Optimal. Leaf size=244 \[ \frac {\log \left (a^2-\sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}-2 a x+x^2\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}}{2 a^2+\sqrt [3]{d} \sqrt [3]{x (-a-b)+a b+x^2}-4 a x+2 x^2}\right )}{d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+6 a^2 x^2+d^{2/3} \left (x (-a-b)+a b+x^2\right )^{2/3}-4 a x^3+x^4\right )}{2 d^{2/3}} \]
________________________________________________________________________________________
Rubi [F] time = 7.40, 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 (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
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 (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+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 (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+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 (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+5 b-4 x) (-a+x)^{2/3}}{\sqrt [3]{-b+x} \left (a^5 \left (1-\frac {b d}{a^5}\right )+\left (-5 a^4+d\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-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 (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}+\frac {4 x (-a+x)^{2/3}}{\sqrt [3]{-b+x} \left (-a^5 \left (1-\frac {b d}{a^5}\right )+5 a^4 \left (1-\frac {d}{5 a^4}\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+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 (-a^5 \left (1-\frac {b d}{a^5}\right )+5 a^4 \left (1-\frac {d}{5 a^4}\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+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 (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 \left (1-\frac {d}{5 a^4}\right ) x+10 a^3 x^2-10 a^2 x^3+5 a x^4-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 d+b d-d x^3+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 d-b d+d x^3-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 ) d+d x^3-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 ) d+d x^3-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 ) d-d x^3+x^{15}\right )}+\frac {x^7}{\sqrt [3]{a-b+x^3} \left (-a \left (1-\frac {b}{a}\right ) d-d x^3+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 ) d+d x^3-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 ) d-d x^3+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 ) d-d x^3+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 ) d+d x^3-x^{15}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 3.87, 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 (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.67, size = 244, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )\right )}{2 d^{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 (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {-a \left (a -5 b \right )-\left (3 a +5 b \right ) x +4 x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{5}+b d -\left (-5 a^{4}+d \right ) x -10 a^{3} x^{2}+10 a^{2} x^{3}-5 a \,x^{4}+x^{5}\right )}\, dx\]
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 (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\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.
[In]
[Out]
________________________________________________________________________________________
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 (5\,a\,x^4-b\,d+x\,\left (d-5\,a^4\right )+a^5-x^5-10\,a^2\,x^3+10\,a^3\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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} + 5 a^{4} x - 10 a^{3} x^{2} + 10 a^{2} x^{3} - 5 a x^{4} + b d - d x + x^{5}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________