Optimal. Leaf size=383 \[ \frac {\left (a+\sqrt {b}\right ) \log \left (\sqrt [3]{k x^3+(-k-1) x^2+x}-\sqrt [6]{b} x\right )}{2 b^{5/6}}+\frac {\left (\sqrt {b}-a\right ) \log \left (\sqrt [6]{b} x+\sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{2 b^{5/6}}+\frac {\left (a-\sqrt {b}\right ) \log \left (-\sqrt [6]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+\sqrt [3]{b} x^2+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{4 b^{5/6}}+\frac {\left (-a-\sqrt {b}\right ) \log \left (\sqrt [6]{b} x \sqrt [3]{k x^3+(-k-1) x^2+x}+\sqrt [3]{b} x^2+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{4 b^{5/6}}-\frac {\sqrt {3} \left (a-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} x}{\sqrt [6]{b} x-2 \sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{2 b^{5/6}}+\frac {\sqrt {3} \left (a+\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} x}{\sqrt [6]{b} x+2 \sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{2 b^{5/6}} \]
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Rubi [F] time = 35.03, 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 (1-(1+k) x+(a+k) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (-b+k^2\right ) x^4\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 (1-(1+k) x+(a+k) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (-b+k^2\right ) x^4\right )} \, dx &=\int \frac {x (-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (-b+k^2\right ) x^4\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 (1-(1+k) x+(a+k) x^2\right )}{(1-x)^{2/3} (1-k x)^{2/3} \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (-b+k^2\right ) x^4\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^3 \left (-2+(1+k) x^3\right ) \left (1-(1+k) x^3+(a+k) x^6\right )}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+\left (1+4 k+k^2\right ) x^6-2 \left (k+k^2\right ) x^9+\left (-b+k^2\right ) x^{12}\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 {(1+k) (a+k)}{\left (b-k^2\right ) \left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3}}+\frac {(1+k) (a+k)-2 \left (b+k+k^2+k^3+a (1+k)^2\right ) x^3+(1+k) \left (3 b+k+k^2+k^3+a \left (1+4 k+k^2\right )\right ) x^6-\left (b \left (1+2 a+4 k+k^2\right )+k \left (k+k^3+2 a \left (1+k+k^2\right )\right )\right ) x^9}{\left (b-k^2\right ) \left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+\left (1+4 k+k^2\right ) x^6-2 \left (k+k^2\right ) x^9+\left (-b+k^2\right ) x^{12}\right )}\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 \frac {(1+k) (a+k)-2 \left (b+k+k^2+k^3+a (1+k)^2\right ) x^3+(1+k) \left (3 b+k+k^2+k^3+a \left (1+4 k+k^2\right )\right ) x^6-\left (b \left (1+2 a+4 k+k^2\right )+k \left (k+k^3+2 a \left (1+k+k^2\right )\right )\right ) x^9}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+\left (1+4 k+k^2\right ) x^6-2 \left (k+k^2\right ) x^9+\left (-b+k^2\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}-\frac {\left (3 (1+k) (a+k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}\\ &=-\frac {3 (1+k) (a+k) x \left (\frac {1-x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(1-k) x}{1-k x}\right )}{\left (b-k^2\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 {(1+k) (a+k)}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )}+\frac {2 \left (-b-k-k^2-k^3-a (1+k)^2\right ) x^3}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )}+\frac {(1+k) \left (3 b+k+k^2+k^3+a \left (1+4 k+k^2\right )\right ) x^6}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )}+\frac {\left (-b \left (1+2 a+4 k+k^2\right )-k \left (k+k^3+2 a \left (1+k+k^2\right )\right )\right ) x^9}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}\\ &=-\frac {3 (1+k) (a+k) x \left (\frac {1-x}{1-k x}\right )^{2/3} (1-k x) \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {(1-k) x}{1-k x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}+\frac {\left (3 (1+k) (a+k) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}-\frac {\left (6 \left (b+k+k^2+k^3+a (1+k)^2\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}+\frac {\left (3 (1+k) \left (3 b+k+k^2+k^3+a \left (1+4 k+k^2\right )\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}+\frac {\left (3 \left (-b \left (1+2 a+4 k+k^2\right )-k \left (k+k^3+2 a \left (1+k+k^2\right )\right )\right ) (1-x)^{2/3} x^{2/3} (1-k x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (1-x^3\right )^{2/3} \left (1-k x^3\right )^{2/3} \left (1-2 (1+k) x^3+(1+k (4+k)) x^6-2 k (1+k) x^9-b \left (1-\frac {k^2}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (b-k^2\right ) ((1-x) x (1-k x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 1.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2 x+(1+k) x^2\right ) \left (1-(1+k) x+(a+k) x^2\right )}{((1-x) x (1-k x))^{2/3} \left (1-2 (1+k) x+\left (1+4 k+k^2\right ) x^2-2 \left (k+k^2\right ) x^3+\left (-b+k^2\right ) x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.79, size = 383, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \left (a-\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} x}{\sqrt [6]{b} x-2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}+\frac {\sqrt {3} \left (a+\sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b} x}{\sqrt [6]{b} x+2 \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{2 b^{5/6}}+\frac {\left (a+\sqrt {b}\right ) \log \left (-\sqrt [6]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{2 b^{5/6}}+\frac {\left (-a+\sqrt {b}\right ) \log \left (\sqrt [6]{b} x+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{2 b^{5/6}}+\frac {\left (a-\sqrt {b}\right ) \log \left (\sqrt [3]{b} x^2-\sqrt [6]{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 )}{4 b^{5/6}}+\frac {\left (-a-\sqrt {b}\right ) \log \left (\sqrt [3]{b} x^2+\sqrt [6]{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 )}{4 b^{5/6}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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} - 2 \, x\right )}}{{\left ({\left (k^{2} - b\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 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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (-2 x +\left (1+k \right ) x^{2}\right ) \left (1-\left (1+k \right ) x +\left (a +k \right ) x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (1-2 \left (1+k \right ) x +\left (k^{2}+4 k +1\right ) x^{2}-2 \left (k^{2}+k \right ) x^{3}+\left (k^{2}-b \right ) x^{4}\right )}\, dx\]
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} - 2 \, x\right )}}{{\left ({\left (k^{2} - b\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 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 = -0.00 \begin {gather*} \int \frac {\left (2\,x-x^2\,\left (k+1\right )\right )\,\left (\left (a+k\right )\,x^2+\left (-k-1\right )\,x+1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (x^4\,\left (b-k^2\right )-x^2\,\left (k^2+4\,k+1\right )+2\,x\,\left (k+1\right )+2\,x^3\,\left (k^2+k\right )-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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