Optimal. Leaf size=87 \[ \frac {1}{3} \text {RootSum}\left [-\text {$\#$1}^6-\text {$\#$1}^3 c+b\& ,\frac {3 \log \left (\text {$\#$1} \sqrt [3]{k x^3+(-k-1) x^2+x}+x-1\right )-\log \left (k x^3+(-k-1) x^2+x\right )}{2 \text {$\#$1}^5+\text {$\#$1}^2 c}\& \right ] \]
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Rubi [F] time = 26.02, 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 (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(4-c) x+(-6+b+2 c+c k) x^2+(4-c-2 b k-2 c k) x^3+\left (-1+c k+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 {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(4-c) x+(-6+b+2 c+c k) x^2+(4-c-2 b k-2 c k) x^3+\left (-1+c k+b k^2\right ) x^4\right )} \, dx &=\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {x^{2/3} (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{1-x} \sqrt [3]{1-k x} \left (-1+(4-c) x+(-6+b+2 c+c k) x^2+(4-c-2 b k-2 c k) x^3+\left (-1+c k+b k^2\right ) x^4\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 {x^{2/3} (1-k x)^{2/3} (-1+(-1+2 k) x)}{\sqrt [3]{1-x} \left (-1+(4-c) x+(-6+b+2 c+c k) x^2+(4-c-2 b k-2 c k) x^3+\left (-1+c k+b k^2\right ) x^4\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-k x^3\right )^{2/3} \left (-1+(-1+2 k) x^3\right )}{\sqrt [3]{1-x^3} \left (-1+(4-c) x^3+(-6+b+2 c+c k) x^6+(4-c-2 b k-2 c k) x^9+\left (-1+c k+b k^2\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \operatorname {Subst}\left (\int \left (\frac {x^4 \left (1-k x^3\right )^{2/3}}{\sqrt [3]{1-x^3} \left (1-4 \left (1-\frac {c}{4}\right ) x^3+6 \left (1+\frac {1}{6} (-b-c (2+k))\right ) x^6-4 \left (1-\frac {c}{4}-\frac {1}{2} (b+c) k\right ) x^9+(1-k (c+b k)) x^{12}\right )}+\frac {(1-2 k) x^7 \left (1-k x^3\right )^{2/3}}{\sqrt [3]{1-x^3} \left (1-4 \left (1-\frac {c}{4}\right ) x^3+6 \left (1+\frac {1}{6} (-b-c (2+k))\right ) x^6-4 \left (1-\frac {c}{4}-\frac {1}{2} (b+c) k\right ) x^9+(1-k (c+b k)) x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=-\frac {\left (3 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-k x^3\right )^{2/3}}{\sqrt [3]{1-x^3} \left (1-4 \left (1-\frac {c}{4}\right ) x^3+6 \left (1+\frac {1}{6} (-b-c (2+k))\right ) x^6-4 \left (1-\frac {c}{4}-\frac {1}{2} (b+c) k\right ) x^9+(1-k (c+b k)) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}-\frac {\left (3 (1-2 k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (1-k x^3\right )^{2/3}}{\sqrt [3]{1-x^3} \left (1-4 \left (1-\frac {c}{4}\right ) x^3+6 \left (1+\frac {1}{6} (-b-c (2+k))\right ) x^6-4 \left (1-\frac {c}{4}-\frac {1}{2} (b+c) k\right ) x^9+(1-k (c+b k)) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\\ \end {align*}
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Mathematica [F] time = 2.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x (-1+k x) (-1+(-1+2 k) x)}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(4-c) x+(-6+b+2 c+c k) x^2+(4-c-2 b k-2 c k) x^3+\left (-1+c k+b k^2\right ) x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.05, size = 87, normalized size = 1.00 \begin {gather*} \frac {1}{3} \text {RootSum}\left [b-c \text {$\#$1}^3-\text {$\#$1}^6\&,\frac {-\log \left (x+(-1-k) x^2+k x^3\right )+3 \log \left (-1+x+\sqrt [3]{x+(-1-k) x^2+k x^3} \text {$\#$1}\right )}{c \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \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 (2 \, k - 1\right )} x - 1\right )} {\left (k x - 1\right )} x}{{\left ({\left (b k^{2} + c k - 1\right )} x^{4} - {\left (2 \, b k + 2 \, c k + c - 4\right )} x^{3} + {\left (c k + b + 2 \, c - 6\right )} x^{2} - {\left (c - 4\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{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 \[\int \frac {x \left (k x -1\right ) \left (-1+\left (-1+2 k \right ) x \right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-1+\left (4-c \right ) x +\left (c k +b +2 c -6\right ) x^{2}+\left (-2 b k -2 c k -c +4\right ) x^{3}+\left (b \,k^{2}+c k -1\right ) x^{4}\right )}\, dx\]
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 (2 \, k - 1\right )} x - 1\right )} {\left (k x - 1\right )} x}{{\left ({\left (b k^{2} + c k - 1\right )} x^{4} - {\left (2 \, b k + 2 \, c k + c - 4\right )} x^{3} + {\left (c k + b + 2 \, c - 6\right )} x^{2} - {\left (c - 4\right )} x - 1\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{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\,\left (x\,\left (2\,k-1\right )-1\right )\,\left (k\,x-1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (-b\,k^2-c\,k+1\right )\,x^4+\left (c+2\,b\,k+2\,c\,k-4\right )\,x^3+\left (6-2\,c-c\,k-b\right )\,x^2+\left (c-4\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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