Optimal. Leaf size=238 \[ \frac {\left (a+b^2\right ) \log \left (\sqrt [3]{k x^3+(-k-1) x^2+x}-\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {\left (-\sqrt {3} a-\sqrt {3} b^2\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{\sqrt [3]{b}}+\frac {\left (-a-b^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{2 \sqrt [3]{b}}+\frac {3 \left (k x^3-k x^2-x^2+x\right )^{2/3} \left (5 b x^2+2 k x^2-2 k x-2 x+2\right )}{10 x^4} \]
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Rubi [F] time = 32.19, 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 (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \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.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \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 (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{\sqrt [3]{1-x} x^{13/3} \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 {8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )}{(b-k)^4 \sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x}}+\frac {(1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right )}{(b-k)^3 \sqrt [3]{1-x} x^{10/3} \sqrt [3]{1-k x}}+\frac {a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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) \left (a+k^2\right )}{(b-k) \sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}}-\frac {\left (a+b^2\right ) \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right ) x}{(b-k)^4 \sqrt [3]{1-x} x^{13/3} \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 {\left (a+b^2\right ) \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right ) x}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (1+(-1-k) x+(-b+k) x^2\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left ((1+k) \left (a+k^2\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}{(b-k) \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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 {\left ((1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 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^{10/3} \sqrt [3]{1-k x}} \, dx}{(b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x}} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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) \left (a+k^2\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 )}{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 {-\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )}+\frac {\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )}{\sqrt [3]{1-x} x^{13/3} \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)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left ((1+k) \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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 {\left (3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {-\frac {5}{3} (1+k)+k x}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}} \, dx}{7 (b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {-\frac {8}{3} (1+k)+2 k x}{\sqrt [3]{1-x} x^{10/3} \sqrt [3]{1-k x}} \, dx}{10 (b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {12 (1+k) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{35 (b-k)^4 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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 {15 (1+k)^2 \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{28 (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (a+k^2\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 )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (a \left (1+2 b+k^2\right )+k \left (2 b \left (1+3 k+k^2\right )-k \left (1+4 k+k^2\right )\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 (9 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {2 \left (5+4 k+5 k^2\right )}{9 \sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{28 (b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\left (\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (9 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {\frac {2}{9} \left (20+19 k+20 k^2\right )-\frac {8}{3} k (1+k) x}{\sqrt [3]{1-x} x^{7/3} \sqrt [3]{1-k x}} \, dx}{70 (b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a+b^2\right ) \left (1+5 b^2+5 b^2 k+k^5+5 b \left (1+k^2\right )+5 b k \left (1+k^2\right )+\frac {1+4 b^3-k-k^5+k^6+b^2 \left (13+14 k+13 k^2\right )+b \left (7+2 k+2 k^2+2 k^3+7 k^4\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^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {12 (1+k) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{35 (b-k)^4 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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 {15 (1+k)^2 \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{28 (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (20+19 k+20 k^2\right ) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{140 (b-k)^4 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (a+k^2\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 )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (a \left (1+2 b+k^2\right )+k \left (2 b \left (1+3 k+k^2\right )-k \left (1+4 k+k^2\right )\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 ((1+k) \left (5+4 k+5 k^2\right ) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 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}{14 (b-k)^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (\left (\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (27 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int -\frac {20 (1+k) \left (4-k+4 k^2\right )}{27 \sqrt [3]{1-x} x^{4/3} \sqrt [3]{1-k x}} \, dx}{280 (b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a+b^2\right ) \left (1+5 b^2+5 b^2 k+k^5+5 