\(\int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} (a+b^4 d-(1+4 b^3 d) x+6 b^2 d x^2-4 b d x^3+d x^4)} \, dx\) [2938]

Optimal result

Integrand size = 72, antiderivative size = 347 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{2 b^2 \sqrt [3]{d}-4 b \sqrt [3]{d} x+2 \sqrt [3]{d} x^2+\sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (b^2 \sqrt {d}-2 b \sqrt {d} x+\sqrt {d} x^2-\sqrt [6]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (b^4 d-4 b^3 d x+6 b^2 d x^2-4 b d x^3+d x^4+\left (b^2 d^{2/3}-2 b d^{2/3} x+d^{2/3} x^2\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+\sqrt [3]{d} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]

[Out]

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

Rubi [F]

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

[In]

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

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

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 811, normalized size of antiderivative = 2.34 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - 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 {2}{3}} - 2 \, {\left (2 \, b^{3} d - 1\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} d - 2 \, b d x + d x^{2}\right )} + {\left (b^{4} d - 4 \, b^{3} d x + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4}\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 {2}{3}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 2 \, a}{b^{4} d + 6 \, b^{2} d x^{2} - 4 \, b d x^{3} + d x^{4} - {\left (4 \, b^{3} d + 1\right )} x + a}\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 {1}{3}} + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\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}}}{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 {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}}}{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}}\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 {1}{3}} + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\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}}}{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 {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}}}{b^{2} - 2 \, b x + x^{2}}\right )}{2 \, d}\right ] \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (a+b^4 d-\left (1+4 b^3 d\right ) x+6 b^2 d x^2-4 b d x^3+d x^4\right )} \, dx=\int -\frac {\left (b-x\right )\,\left (b-4\,a+3\,x\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a+b^4\,d+d\,x^4-x\,\left (4\,d\,b^3+1\right )+6\,b^2\,d\,x^2-4\,b\,d\,x^3\right )} \,d x \]

[In]

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

[Out]

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