Optimal. Leaf size=162 \[ -\frac {\log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\sqrt [3]{b} x^3 \sqrt [3]{k x^3+(-k-1) x^2+x}+x^6\right )}{2 \sqrt [3]{b}}+\frac {\log \left (x^3-\sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{\sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^3}{2 \sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}+x^3}\right )}{\sqrt [3]{b}} \]
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Rubi [F] time = 27.47, 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 {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx &=\frac {\left ((1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {x^{13/3} \left (8-7 (1+k) x+6 k x^2\right )}{(1-x)^{2/3} (1-k x)^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{15} \left (8-7 (1+k) x^3+6 k x^6\right )}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (3 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {7 (1+k) x^{18}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (b-b (1+k) x^3+b k x^6-x^{24}\right )}+\frac {8 x^{15}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )}+\frac {6 k x^{21}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\\ &=\frac {\left (24 (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{15}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}+\frac {\left (18 k (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{21}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (-b+b (1+k) x^3-b k x^6+x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}+\frac {\left (21 (1+k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{18}}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (b-b (1+k) x^3+b k x^6-x^{24}\right )} \, dx,x,\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 8.30, size = 162, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^3}{x^3+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (x^3-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (x^6+\sqrt [3]{b} x^3 \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\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 (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{5}}{{\left (x^{8} - b k x^{2} + b {\left (k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (8-7 \left (1+k \right ) x +6 k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (-b +b \left (1+k \right ) x -b k \,x^{2}+x^{8}\right )}\, dx\]
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 (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{5}}{{\left (x^{8} - b k x^{2} + b {\left (k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}}}\,{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 {x^5\,\left (6\,k\,x^2-7\,x\,\left (k+1\right )+8\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (-x^8+b\,k\,x^2-b\,\left (k+1\right )\,x+b\right )} \,d x \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 {x^{5} \left (6 k x^{2} - 7 k x - 7 x + 8\right )}{\left (x \left (x - 1\right ) \left (k x - 1\right )\right )^{\frac {2}{3}} \left (- b k x^{2} + b k x + b x - b + x^{8}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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