Optimal. Leaf size=201 \[ -\frac {\log \left (b^{2/3} \left (k x^3+(-k-1) x^2+x\right )^{2/3}+\sqrt [3]{k x^3+(-k-1) x^2+x} \left (\sqrt [3]{b}-\sqrt [3]{b} k x\right )+k^2 x^2-2 k x+1\right )}{2 b^{2/3}}+\frac {\log \left (\sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}+k x-1\right )}{b^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}}{\sqrt [3]{b} \sqrt [3]{k x^3+(-k-1) x^2+x}-2 k x+2}\right )}{b^{2/3}} \]
________________________________________________________________________________________
Rubi [F] time = 3.67, 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 {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(b+2 k) x+\left (b+k^2\right ) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(b+2 k) x+\left (b+k^2\right ) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {-1+(2-k) x}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (1-(b+2 k) x+\left (b+k^2\right ) 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 {2-k-\frac {k \sqrt {-4+b+4 k}}{\sqrt {b}}}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (-b-2 k-\sqrt {b} \sqrt {-4+b+4 k}+2 \left (b+k^2\right ) x\right )}+\frac {2-k+\frac {k \sqrt {-4+b+4 k}}{\sqrt {b}}}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (-b-2 k+\sqrt {b} \sqrt {-4+b+4 k}+2 \left (b+k^2\right ) x\right )}\right ) \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ &=\frac {\left (\left (2-k \left (1-\frac {\sqrt {-4+b+4 k}}{\sqrt {b}}\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (-b-2 k+\sqrt {b} \sqrt {-4+b+4 k}+2 \left (b+k^2\right ) x\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}+\frac {\left (\left (2-k \left (1+\frac {\sqrt {-4+b+4 k}}{\sqrt {b}}\right )\right ) \sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x}\right ) \int \frac {1}{\sqrt [3]{1-x} \sqrt [3]{x} \sqrt [3]{1-k x} \left (-b-2 k-\sqrt {b} \sqrt {-4+b+4 k}+2 \left (b+k^2\right ) x\right )} \, dx}{\sqrt [3]{(1-x) x (1-k x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 5.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+(2-k) x}{\sqrt [3]{(1-x) x (1-k x)} \left (1-(b+2 k) x+\left (b+k^2\right ) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.61, size = 201, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2-2 k x+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{b^{2/3}}+\frac {\log \left (-1+k x+\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{b^{2/3}}-\frac {\log \left (1-2 k x+k^2 x^2+\left (\sqrt [3]{b}-\sqrt [3]{b} k x\right ) \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 b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (k - 2\right )} x + 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (k^{2} + b\right )} x^{2} - {\left (b + 2 \, k\right )} x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {-1+\left (2-k \right ) x}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (1-\left (b +2 k \right ) x +\left (k^{2}+b \right ) x^{2}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (k - 2\right )} x + 1}{\left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {1}{3}} {\left ({\left (k^{2} + b\right )} x^{2} - {\left (b + 2 \, k\right )} x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {x\,\left (k-2\right )+1}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (\left (k^2+b\right )\,x^2+\left (-b-2\,k\right )\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________