b \left (1+k^2\right )+5 b k \left (1+k^2\right )+\frac {1+4 b^3-k-k^5+k^6+b^2 \left (13+14 k+13 k^2\right )+b \left (7+2 k+2 k^2+2 k^3+7 k^4\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^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {12 (1+k) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{35 (b-k)^4 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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 {15 (1+k)^2 \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{28 (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (20+19 k+20 k^2\right ) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{140 (b-k)^4 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (a+k^2\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 )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (5+4 k+5 k^2\right ) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 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 )}{28 (1-k) (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (a \left (1+2 b+k^2\right )+k \left (2 b \left (1+3 k+k^2\right )-k \left (1+4 k+k^2\right )\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 (\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left ((1+k) \left (4-k+4 k^2\right ) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\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}{14 (b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a+b^2\right ) \left (1+5 b^2+5 b^2 k+k^5+5 b \left (1+k^2\right )+5 b k \left (1+k^2\right )+\frac {1+4 b^3-k-k^5+k^6+b^2 \left (13+14 k+13 k^2\right )+b \left (7+2 k+2 k^2+2 k^3+7 k^4\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^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {3 \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{10 (b-k)^4 x^3 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{7 (b-k)^3 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {12 (1+k) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{35 (b-k)^4 x^2 \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (a \left (1+2 b+k^2\right )+2 b k \left (1+3 k+k^2\right )-k^2 \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 {15 (1+k)^2 \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 k+k^2\right )\right ) (1-x) (1-k x)}{28 (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 \left (20+19 k+20 k^2\right ) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\right )\right ) (1-x) (1-k x)}{140 (b-k)^4 x \sqrt [3]{(1-x) x (1-k x)}}+\frac {3 (1+k) \left (a+k^2\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 )}{2 (1-k) (b-k) x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (5+4 k+5 k^2\right ) \left (9 b k^2-3 k^3-a \left (1+3 b-k+k^2\right )-b^2 \left (1+8 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 )}{28 (1-k) (b-k)^3 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (4-k+4 k^2\right ) \left (8 b k^3-2 k^4+4 b^3 \left (1+3 k+k^2\right )+b^2 \left (1-12 k^2+k^4\right )+a \left (1+2 b^2+k^4+4 b \left (1+k+k^2\right )\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 )}{28 (1-k) (b-k)^4 x \sqrt [3]{(1-x) x (1-k x)}}-\frac {3 (1+k) \left (a \left (1+2 b+k^2\right )+k \left (2 b \left (1+3 k+k^2\right )-k \left (1+4 k+k^2\right )\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 (\frac {\left (a+b^2\right ) \left (1+7 b+13 b^2+4 b^3-k+2 b k+14 b^2 k+2 b k^2+13 b^2 k^2+2 b k^3+7 b k^4-k^5+k^6\right )}{\sqrt {1+4 b-2 k+k^2}}-\left (a+b^2\right ) (1+k) \left (1+5 b^2-k+k^2-k^3+k^4+5 b \left (1+k^2\right )\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} x^{13/3} \sqrt [3]{1-k x} \left (-1-k+\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (a+b^2\right ) \left (1+5 b^2+5 b^2 k+k^5+5 b \left (1+k^2\right )+5 b k \left (1+k^2\right )+\frac {1+4 b^3-k-k^5+k^6+b^2 \left (13+14 k+13 k^2\right )+b \left (7+2 k+2 k^2+2 k^3+7 k^4\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^{13/3} \sqrt [3]{1-k x} \left (-1-k-\sqrt {1+4 b-2 k+k^2}+2 (-b+k) x\right )} \, dx}{(b-k)^4 \sqrt [3]{(1-x) x (1-k x)}}\\ \end {align*}
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Mathematica [F] time = 6.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-2+(1+k) x) \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (a+k^2\right ) x^4\right )}{x^4 \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 = 0.52, size = 238, normalized size = 1.00 \begin {gather*} \frac {3 \left (2-2 x-2 k x+5 b x^2+2 k x^2\right ) \left (x-x^2-k x^2+k x^3\right )^{2/3}}{10 x^4}+\frac {\left (-\sqrt {3} a-\sqrt {3} b^2\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 {\left (a+b^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}+\frac {\left (-a-b^2\right ) \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 (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 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^{4}}\,{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 \[\int \frac {\left (-2+\left (1+k \right ) x \right ) \left (1-2 \left (1+k \right ) x +\left (k^{2}+4 k +1\right ) x^{2}-2 \left (k^{2}+k \right ) x^{3}+\left (k^{2}+a \right ) x^{4}\right )}{x^{4} \left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-\left (1+k \right ) x +\left (-b +k \right ) x^{2}\right )}\, dx\]
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 (k^{2} + a\right )} x^{4} - 2 \, {\left (k^{2} + k\right )} x^{3} + {\left (k^{2} + 4 \, k + 1\right )} x^{2} - 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^{4}}\,{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 (x^2\,\left (k^2+4\,k+1\right )-2\,x\,\left (k+1\right )+x^4\,\left (k^2+a\right )-2\,x^3\,\left (k^2+k\right )+1\right )}{x^4\,{\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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