3.29.44 \(\int \frac {(-4 a+b+3 x) (-b^3+3 b^2 x-3 b x^2+x^3)}{((-a+x) (-b+x)^2)^{2/3} (b^4+a d-(4 b^3+d) x+6 b^2 x^2-4 b x^3+x^4)} \, dx\)

Optimal. Leaf size=291 \[ -\frac {\log \left (a^2 d+\left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3} \left (d^{2/3} x-a d^{2/3}\right )+\sqrt [3]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{4/3}-2 a d x+d x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [6]{d} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}+a \sqrt {d}-\sqrt {d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}}{\left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]

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Rubi [F]  time = 7.49, 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 (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[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(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)^(2/3)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][(a - b + x^3)^(5/3)/(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))^(2/3) + (9*(-a + x)^(2/3
)*(-b + x)^(4/3)*Defer[Subst][Defer[Int][(x^3*(a - b + x^3)^(5/3))/(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))^(2/3) - (3*(4*a - b)*(-a + x)^(2/3)*
(-b + x)^(4/3)*Defer[Subst][Defer[Int][(a - b + x^3)^(5/3)/(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))^(2/3)

Rubi steps

\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 (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 ((-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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (-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 (-b+x)^{5/3}}{(-a+x)^{2/3} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )}\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 (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+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 (-b^4-a d+\left (4 b^3+d\right ) x-6 b^2 x^2+4 b x^3-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}}{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 )}{\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}}{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 )}{\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}}{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 )}{\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}}{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 )}{\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 )+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^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 )+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 )}{\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}}{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 )}{\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 )+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 )}{\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 )+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 )}{\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}}{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 )}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 2.88, 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 (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]

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

[Out]

Integrate[((-4*a + b + 3*x)*(-b^3 + 3*b^2*x - 3*b*x^2 + x^3))/(((-a + x)*(-b + x)^2)^(2/3)*(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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IntegrateAlgebraic [A]  time = 3.11, size = 291, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt {d}-\sqrt {d} x+\sqrt [6]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d-2 a d x+d x^2+\left (-a d^{2/3}+d^{2/3} x\right ) \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{4/3}\right )}{2 \sqrt [3]{d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 1.97, size = 798, normalized size = 2.74 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} - 2 \, a d - 2 \, {\left (2 \, b^{3} - d\right )} x + \sqrt {3} {\left ({\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 )} d^{\frac {2}{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 {2}{3}} d + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} d^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 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}} {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {1}{3}}}{b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} + a d - {\left (4 \, b^{3} + d\right )} x}\right ) - d^{\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 )} d^{\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} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} d^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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^{\frac {2}{3}}\right )}}{3 \, {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\frac {1}{3}}}\right ) + d^{\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 )} d^{\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} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} d^{\frac {1}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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} + 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 {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((-4*a+b+3*x)*(-b^3+3*b^2*x-3*b*x^2+x^3)/((-a+x)*(-b+x)^2)^(2/3)/(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 (b^{3} - 3 \, b^{2} x + 3 \, b x^{2} - x^{3}\right )} {\left (4 \, a - b - 3 \, 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 {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

int(-((b - 4*a + 3*x)*(3*b*x^2 - 3*b^2*x + b^3 - x^3))/((-(a - x)*(b - x)^2)^(2/3)*(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((-4*a+b+3*x)*(-b**3+3*b**2*x-3*b*x**2+x**3)/((-a+x)*(-b+x)**2)**(2/3)/(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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