Optimal. Leaf size=272 \[ \frac {\log \left (a^2 \sqrt {d}-\sqrt [6]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a \sqrt {d} x+\sqrt {d} x^2\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x (-a-b)+a b+x^2}}{2 a^2 \sqrt [3]{d}+\sqrt [3]{x (-a-b)+a b+x^2}-4 a \sqrt [3]{d} x+2 \sqrt [3]{d} x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^4 d-4 a^3 d x+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )+6 a^2 d x^2+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-4 a d x^3+d x^4\right )}{2 \sqrt [3]{d}} \]
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Rubi [F] time = 7.48, 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 (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{2/3} (a-5 b+4 x)}{\sqrt [3]{-b+x} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{2/3} (a-5 b+4 x)}{\sqrt [3]{-b+x} \left (b \left (1-\frac {a^5 d}{b}\right )-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {5 \left (1-\frac {a}{5 b}\right ) b (-a+x)^{2/3}}{\sqrt [3]{-b+x} \left (-b \left (1-\frac {a^5 d}{b}\right )+\left (1-5 a^4 d\right ) x+10 a^3 d x^2-10 a^2 d x^3+5 a d x^4-d x^5\right )}+\frac {4 x (-a+x)^{2/3}}{\sqrt [3]{-b+x} \left (b \left (1-\frac {a^5 d}{b}\right )-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (4 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {x (-a+x)^{2/3}}{\sqrt [3]{-b+x} \left (b \left (1-\frac {a^5 d}{b}\right )-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left ((-a+5 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{2/3}}{\sqrt [3]{-b+x} \left (-b \left (1-\frac {a^5 d}{b}\right )+\left (1-5 a^4 d\right ) x+10 a^3 d x^2-10 a^2 d x^3+5 a d x^4-d x^5\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (12 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (-a+b-x^3+d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+5 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{a-b+x^3} \left (a-b+x^3-d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (12 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (-a-x^3\right )}{\sqrt [3]{a-b+x^3} \left (a \left (1-\frac {b}{a}\right )+x^3-d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+5 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{a-b+x^3} \left (a \left (1-\frac {b}{a}\right )+x^3-d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (12 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^4}{\sqrt [3]{a-b+x^3} \left (-a \left (1-\frac {b}{a}\right )-x^3+d x^{15}\right )}+\frac {x^7}{\sqrt [3]{a-b+x^3} \left (-a \left (1-\frac {b}{a}\right )-x^3+d x^{15}\right )}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+5 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{a-b+x^3} \left (a \left (1-\frac {b}{a}\right )+x^3-d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (12 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (-a \left (1-\frac {b}{a}\right )-x^3+d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (12 a \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{a-b+x^3} \left (-a \left (1-\frac {b}{a}\right )-x^3+d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+5 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{a-b+x^3} \left (a \left (1-\frac {b}{a}\right )+x^3-d x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 4.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (b-a^5 d-\left (1-5 a^4 d\right ) x-10 a^3 d x^2+10 a^2 d x^3-5 a d x^4+d x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.71, size = 272, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a^2 \sqrt [3]{d}-4 a \sqrt [3]{d} x+2 \sqrt [3]{d} x^2+\sqrt [3]{a b+(-a-b) x+x^2}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^2 \sqrt {d}-2 a \sqrt {d} x+\sqrt {d} x^2-\sqrt [6]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^4 d-4 a^3 d x+6 a^2 d x^2-4 a d x^3+d x^4+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )\right )}{2 \sqrt [3]{d}} \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 {{\left (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {-a \left (a -5 b \right )-\left (3 a +5 b \right ) x +4 x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (b -a^{5} d -\left (-5 a^{4} d +1\right ) x -10 a^{3} d \,x^{2}+10 a^{2} d \,x^{3}-5 a d \,x^{4}+d \,x^{5}\right )}\, dx\]
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 (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} d + 10 \, a^{3} d x^{2} - 10 \, a^{2} d x^{3} + 5 \, a d x^{4} - d x^{5} - {\left (5 \, a^{4} d - 1\right )} x - b\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\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 {-4\,x^2+\left (3\,a+5\,b\right )\,x+a\,\left (a-5\,b\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (b-a^5\,d+d\,x^5+x\,\left (5\,a^4\,d-1\right )+10\,a^2\,d\,x^3-10\,a^3\,d\,x^2-5\,a\,d\,x^4\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 (- a + x\right ) \left (a - 5 b + 4 x\right )}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )} \left (- a^{5} d + 5 a^{4} d x - 10 a^{3} d x^{2} + 10 a^{2} d x^{3} - 5 a d x^{4} + b + d x^{5} - x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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