Optimal. Leaf size=203 \[ \frac {(-a-b) \log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\sqrt [3]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+x^2\right )}{2 \sqrt [3]{b}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{\sqrt [3]{b}}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}+x}\right )}{\sqrt [3]{b}}+\frac {3 x^2}{2 \left (k x^3-k x^2-x^2+x\right )^{2/3}} \]
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Rubi [F] time = 54.85, 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 {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx &=\int \frac {x (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) 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 {\sqrt [3]{x} (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{2/3} (-1+x) (1-k x)^{2/3} (-1+k x) \left (b-b (1+k) 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]{x} (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{5/3} (1-k x)^{2/3} (-1+k x) \left (b-b (1+k) 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]{x} (-2+(1+k) x) \left (a-a (1+k) x+(1+a k) x^2\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (b-b (1+k) 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 (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) \sqrt [3]{x}}{(1-b k)^2 (1-x)^{5/3} (1-k x)^{5/3}}-\frac {(1+k) (1+a k) x^{4/3}}{(1-b k) (1-x)^{5/3} (1-k x)^{5/3}}-\frac {\sqrt [3]{x} \left ((a+b) \left (2+b+b k^2\right )-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right ) x\right )}{(-1+b k)^2 (1-x)^{5/3} (1-k x)^{5/3} \left (b-b (1+k) x+(-1+b k) x^2\right )}\right ) \, dx}{((1-x) x (1-k x))^{2/3}}\\ &=-\frac {\left ((1+k) (1+a k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {x^{4/3}}{(1-x)^{5/3} (1-k x)^{5/3}} \, dx}{(1-b k) ((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]{x} \left ((a+b) \left (2+b+b k^2\right )-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right ) x\right )}{(1-x)^{5/3} (1-k x)^{5/3} \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left (\left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3}} \, dx}{(1-b k)^2 ((1-x) x (1-k x))^{2/3}}\\ &=-\frac {3 (1+k) (1+a k) x^2}{2 (1-k) (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x^2}{2 (1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left (2 (1+k) (1+a k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{2/3} (1-k x)^{5/3}} \, dx}{(1-k) (1-b k) ((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 (-\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )\right ) \sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)-\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )}+\frac {\left (\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )\right ) \sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)+\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )}\right ) \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {\left ((1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{2/3} (1-k x)^{5/3}} \, dx}{(1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}\\ &=-\frac {3 (1+k) (1+a k) (1-x) x}{(1-k)^2 (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) x}{2 (1-k)^2 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {3 (1+k) (1+a k) x^2}{2 (1-k) (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x^2}{2 (1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left ((1+k) (1+a k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3}} \, dx}{(1-k)^2 (1-b k) ((1-x) x (1-k x))^{2/3}}-\frac {\left ((1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {1}{(1-x)^{2/3} x^{2/3} (1-k x)^{2/3}} \, dx}{2 (1-k)^2 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {\left (\left (\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)+\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )} \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left ((a+b) \left (3+b+3 k+b k^3+\frac {4+b^2 (1-k)^2 \left (1+k+k^2\right )+b \left (5+2 k+5 k^2\right )}{\sqrt {b} \sqrt {4+b (1-k)^2}}\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)-\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )} \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}\\ &=-\frac {3 (1+k) (1+a k) (1-x) x}{(1-k)^2 (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) x}{2 (1-k)^2 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {3 (1+k) (1+a k) x^2}{2 (1-k) (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) x^2}{2 (1-k) (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {3 (1+k) (1+a k) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {1-x}{1-k x}\right )}{(1-k)^3 (1-b k) ((1-x) x (1-k x))^{2/3}}+\frac {3 (1+k) \left (2+a+b+4 a k+a (1-2 b) k^2+b k^2\right ) (1-x) \left (\frac {(1-k) x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {1-x}{1-k x}\right )}{2 (1-k)^3 (1-b k)^2 ((1-x) x (1-k x))^{2/3}}-\frac {\left (\left (\frac {(a+b) \left (4+5 b+b^2+2 b k-b^2 k+5 b k^2-b^2 k^3+b^2 k^4\right )}{\sqrt {b} \sqrt {4+b-2 b k+b k^2}}-(a+b) (1+k) \left (3+b \left (1-k+k^2\right )\right )\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)+\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )} \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}+\frac {\left ((a+b) \left (3+b+3 k+b k^3+\frac {4+b^2 (1-k)^2 \left (1+k+k^2\right )+b \left (5+2 k+5 k^2\right )}{\sqrt {b} \sqrt {4+b (1-k)^2}}\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(1-x)^{5/3} (1-k x)^{5/3} \left (-b (1+k)-\sqrt {b} \sqrt {4+b-2 b k+b k^2}+2 (-1+b k) x\right )} \, dx}{(-1+b k)^2 ((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 14.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 x+(1+k) x^2\right ) \left (a-a (1+k) x+(1+a k) x^2\right )}{(-1+x) ((1-x) x (1-k x))^{2/3} (-1+k x) \left (b-b (1+k) x+(-1+b k) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.77, size = 203, normalized size = 1.00 \begin {gather*} \frac {3 x^2}{2 \left (x-x^2-k x^2+k x^3\right )^{2/3}}+\frac {\left (\sqrt {3} a+\sqrt {3} b\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {(a+b) \log \left (x-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {(-a-b) \log \left (x^2+\sqrt [3]{b} x \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 (a {\left (k + 1\right )} x - {\left (a k + 1\right )} x^{2} - a\right )} {\left ({\left (k + 1\right )} x^{2} - 2 \, x\right )}}{{\left (b {\left (k + 1\right )} x - {\left (b k - 1\right )} x^{2} - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left (k x - 1\right )} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 x +\left (1+k \right ) x^{2}\right ) \left (a -a \left (1+k \right ) x +\left (a k +1\right ) x^{2}\right )}{\left (-1+x \right ) \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (k x -1\right ) \left (b -b \left (1+k \right ) x +\left (b k -1\right ) x^{2}\right )}\, dx\]
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 (a {\left (k + 1\right )} x - {\left (a k + 1\right )} x^{2} - a\right )} {\left ({\left (k + 1\right )} x^{2} - 2 \, x\right )}}{{\left (b {\left (k + 1\right )} x - {\left (b k - 1\right )} x^{2} - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}} {\left (k x - 1\right )} {\left (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\,x-x^2\,\left (k+1\right )\right )\,\left (\left (a\,k+1\right )\,x^2-a\,\left (k+1\right )\,x+a\right )}{\left (k\,x-1\right )\,\left (x-1\right )\,{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (\left (b\,k-1\right )\,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(-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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