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

Optimal. Leaf size=319 \[ \frac {\log \left (-\sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}+b^2-2 b x+x^2\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}}{\sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}+2 b^2-4 b x+2 x^2}\right )}{d^{2/3}}-\frac {\log \left (d^{2/3} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}+\sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3} \left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )+b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4\right )}{2 d^{2/3}} \]

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Rubi [F]  time = 7.08, 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 {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(9*a*(-a + x)^(1/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x*(a - b + x^3)^(1/3))/(a^4*(1 + (b*(-4*a^3 + 6*a^
2*b - 4*a*b^2 + b^3))/a^4) + 4*a^3*(1 - (12*a^2*b - 12*a*b^2 + 4*b^3 + d)/(4*a^3))*x^3 + 6*a^2*(1 + (b*(-2*a +
b))/a^2)*x^6 + 4*a*(1 - b/a)*x^9 + x^12), x], x, (-a + x)^(1/3)])/(-((a - x)*(b - x)^2))^(1/3) + (9*(-a + x)^
(1/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x^4*(a - b + x^3)^(1/3))/(a^4*(1 + (b*(-4*a^3 + 6*a^2*b - 4*a*b^
2 + b^3))/a^4) + 4*a^3*(1 - (12*a^2*b - 12*a*b^2 + 4*b^3 + d)/(4*a^3))*x^3 + 6*a^2*(1 + (b*(-2*a + b))/a^2)*x^
6 + 4*a*(1 - b/a)*x^9 + x^12), x], x, (-a + x)^(1/3)])/(-((a - x)*(b - x)^2))^(1/3) - (3*(4*a - b)*(-a + x)^(1
/3)*(-b + x)^(2/3)*Defer[Subst][Defer[Int][(x*(a - b + x^3)^(1/3))/(b^4*(1 + (a*d)/b^4) - (4*b^3 + d)*(a + x^3
) + 6*b^2*(a + x^3)^2 - 4*b*(a + x^3)^3 + (a + x^3)^4), x], x, (-a + x)^(1/3)])/(-((a - x)*(b - x)^2))^(1/3)

Rubi steps

\begin {align*} \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x} (-4 a+b+3 x)}{\sqrt [3]{-a+x} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {4 a \left (1-\frac {b}{4 a}\right ) \sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-x^4\right )}+\frac {3 x \sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (3 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {x \sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left ((4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-b+x}}{\sqrt [3]{-a+x} \left (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-x^4\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \left (a+x^3\right ) \sqrt [3]{a-b+x^3}}{b^4+a d-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4+a d-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \left (a+x^3\right ) \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \left (\frac {a x \sqrt [3]{a-b+x^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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}}+\frac {x^4 \sqrt [3]{a-b+x^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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}}\right ) \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (9 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt [3]{a-b+x^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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left (9 a \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^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 )+4 a^3 \left (1-\frac {12 a^2 b-12 a b^2+4 b^3+d}{4 a^3}\right ) x^3+6 a^2 \left (1+\frac {b (-2 a+b)}{a^2}\right ) x^6+4 a \left (1-\frac {b}{a}\right ) x^9+x^{12}} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left (3 (4 a-b) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x \sqrt [3]{a-b+x^3}}{b^4 \left (1+\frac {a d}{b^4}\right )-\left (4 b^3+d\right ) \left (a+x^3\right )+6 b^2 \left (a+x^3\right )^2-4 b \left (a+x^3\right )^3+\left (a+x^3\right )^4} \, dx,x,\sqrt [3]{-a+x}\right )}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ \end {align*}

