Optimal. Leaf size=217 \[ -\frac {\log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{4/3}+\left (k x^3+(-k-1) x^2+x\right )^{2/3} \left (\sqrt [3]{b} x-\sqrt [3]{b} k x^2\right )+k^2 x^4-2 k x^3+x^2\right )}{2 b^{2/3}}+\frac {\log \left (\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+k x^2-x\right )}{b^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}}{\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}-2 k x^2+2 x}\right )}{b^{2/3}} \]
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Rubi [F] time = 3.56, 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 (b-(1+2 b) x+(b+k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {-1+2 k x+(1-2 k) x^2}{((1-x) x (1-k x))^{2/3} \left (b-(1+2 b) x+(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 (b-(1+2 b) x+(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 (b-(1+2 b) x+(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+2 k-\sqrt {1+4 b-4 b k}\right ) \sqrt [3]{1-x}}{x^{2/3} (1-k x)^{2/3} \left (-1-2 b-\sqrt {1+4 b-4 b k}+2 (b+k) x\right )}+\frac {\left (-1+2 k+\sqrt {1+4 b-4 b k}\right ) \sqrt [3]{1-x}}{x^{2/3} (1-k x)^{2/3} \left (-1-2 b+\sqrt {1+4 b-4 b k}+2 (b+k) x\right )}\right ) \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (\left (-1+2 k-\sqrt {1+4 b-4 b 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 (-1-2 b-\sqrt {1+4 b-4 b k}+2 (b+k) x\right )} \, dx}{((1-x) x (1-k x))^{2/3}}+\frac {\left (\left (-1+2 k+\sqrt {1+4 b-4 b 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 (-1-2 b+\sqrt {1+4 b-4 b k}+2 (b+k) x\right )} \, dx}{((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 8.40, 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 (b-(1+2 b) x+(b+k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.78, size = 217, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{4/3}+\left (k x^3+(-k-1) x^2+x\right )^{2/3} \left (\sqrt [3]{b} x-\sqrt [3]{b} k x^2\right )+k^2 x^4-2 k x^3+x^2\right )}{2 b^{2/3}}+\frac {\log \left (\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+k x^2-x\right )}{b^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}}{\sqrt [3]{b} \left (k x^3+(-k-1) x^2+x\right )^{2/3}-2 k x^2+2 x}\right )}{b^{2/3}} \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\right )} x^{2} - {\left (2 \, b + 1\right )} x + b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \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 (b -\left (1+2 b \right ) x +\left (b +k \right ) x^{2}\right )}\, 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 {{\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\right )} x^{2} - {\left (2 \, b + 1\right )} x + b\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 (\left (b+k\right )\,x^2+\left (-2\,b-1\right )\,x+b\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}} \,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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