Optimal. Leaf size=109 \[ -\text {RootSum}\left [b-c \text {$\#$1}^3-\text {$\#$1}^6\& ,\frac {a \log (x)-a \log \left (\sqrt [3]{x+(-1-k) x^2+k x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{x+(-1-k) x^2+k x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{c \text {$\#$1}+2 \text {$\#$1}^4}\& \right ] \]
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Rubi [F]
time = 34.54, 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 )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\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 )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\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} \sqrt [3]{x} \sqrt [3]{1-k x} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+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 ) \text {Subst}\left (\int \frac {x \left (-2+(1+k) x^3\right ) \left (1-(1+k) x^3+(a+k) x^6\right )}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+\left (1+c+4 k+k^2\right ) x^6-\left (c+2 k+c k+2 k^2\right ) x^9+\left (-b+c k+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 ) \text {Subst}\left (\int \left (\frac {3 (1+k) x^4}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+(1+c+k (4+k)) x^6-c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9-b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )}+\frac {2 (-a-k) \left (1+\frac {(1+k)^2}{2 (a+k)}\right ) x^7}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+(1+c+k (4+k)) x^6-c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9-b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )}+\frac {(1+k) (a+k) x^{10}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+(1+c+k (4+k)) x^6-c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9-b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )}+\frac {2 x}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+2 (1+k) x^3-(1+c+k (4+k)) x^6+c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9+b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {\left (6 \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (-1+2 (1+k) x^3-(1+c+k (4+k)) x^6+c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9+b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (9 (1+k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+(1+c+k (4+k)) x^6-c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9-b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (3 (1+k) (a+k) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+(1+c+k (4+k)) x^6-c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9-b \left (1-\frac {k (c+k)}{b}\right ) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (3 \left (-1-2 a-4 k-k^2\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt [3]{1-x^3} \sqrt [3]{1-k x^3} \left (1-2 (1+k) x^3+(1+c+k (4+k)) x^6-c \left (1+\frac {k (2+c+2 k)}{c}\right ) x^9-b \left (1-\frac {k (c+k)}{b}\right ) 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 = 73.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-2+(1+k) x) \left (1-(1+k) x+(a+k) x^2\right )}{\sqrt [3]{(1-x) x (1-k x)} \left (1-2 (1+k) x+\left (1+c+4 k+k^2\right ) x^2-\left (c+2 k+c k+2 k^2\right ) x^3+\left (-b+c k+k^2\right ) x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-2+\left (1+k \right ) x \right ) \left (1-\left (1+k \right ) x +\left (a +k \right ) x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-2 \left (1+k \right ) x +\left (k^{2}+c +4 k +1\right ) x^{2}-\left (c k +2 k^{2}+c +2 k \right ) x^{3}+\left (c k +k^{2}-b \right ) x^{4}\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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Sympy [F(-1)] Timed out
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*} \text {could not integrate} \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 {\left (x\,\left (k+1\right )-2\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 )}^{1/3}\,\left (x^4\,\left (k^2+c\,k-b\right )-x^3\,\left (c+2\,k+c\,k+2\,k^2\right )+x^2\,\left (k^2+4\,k+c+1\right )-2\,x\,\left (k+1\right )+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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