3.31.8 \(\int \frac {(a-2 b+x) (a^2-2 a x+x^2)}{\sqrt [3]{(-a+x) (-b+x)} (-b^2+a^4 d+2 (b-2 a^3 d) x+(-1+6 a^2 d) x^2-4 a d x^3+d x^4)} \, dx\)

Optimal. Leaf size=405 \[ \frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [3]{x (-a-b)+a b+x^2}-a \sqrt [6]{d} x\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}+a \sqrt [3]{x (-a-b)+a b+x^2}-a \sqrt [6]{d} x\right )}{2 d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {a^2}{\sqrt {3}}+\frac {2 \left (x (-a-b)+a b+x^2\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}}{(a-x)^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \left (x (-a-b)+a b+x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^3 \sqrt [6]{d}-a^2 \sqrt [6]{d} x\right )\right )}{4 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \left (x (-a-b)+a b+x^2\right )^{2/3}+a^2 \sqrt [3]{d} x^2+\sqrt [3]{x (-a-b)+a b+x^2} \left (a^2 \sqrt [6]{d} x-a^3 \sqrt [6]{d}\right )\right )}{4 d^{2/3}} \]

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Rubi [F]  time = 9.11, 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-2 b+x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((a - 2*b + x)*(a^2 - 2*a*x + x^2))/(((-a + x)*(-b + x))^(1/3)*(-b^2 + a^4*d + 2*(b - 2*a^3*d)*x + (-1 + 6
*a^2*d)*x^2 - 4*a*d*x^3 + d*x^4)),x]

[Out]

(-3*(a - 2*b)*(-a + x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^7/((a - b + x^3)^(1/3)*(a^2*(1 + (b*(-2*
a + b))/a^2) + (2*a - 2*b)*x^3 + x^6 - d*x^12)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(1/3) - (3*a*(-a +
x)^(1/3)*(-b + x)^(1/3)*Defer[Subst][Defer[Int][x^7/((a - b + x^3)^(1/3)*(a^2*(1 + (b*(-2*a + b))/a^2) + 2*a*(
1 - b/a)*x^3 + x^6 - d*x^12)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(1/3) - (3*(-a + x)^(1/3)*(-b + x)^(1
/3)*Defer[Subst][Defer[Int][x^10/((a - b + x^3)^(1/3)*(a^2*(1 + (b*(-2*a + b))/a^2) + 2*a*(1 - b/a)*x^3 + x^6
- d*x^12)), x], x, (-a + x)^(1/3)])/((a - x)*(b - x))^(1/3)

Rubi steps

\begin {align*} \int \frac {(a-2 b+x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx &=\int \frac {(-a+x)^2 (a-2 b+x)}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{5/3} (a-2 b+x)}{\sqrt [3]{-b+x} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+2 b-x) (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (b^2-a^4 d-2 \left (b-2 a^3 d\right ) x+\left (1-6 a^2 d\right ) x^2+4 a d x^3-d x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {2 \left (1-\frac {a}{2 b}\right ) b (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (b^2-a^4 d-2 \left (b-2 a^3 d\right ) x+\left (1-6 a^2 d\right ) x^2+4 a d x^3-d x^4\right )}+\frac {x (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x-\left (1-6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {x (-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x-\left (1-6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left ((-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {(-a+x)^{5/3}}{\sqrt [3]{-b+x} \left (b^2-a^4 d-2 \left (b-2 a^3 d\right ) x+\left (1-6 a^2 d\right ) x^2+4 a d x^3-d x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=-\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+2 a x^3-2 b x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+2 a x^3-2 b x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=-\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (a^2-2 a b+b^2+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=-\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (a+x^3\right )}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=-\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \left (\frac {a x^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}+\frac {x^{10}}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ &=-\frac {\left (3 \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}-\frac {\left (3 a \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+2 a \left (1-\frac {b}{a}\right ) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}+\frac {\left (3 (-a+2 b) \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{a-b+x^3} \left (a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right )+(2 a-2 b) x^3+x^6-d x^{12}\right )} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)}}\\ \end {align*}

