Optimal. Leaf size=276 \[ -\frac {\log \left (a^2+d^{2/3} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{4/3}+\left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}+a-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}}{\sqrt [3]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}-2 a+2 x}\right )}{d^{2/3}} \]
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Rubi [F] time = 8.63, 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 {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{(-a+x)^{2/3} (-b+x)^{4/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-4 a+b+3 x) \left (b^2-2 b x+x^2\right )}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{5/3} (-4 a+b+3 x)}{(-a+x)^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {4 a \left (1-\frac {b}{4 a}\right ) (-b+x)^{5/3}}{(-a+x)^{2/3} \left (-a-b^4 d+\left (1+4 b^3 d\right ) x-6 b^2 d x^2+4 b d x^3-d x^4\right )}+\frac {3 x (-b+x)^{5/3}}{(-a+x)^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (3 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {x (-b+x)^{5/3}}{(-a+x)^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left ((4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{5/3}}{(-a+x)^{2/3} \left (-a-b^4 d+\left (1+4 b^3 d\right ) x-6 b^2 d x^2+4 b d x^3-d x^4\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a+x^3\right ) \left (a-b+x^3\right )^{5/3}}{a+b^4 d-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a+b^4 d-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a+x^3\right ) \left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \left (\frac {a \left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}}+\frac {x^3 \left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (9 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (9 a (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a^4 \left (1+\frac {b \left (-4 a^3+6 a^2 b-4 a b^2+b^3\right )}{a^4}\right ) d-\left (1-4 (a-b)^3 d\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) d x^6+4 a \left (1-\frac {b}{a}\right ) d x^9+d x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left (3 (4 a-b) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {\left (a-b+x^3\right )^{5/3}}{a \left (1+\frac {b^4 d}{a}\right )-\left (1+4 b^3 d\right ) \left (a+x^3\right )+6 b^2 d \left (a+x^3\right )^2-4 b d \left (a+x^3\right )^3+d \left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 2.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-4 a+b+3 x) \left (-b^3+3 b^2 x-3 b x^2+x^3\right )}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.15, size = 276, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}{-2 a+2 x+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}\right )}{d^{2/3}}+\frac {\log \left (a-x+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{d^{2/3}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{4/3}\right )}{2 d^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 351, normalized size = 1.27 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b^{2} d^{2} - 2 \, b d^{2} x + d^{2} x^{2}\right )}}\right ) + {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d + {\left (b^{4} d - 4 \, b^{3} d x + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4}\right )} {\left (d^{2}\right )}^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) - 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b^{2} - 2 \, b x + x^{2}}\right )}{2 \, d^{2}} \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 (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, x\right )}}{{\left (b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-4 a +b +3 x \right ) \left (-b^{3}+3 b^{2} x -3 b \,x^{2}+x^{3}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a +b^{4} d -\left (4 b^{3} d +1\right ) x +6 b^{2} d \,x^{2}-4 b d \,x^{3}+d \,x^{4}\right )}\, dx\]
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 (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, x\right )}}{{\left (b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{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 {\left (b-4\,a+3\,x\right )\,\left (b^3-3\,b^2\,x+3\,b\,x^2-x^3\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a+b^4\,d+d\,x^4-x\,\left (4\,d\,b^3+1\right )+6\,b^2\,d\,x^2-4\,b\,d\,x^3\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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