Optimal. Leaf size=279 \[ -\frac {\log \left (b^{2/3} k^4 x^4-4 b^{2/3} k^3 x^3+6 b^{2/3} k^2 x^2-4 b^{2/3} k x+b^{2/3}+\sqrt [3]{k x^3+(-k-1) x^2+x} \left (\sqrt [3]{b} k^2 x^2-2 \sqrt [3]{b} k x+\sqrt [3]{b}\right )+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{2 \sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b} k^2 x^2+2 \sqrt [3]{b} k x-\sqrt [3]{b}+\sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{\sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{k x^3+(-k-1) x^2+x}}{2 \sqrt [3]{b} k^2 x^2-4 \sqrt [3]{b} k x+2 \sqrt [3]{b}+\sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{\sqrt [3]{b}} \]
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Rubi [F] time = 11.02, 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 {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx &=\int \frac {-1+(2-3 k) x+k (1+4 k) x^2-3 k^2 x^3}{\sqrt [3]{(-1+x) x (-1+k x)} \left ((-1+x) x-b (-1+k x)^5\right )} \, dx\\ &=\frac {\left (\sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{-1+k x}\right ) \int \frac {-1+(2-3 k) x+k (1+4 k) x^2-3 k^2 x^3}{\sqrt [3]{-1+x} \sqrt [3]{x} \sqrt [3]{-1+k x} \left ((-1+x) x-b (-1+k x)^5\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 {(-1+k x)^{2/3} \left (1+(-2+4 k) x-3 k x^2\right )}{\sqrt [3]{-1+x} \sqrt [3]{x} \left ((-1+x) x-b (-1+k x)^5\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 \left (-1+k x^3\right )^{2/3} \left (1+(-2+4 k) x^3-3 k x^6\right )}{\sqrt [3]{-1+x^3} \left (x^3 \left (-1+x^3\right )-b \left (-1+k x^3\right )^5\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 \left (-1+k x^3\right )^{2/3}}{\sqrt [3]{-1+x^3} \left (b-(1+5 b k) x^3+\left (1+10 b k^2\right ) x^6-10 b k^3 x^9+5 b k^4 x^{12}-b k^5 x^{15}\right )}+\frac {2 (-1+2 k) x^4 \left (-1+k x^3\right )^{2/3}}{\sqrt [3]{-1+x^3} \left (b-(1+5 b k) x^3+\left (1+10 b k^2\right ) x^6-10 b k^3 x^9+5 b k^4 x^{12}-b k^5 x^{15}\right )}+\frac {3 k x^7 \left (-1+k x^3\right )^{2/3}}{\sqrt [3]{-1+x^3} \left (-b+(1+5 b k) x^3-\left (1+10 b k^2\right ) x^6+10 b k^3 x^9-5 b k^4 x^{12}+b k^5 x^{15}\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 \left (-1+k x^3\right )^{2/3}}{\sqrt [3]{-1+x^3} \left (b-(1+5 b k) x^3+\left (1+10 b k^2\right ) x^6-10 b k^3 x^9+5 b k^4 x^{12}-b k^5 x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(-1+x) x (-1+k x)}}-\frac {\left (6 (1-2 k) \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 (b-(1+5 b k) x^3+\left (1+10 b k^2\right ) x^6-10 b k^3 x^9+5 b k^4 x^{12}-b k^5 x^{15}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(-1+x) x (-1+k x)}}+\frac {\left (9 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 (-b+(1+5 b k) x^3-\left (1+10 b k^2\right ) x^6+10 b k^3 x^9-5 b k^4 x^{12}+b k^5 x^{15}\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 \left (-1+k x^3\right )^{2/3}}{\sqrt [3]{-1+x^3} \left (-x^3+x^6-b \left (-1+k x^3\right )^5\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(-1+x) x (-1+k x)}}-\frac {\left (6 (1-2 k) \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 (-x^3+x^6-b \left (-1+k x^3\right )^5\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(-1+x) x (-1+k x)}}+\frac {\left (9 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 (x^3-x^6+b \left (-1+k x^3\right )^5\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{(-1+x) x (-1+k x)}}\\ \end {align*}
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Mathematica [F] time = 5.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+(-2+3 k) x-\left (k+4 k^2\right ) x^2+3 k^2 x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-b+(1+5 b k) x-\left (1+10 b k^2\right ) x^2+10 b k^3 x^3-5 b k^4 x^4+b k^5 x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.08, size = 279, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b^{2/3} k^4 x^4-4 b^{2/3} k^3 x^3+6 b^{2/3} k^2 x^2-4 b^{2/3} k x+b^{2/3}+\sqrt [3]{k x^3+(-k-1) x^2+x} \left (\sqrt [3]{b} k^2 x^2-2 \sqrt [3]{b} k x+\sqrt [3]{b}\right )+\left (k x^3+(-k-1) x^2+x\right )^{2/3}\right )}{2 \sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b} k^2 x^2+2 \sqrt [3]{b} k x-\sqrt [3]{b}+\sqrt [3]{k x^3+(-k-1) x^2+x}\right )}{\sqrt [3]{b}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{k x^3+(-k-1) x^2+x}}{2 \sqrt [3]{b} k^2 x^2-4 \sqrt [3]{b} k x+2 \sqrt [3]{b}+\sqrt [3]{k x^3+(-k-1) x^2+x}}\right )}{\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(-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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maple [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+\left (-2+3 k \right ) x -\left (4 k^{2}+k \right ) x^{2}+3 k^{2} x^{3}}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {1}{3}} \left (-b +\left (5 b k +1\right ) x -\left (10 b \,k^{2}+1\right ) x^{2}+10 b \,k^{3} x^{3}-5 b \,k^{4} x^{4}+b \,k^{5} x^{5}\right )}\, dx \end {gather*}
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 {3 \, k^{2} x^{3} - {\left (4 \, k^{2} + k\right )} x^{2} + {\left (3 \, k - 2\right )} x + 1}{{\left (b k^{5} x^{5} - 5 \, b k^{4} x^{4} + 10 \, b k^{3} x^{3} - {\left (10 \, b k^{2} + 1\right )} x^{2} + {\left (5 \, b k + 1\right )} x - b\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 = -0.00 \begin {gather*} -\int \frac {x\,\left (3\,k-2\right )+3\,k^2\,x^3-x^2\,\left (4\,k^2+k\right )+1}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{1/3}\,\left (b+x^2\,\left (10\,b\,k^2+1\right )-x\,\left (5\,b\,k+1\right )-10\,b\,k^3\,x^3+5\,b\,k^4\,x^4-b\,k^5\,x^5\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (k x - 1\right ) \left (3 k x^{2} - 4 k x + 2 x - 1\right )}{\sqrt [3]{x \left (x - 1\right ) \left (k x - 1\right )} \left (b k^{5} x^{5} - 5 b k^{4} x^{4} + 10 b k^{3} x^{3} - 10 b k^{2} x^{2} + 5 b k x - b - x^{2} + x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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