{2}}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (1-\left (2+b \right ) x +\left (b k +1\right ) x^{2}\right )}\, dx\]
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*} -\int \frac {{\left (2 \, k - 1\right )} x^{2} - 2 \, k x + 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left ({\left (b k + 1\right )} x^{2} - {\left (b + 2\right )} 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 = -0.00 \begin {gather*} \int -\frac {\left (2\,k-1\right )\,x^2-2\,k\,x+1}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (\left (b\,k+1\right )\,x^2+\left (-b-2\right )\,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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