Optimal. Leaf size=209 \[ \frac {(-a-b) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{2 \sqrt [3]{b}}+\frac {(a+b) \log \left (\sqrt [3]{k x^3+(-k-1) x^2+x}-\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {\left (-\sqrt {3} a-\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{\sqrt [3]{b}}+\frac {3 \left (k x^3+(-k-1) x^2+x\right )^{2/3}}{2 x^2} \]
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Rubi [F] time = 23.85, 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 {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x^2 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-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 {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x^2 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \left (-\frac {2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )}{(b-k)^2 \sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}}-\frac {(1+k) (a+k)}{(b-k) \sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}}-\frac {(a+b) \left (1+2 b+k^2\right )-(a+b) (1+k) \left (1+3 b-k+k^2\right ) x}{(b-k)^2 \sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (1+(-1-k) x+(-b+k) x^2\right )}\right ) \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {(a+b) \left (1+2 b+k^2\right )-(a+b) (1+k) \left (1+3 b-k+k^2\right ) x}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (1+(-1-k) x+(-b+k) x^2\right )} \, dx}{(b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left ((1+k) (a+k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{(b-k) \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}} \, dx}{(b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {3 \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x) (1-k x)}{4 (b-k)^2 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) (a+k) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{4/3} (1-k x) \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};\frac {1-x}{1-k x}\right )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \left (\frac {-\left ((a+b) (1+k) \left (1+3 b-k+k^2\right )\right )-\frac {(a+b) \left (1+5 b+4 b^2-k+2 b k+5 b k^2-k^3+k^4\right )}{\sqrt {1+4 b-2 k+k^2}}}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}+\frac {-\left ((a+b) (1+k) \left (1+3 b-k+k^2\right )\right )+\frac {(a+b) \left (1+5 b+4 b^2-k+2 b k+5 b k^2-k^3+k^4\right )}{\sqrt {1+4 b-2 k+k^2}}}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}\right ) \, dx}{(b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left ((1+k) \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{2 (b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {3 \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x) (1-k x)}{4 (b-k)^2 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) (a+k) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{4/3} (1-k x) \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};\frac {1-x}{1-k x}\right )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (2 k^2-a \left (1+2 b+k^2\right )-b \left (1+4 k+k^2\right )\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{4/3} (1-k x) \, _2F_1\left (\frac {2}{3},\frac {4}{3};\frac {5}{3};\frac {1-x}{1-k x}\right )}{4 (1-k) (b-k)^2 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\left (-\left ((a+b) (1+k) \left (1+3 b-k+k^2\right )\right )+\frac {(a+b) \left (1+5 b+4 b^2-k+2 b k+5 b k^2-k^3+k^4\right )}{\sqrt {1+4 b-2 k+k^2}}\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left ((a+b) \left (1+k^3+3 b (1+k)+\frac {4 b^2+(1-k)^2 \left (1+k+k^2\right )+b \left (5+2 k+5 k^2\right )}{\sqrt {4 b+(-1+k)^2}}\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^2 \sqrt [3]{(1-x) x (1-k x)}}\\ \end {align*}
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Mathematica [F] time = 12.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{x^2 \sqrt [3]{(1-x) x (1-k x)} \left (1-(1+k) x+(-b+k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.39, size = 209, normalized size = 1.00 \begin {gather*} \frac {3 \left (x+(-1-k) x^2+k x^3\right )^{2/3}}{2 x^2}+\frac {\left (-\sqrt {3} a-\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {(a+b) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {(-a-b) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{x+(-1-k) x^2+k x^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 ({\left (a + k\right )} x^{2} - {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{2}}\,{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 \[\int \frac {\left (-2+\left (1+k \right ) x \right ) \left (1-\left (1+k \right ) x +\left (a +k \right ) x^{2}\right )}{x^{2} \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-\left (1+k \right ) x +\left (-b +k \right ) x^{2}\right )}\, dx\]
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 ({\left (a + k\right )} x^{2} - {\left (k + 1\right )} x + 1\right )} {\left ({\left (k + 1\right )} x - 2\right )}}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (b - k\right )} x^{2} + {\left (k + 1\right )} x - 1\right )} x^{2}}\,{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 (x\,\left (k+1\right )-2\right )\,\left (\left (a+k\right )\,x^2+\left (-k-1\right )\,x+1\right )}{x^2\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (b-k\right )\,x^2+\left (k+1\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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