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Mathematica [F]  time = 4.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a-2 b+x) \left (a^2-2 a x+x^2\right )}{\sqrt [3]{(-a+x) (-b+x)} \left (-b^2+a^4 d+2 \left (b-2 a^3 d\right ) x+\left (-1+6 a^2 d\right ) x^2-4 a d x^3+d x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((a - 2*b + x)*(a^2 - 2*a*x + x^2))/(((-a + x)*(-b + x))^(1/3)*(-b^2 + a^4*d + 2*(b - 2*a^3*d)*x + (
-1 + 6*a^2*d)*x^2 - 4*a*d*x^3 + d*x^4)),x]

[Out]

Integrate[((a - 2*b + x)*(a^2 - 2*a*x + x^2))/(((-a + x)*(-b + x))^(1/3)*(-b^2 + a^4*d + 2*(b - 2*a^3*d)*x + (
-1 + 6*a^2*d)*x^2 - 4*a*d*x^3 + d*x^4)), x]

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IntegrateAlgebraic [A]  time = 3.54, size = 405, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {a^2}{\sqrt {3}}-\frac {2 a x}{\sqrt {3}}+\frac {x^2}{\sqrt {3}}+\frac {2 \left (a b+(-a-b) x+x^2\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{(a-x)^2}\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [6]{d} x-a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{2 d^{2/3}}+\frac {\log \left (a^2 \sqrt [6]{d}-a \sqrt [6]{d} x+a \sqrt [3]{a b+(-a-b) x+x^2}\right )}{2 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2+\left (a^3 \sqrt [6]{d}-a^2 \sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{4 d^{2/3}}-\frac {\log \left (a^4 \sqrt [3]{d}-2 a^3 \sqrt [3]{d} x+a^2 \sqrt [3]{d} x^2+\left (-a^3 \sqrt [6]{d}+a^2 \sqrt [6]{d} x\right ) \sqrt [3]{a b+(-a-b) x+x^2}+a^2 \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{4 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a - 2*b + x)*(a^2 - 2*a*x + x^2))/(((-a + x)*(-b + x))^(1/3)*(-b^2 + a^4*d + 2*(b - 2*a^3
*d)*x + (-1 + 6*a^2*d)*x^2 - 4*a*d*x^3 + d*x^4)),x]

[Out]

-1/2*(Sqrt[3]*ArcTan[(a^2/Sqrt[3] - (2*a*x)/Sqrt[3] + x^2/Sqrt[3] + (2*(a*b + (-a - b)*x + x^2)^(2/3))/(Sqrt[3
]*d^(1/3)))/(a - x)^2])/d^(2/3) + Log[a^2*d^(1/6) - a*d^(1/6)*x - a*(a*b + (-a - b)*x + x^2)^(1/3)]/(2*d^(2/3)
) + Log[a^2*d^(1/6) - a*d^(1/6)*x + a*(a*b + (-a - b)*x + x^2)^(1/3)]/(2*d^(2/3)) - Log[a^4*d^(1/3) - 2*a^3*d^
(1/3)*x + a^2*d^(1/3)*x^2 + (a^3*d^(1/6) - a^2*d^(1/6)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + a^2*(a*b + (-a - b)
*x + x^2)^(2/3)]/(4*d^(2/3)) - Log[a^4*d^(1/3) - 2*a^3*d^(1/3)*x + a^2*d^(1/3)*x^2 + (-(a^3*d^(1/6)) + a^2*d^(
1/6)*x)*(a*b + (-a - b)*x + x^2)^(1/3) + a^2*(a*b + (-a - b)*x + x^2)^(2/3)]/(4*d^(2/3))

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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.

