Optimal. Leaf size=291 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}-2 a+2 x}\right )}{d^{2/3}} \]
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Rubi [F] time = 8.96, 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 {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x} \left (a b+(-3 a+b) x+x^2\right )}{\sqrt [3]{x} \sqrt [3]{-a+x} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-b+x^3} \left (a b+(-3 a+b) x^3+x^6\right )}{\sqrt [3]{-a+x^3} \left (-a^2+\left (2 a+b^2 d\right ) x^3-(1+2 b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {(3 a-b) x^4 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (a^2-2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3+(1+2 b d) x^6-d x^9\right )}+\frac {a b x \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )}+\frac {x^7 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^7 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (3 (3 a-b) \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (a^2-2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3+(1+2 b d) x^6-d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)^2}}+\frac {\left (3 a b \sqrt [3]{x} \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-b+x^3}}{\sqrt [3]{-a+x^3} \left (-a^2+2 a \left (1+\frac {b^2 d}{2 a}\right ) x^3-(1+2 b d) x^6+d x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x (-a+x) (-b+x)^2}}\\ \end {align*}
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Mathematica [F] time = 3.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b^2+(4 a-b) b x-3 a x^2+x^3}{\sqrt [3]{x (-a+x) (-b+x)^2} \left (-a^2+\left (2 a+b^2 d\right ) x-(1+2 b d) x^2+d x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.72, size = 291, normalized size = 1.00 \begin {gather*} -\frac {\log \left (a^2+d^{2/3} \left (x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4\right )^{2/3}+\sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}}{\sqrt [3]{d} \sqrt [3]{x^2 \left (2 a b+b^2\right )-a b^2 x+x^3 (-a-2 b)+x^4}-2 a+2 x}\right )}{d^{2/3}} \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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {a b^{2} - {\left (4 \, a - b\right )} b x + 3 \, a x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (2 \, b d + 1\right )} x^{2} - a^{2} + {\left (b^{2} d + 2 \, a\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a \,b^{2}+\left (4 a -b \right ) b x -3 a \,x^{2}+x^{3}}{\left (x \left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-a^{2}+\left (b^{2} d +2 a \right ) x -\left (2 b d +1\right ) x^{2}+d \,x^{3}\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 {a b^{2} - {\left (4 \, a - b\right )} b x + 3 \, a x^{2} - x^{3}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2} x\right )^{\frac {1}{3}} {\left (d x^{3} - {\left (2 \, b d + 1\right )} x^{2} - a^{2} + {\left (b^{2} d + 2 \, a\right )} x\right )}}\,{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 {a\,b^2+3\,a\,x^2-x^3-b\,x\,\left (4\,a-b\right )}{{\left (-x\,\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (x\,\left (d\,b^2+2\,a\right )+d\,x^3-x^2\,\left (2\,b\,d+1\right )-a^2\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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