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Mathematica [C]  time = 1.76, size = 893, normalized size = 2.80 \begin {gather*} \frac {(a-b) \left (\frac {b-x}{a-x}\right )^{2/3} (x-a) \left (4 \text {RootSum}\left [-a^3 \text {$\#$1}^4+b^3 \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4+d \text {$\#$1}^3-3 d \text {$\#$1}^2+3 d \text {$\#$1}-d\&,\frac {-2 \sqrt {3} \sqrt [3]{\text {$\#$1}} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )+6 \sqrt [3]{\frac {x-b}{x-a}}+2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x-b}{x-a}}\right ) \sqrt [3]{\text {$\#$1}}-\log \left (\left (\frac {x-b}{x-a}\right )^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x-b}{x-a}}+\text {$\#$1}^{2/3}\right ) \sqrt [3]{\text {$\#$1}}}{4 a^3 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3-3 d \text {$\#$1}^2+6 d \text {$\#$1}-3 d}\&\right ]+5 \text {RootSum}\left [-a^3 \text {$\#$1}^4+b^3 \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4+d \text {$\#$1}^3-3 d \text {$\#$1}^2+3 d \text {$\#$1}-d\&,\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right ) \text {$\#$1}^{4/3}-2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x-b}{x-a}}\right ) \text {$\#$1}^{4/3}+\log \left (\left (\frac {x-b}{x-a}\right )^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x-b}{x-a}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{4/3}-6 \sqrt [3]{\frac {x-b}{x-a}} \text {$\#$1}}{4 a^3 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3-3 d \text {$\#$1}^2+6 d \text {$\#$1}-3 d}\&\right ]-\text {RootSum}\left [-a^3 \text {$\#$1}^4+b^3 \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4+d \text {$\#$1}^3-3 d \text {$\#$1}^2+3 d \text {$\#$1}-d\&,\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right ) \text {$\#$1}^{7/3}-2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {x-b}{x-a}}\right ) \text {$\#$1}^{7/3}+\log \left (\left (\frac {x-b}{x-a}\right )^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {x-b}{x-a}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{7/3}-6 \sqrt [3]{\frac {x-b}{x-a}} \text {$\#$1}^2}{4 a^3 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3-3 d \text {$\#$1}^2+6 d \text {$\#$1}-3 d}\&\right ]\right )}{2 \sqrt [3]{(b-x)^2 (x-a)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*((b - x)/(a - x))^(2/3)*(-a + x)*(4*RootSum[-d + 3*d*#1 - 3*d*#1^2 + d*#1^3 - a^3*#1^4 + 3*a^2*b*#1^4
- 3*a*b^2*#1^4 + b^3*#1^4 & , (6*((-b + x)/(-a + x))^(1/3) - 2*Sqrt[3]*ArcTan[(1 + (2*((b - x)/(a - x))^(1/3)
)/#1^(1/3))/Sqrt[3]]*#1^(1/3) + 2*Log[-((-b + x)/(-a + x))^(1/3) + #1^(1/3)]*#1^(1/3) - Log[((-b + x)/(-a + x)
)^(2/3) + ((-b + x)/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(1/3))/(-3*d + 6*d*#1 - 3*d*#1^2 + 4*a^3*#1^3 - 12
*a^2*b*#1^3 + 12*a*b^2*#1^3 - 4*b^3*#1^3) & ] + 5*RootSum[-d + 3*d*#1 - 3*d*#1^2 + d*#1^3 - a^3*#1^4 + 3*a^2*b
*#1^4 - 3*a*b^2*#1^4 + b^3*#1^4 & , (-6*((-b + x)/(-a + x))^(1/3)*#1 + 2*Sqrt[3]*ArcTan[(1 + (2*((b - x)/(a -
x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(4/3) - 2*Log[-((-b + x)/(-a + x))^(1/3) + #1^(1/3)]*#1^(4/3) + Log[((-b + x)
/(-a + x))^(2/3) + ((-b + x)/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(4/3))/(-3*d + 6*d*#1 - 3*d*#1^2 + 4*a^3*
#1^3 - 12*a^2*b*#1^3 + 12*a*b^2*#1^3 - 4*b^3*#1^3) & ] - RootSum[-d + 3*d*#1 - 3*d*#1^2 + d*#1^3 - a^3*#1^4 +
3*a^2*b*#1^4 - 3*a*b^2*#1^4 + b^3*#1^4 & , (-6*((-b + x)/(-a + x))^(1/3)*#1^2 + 2*Sqrt[3]*ArcTan[(1 + (2*((b -
x)/(a - x))^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(7/3) - 2*Log[-((-b + x)/(-a + x))^(1/3) + #1^(1/3)]*#1^(7/3) + Log[
((-b + x)/(-a + x))^(2/3) + ((-b + x)/(-a + x))^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(7/3))/(-3*d + 6*d*#1 - 3*d*#1^2
+ 4*a^3*#1^3 - 12*a^2*b*#1^3 + 12*a*b^2*#1^3 - 4*b^3*#1^3) & ]))/(2*((b - x)^2*(-a + x))^(1/3))

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IntegrateAlgebraic [A]  time = 3.12, size = 319, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{d^{2/3}}+\frac {\log \left (b^2-2 b x+x^2-\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{d^{2/3}}-\frac {\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.99, size = 337, normalized size = 1.06 \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} - 2 \, b x + 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}} d\right )}}{3 \, {\left (b^{2} d - 2 \, b d x + d 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 {2}{3}} d^{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 (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} {\left (d^{2}\right )}^{\frac {2}{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 {1}{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.

[In]

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

[Out]

1/2*(2*sqrt(3)*(d^2)^(1/6)*d*arctan(1/3*sqrt(3)*(d^2)^(1/6)*((b^2 - 2*b*x + x^2)*(d^2)^(1/3) + 2*(-a*b^2 - (a
+ 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*d)/(b^2*d - 2*b*d*x + d*x^2)) - (d^2)^(2/3)*log(((-a*b^2 - (a + 2*b)
*x^2 + x^3 + (2*a*b + b^2)*x)^(2/3)*d^2 + (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*a*b + b^2)*x)^(1/3)*(b^2*d - 2*b*
d*x + d*x^2)*(d^2)^(1/3) + (b^4 - 4*b^3*x + 6*b^2*x^2 - 4*b*x^3 + x^4)*(d^2)^(2/3))/(b^4 - 4*b^3*x + 6*b^2*x^2
- 4*b*x^3 + x^4)) + 2*(d^2)^(2/3)*log(-((b^2 - 2*b*x + x^2)*(d^2)^(1/3) - (-a*b^2 - (a + 2*b)*x^2 + x^3 + (2*
a*b + b^2)*x)^(1/3)*d)/(b^2 - 2*b*x + x^2)))/d^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (-4 a +b +3 x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (b^{4}+a d -\left (4 b^{3}+d \right ) x +6 b^{2} x^{2}-4 b \,x^{3}+x^{4}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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