[In]

integrate((a-2*b+x)*(a^2-2*a*x+x^2)/((-a+x)*(-b+x))^(1/3)/(-b^2+a^4*d+2*(-2*a^3*d+b)*x+(6*a^2*d-1)*x^2-4*a*d*x
^3+d*x^4),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a - 2 \, b + x\right )}}{{\left (a^{4} d - 4 \, a d x^{3} + d x^{4} + {\left (6 \, a^{2} d - 1\right )} x^{2} - b^{2} - 2 \, {\left (2 \, a^{3} d - b\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]

integrate((a-2*b+x)*(a^2-2*a*x+x^2)/((-a+x)*(-b+x))^(1/3)/(-b^2+a^4*d+2*(-2*a^3*d+b)*x+(6*a^2*d-1)*x^2-4*a*d*x
^3+d*x^4),x, algorithm="giac")

[Out]

integrate((a^2 - 2*a*x + x^2)*(a - 2*b + x)/((a^4*d - 4*a*d*x^3 + d*x^4 + (6*a^2*d - 1)*x^2 - b^2 - 2*(2*a^3*d
- b)*x)*((a - x)*(b - x))^(1/3)), x)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (a -2 b +x \right ) \left (a^{2}-2 a x +x^{2}\right )}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-b^{2}+a^{4} d +2 \left (-2 a^{3} d +b \right ) x +\left (6 a^{2} d -1\right ) x^{2}-4 a d \,x^{3}+d \,x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-2*b+x)*(a^2-2*a*x+x^2)/((-a+x)*(-b+x))^(1/3)/(-b^2+a^4*d+2*(-2*a^3*d+b)*x+(6*a^2*d-1)*x^2-4*a*d*x^3+d*x
^4),x)

[Out]

int((a-2*b+x)*(a^2-2*a*x+x^2)/((-a+x)*(-b+x))^(1/3)/(-b^2+a^4*d+2*(-2*a^3*d+b)*x+(6*a^2*d-1)*x^2-4*a*d*x^3+d*x
^4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} - 2 \, a x + x^{2}\right )} {\left (a - 2 \, b + x\right )}}{{\left (a^{4} d - 4 \, a d x^{3} + d x^{4} + {\left (6 \, a^{2} d - 1\right )} x^{2} - b^{2} - 2 \, {\left (2 \, a^{3} d - b\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]

integrate((a-2*b+x)*(a^2-2*a*x+x^2)/((-a+x)*(-b+x))^(1/3)/(-b^2+a^4*d+2*(-2*a^3*d+b)*x+(6*a^2*d-1)*x^2-4*a*d*x
^3+d*x^4),x, algorithm="maxima")

[Out]

integrate((a^2 - 2*a*x + x^2)*(a - 2*b + x)/((a^4*d - 4*a*d*x^3 + d*x^4 + (6*a^2*d - 1)*x^2 - b^2 - 2*(2*a^3*d
- b)*x)*((a - x)*(b - x))^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a^2-2\,a\,x+x^2\right )\,\left (a-2\,b+x\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (x^2\,\left (6\,a^2\,d-1\right )+2\,x\,\left (b-2\,a^3\,d\right )+a^4\,d+d\,x^4-b^2-4\,a\,d\,x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a^2 - 2*a*x + x^2)*(a - 2*b + x))/(((a - x)*(b - x))^(1/3)*(x^2*(6*a^2*d - 1) + 2*x*(b - 2*a^3*d) + a^4*
d + d*x^4 - b^2 - 4*a*d*x^3)),x)

[Out]

int(((a^2 - 2*a*x + x^2)*(a - 2*b + x))/(((a - x)*(b - x))^(1/3)*(x^2*(6*a^2*d - 1) + 2*x*(b - 2*a^3*d) + a^4*
d + d*x^4 - b^2 - 4*a*d*x^3)), x)

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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.

[In]

integrate((a-2*b+x)*(a**2-2*a*x+x**2)/((-a+x)*(-b+x))**(1/3)/(-b**2+a**4*d+2*(-2*a**3*d+b)*x+(6*a**2*d-1)*x**2
-4*a*d*x**3+d*x**4),x)

[Out]

Timed out

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