Integrand size = 67, antiderivative size = 319 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}{2 b^2-4 b x+2 x^2+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{d^{2/3}}+\frac {\log \left (b^2-2 b x+x^2-\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{d^{2/3}}-\frac {\log \left (b^4-4 b^3 x+6 b^2 x^2-4 b x^3+x^4+\left (b^2 \sqrt [3]{d}-2 b \sqrt [3]{d} x+\sqrt [3]{d} x^2\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{2 d^{2/3}} \]
3^(1/2)*arctan(3^(1/2)*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)+ln(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)-1/2*ln(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))/d^(2/3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 33.72 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.80 \[ \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 {(a-b) \left (\frac {b-x}{a-x}\right )^{2/3} (-a+x) \left (4 \text {RootSum}\left [-d+3 d \text {$\#$1}-3 d \text {$\#$1}^2+d \text {$\#$1}^3-a^3 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^4\&,\frac {6 \sqrt [3]{\frac {-b+x}{-a+x}}-2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \sqrt [3]{\text {$\#$1}}+2 \log \left (-\sqrt [3]{\frac {-b+x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \sqrt [3]{\text {$\#$1}}-\log \left (\left (\frac {-b+x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {-b+x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \sqrt [3]{\text {$\#$1}}}{-3 d+6 d \text {$\#$1}-3 d \text {$\#$1}^2+4 a^3 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3}\&\right ]+5 \text {RootSum}\left [-d+3 d \text {$\#$1}-3 d \text {$\#$1}^2+d \text {$\#$1}^3-a^3 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^4\&,\frac {-6 \sqrt [3]{\frac {-b+x}{-a+x}} \text {$\#$1}+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{4/3}-2 \log \left (-\sqrt [3]{\frac {-b+x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{4/3}+\log \left (\left (\frac {-b+x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {-b+x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{4/3}}{-3 d+6 d \text {$\#$1}-3 d \text {$\#$1}^2+4 a^3 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3}\&\right ]-\text {RootSum}\left [-d+3 d \text {$\#$1}-3 d \text {$\#$1}^2+d \text {$\#$1}^3-a^3 \text {$\#$1}^4+3 a^2 b \text {$\#$1}^4-3 a b^2 \text {$\#$1}^4+b^3 \text {$\#$1}^4\&,\frac {-6 \sqrt [3]{\frac {-b+x}{-a+x}} \text {$\#$1}^2+2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{\frac {b-x}{a-x}}}{\sqrt [3]{\text {$\#$1}}}}{\sqrt {3}}\right ) \text {$\#$1}^{7/3}-2 \log \left (-\sqrt [3]{\frac {-b+x}{-a+x}}+\sqrt [3]{\text {$\#$1}}\right ) \text {$\#$1}^{7/3}+\log \left (\left (\frac {-b+x}{-a+x}\right )^{2/3}+\sqrt [3]{\frac {-b+x}{-a+x}} \sqrt [3]{\text {$\#$1}}+\text {$\#$1}^{2/3}\right ) \text {$\#$1}^{7/3}}{-3 d+6 d \text {$\#$1}-3 d \text {$\#$1}^2+4 a^3 \text {$\#$1}^3-12 a^2 b \text {$\#$1}^3+12 a b^2 \text {$\#$1}^3-4 b^3 \text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [3]{(b-x)^2 (-a+x)}} \]
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]
((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 - 1 2*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))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(x-b) (-4 a+b+3 x)}{\sqrt [3]{(x-a) (x-b)^2} \left (a d+b^4-x \left (4 b^3+d\right )+6 b^2 x^2-4 b x^3+x^4\right )} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt [3]{x-a} (x-b)^{2/3} \int -\frac {(4 a-b-3 x) \sqrt [3]{x-b}}{\sqrt [3]{x-a} \left (b^4+6 x^2 b^2-4 x^3 b+x^4+a d-\left (4 b^3+d\right ) x\right )}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \frac {(4 a-b-3 x) \sqrt [3]{x-b}}{\sqrt [3]{x-a} \left (b^4+6 x^2 b^2-4 x^3 b+x^4+a d-\left (4 b^3+d\right ) x\right )}dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \int \left (\frac {\left (1-\frac {4 a}{b}\right ) \sqrt [3]{x-b} b}{\sqrt [3]{x-a} \left (-b^4-6 x^2 b^2+4 x^3 b-x^4-a d+\left (4 b^3+d\right ) x\right )}+\frac {3 x \sqrt [3]{x-b}}{\sqrt [3]{x-a} \left (-b^4-6 x^2 b^2+4 x^3 b-x^4-a d+\left (4 b^3+d\right ) x\right )}\right )dx}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (-9 a \text {Subst}\left (\int \frac {x \sqrt [3]{x^3+a-b}}{x^{12}+4 a \left (1-\frac {b}{a}\right ) x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) x^6+4 a^3 \left (1-\frac {4 b^3-12 a b^2+12 a^2 b+d}{4 a^3}\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right )}dx,x,\sqrt [3]{x-a}\right )-9 \text {Subst}\left (\int \frac {x^4 \sqrt [3]{x^3+a-b}}{x^{12}+4 a \left (1-\frac {b}{a}\right ) x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) x^6+4 a^3 \left (1-\frac {4 b^3-12 a b^2+12 a^2 b+d}{4 a^3}\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right )}dx,x,\sqrt [3]{x-a}\right )+3 (4 a-b) \text {Subst}\left (\int \frac {x \sqrt [3]{x^3+a-b}}{\left (\frac {a d}{b^4}+1\right ) b^4+6 \left (x^3+a\right )^2 b^2-4 \left (x^3+a\right )^3 b+\left (x^3+a\right )^4-\left (4 b^3+d\right ) \left (x^3+a\right )}dx,x,\sqrt [3]{x-a}\right )\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\) |
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]
3.29.98.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
\[\int \frac {\left (-b +x \right ) \left (-4 a +b +3 x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (b^{4}+a d -\left (4 b^{3}+d \right ) x +6 b^{2} x^{2}-4 b \,x^{3}+x^{4}\right )}d x\]
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)
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)
Time = 0.50 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.06 \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d\right )}}{3 \, {\left (b^{2} d - 2 \, b d x + d x^{2}\right )}}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d^{2} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + {\left (b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}\right )} {\left (d^{2}\right )}^{\frac {2}{3}}}{b^{4} - 4 \, b^{3} x + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} d}{b^{2} - 2 \, b x + x^{2}}\right )}{2 \, d^{2}} \]
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")
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
Timed out. \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\text {Timed out} \]
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)
\[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int { \frac {{\left (4 \, a - b - 3 \, x\right )} {\left (b - x\right )}}{{\left (b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} + a d - {\left (4 \, b^{3} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}}} \,d x } \]
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")
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)
\[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int { \frac {{\left (4 \, a - b - 3 \, x\right )} {\left (b - x\right )}}{{\left (b^{4} + 6 \, b^{2} x^{2} - 4 \, b x^{3} + x^{4} + a d - {\left (4 \, b^{3} + d\right )} x\right )} \left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}}} \,d x } \]
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")
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)
Timed out. \[ \int \frac {(-b+x) (-4 a+b+3 x)}{\sqrt [3]{(-a+x) (-b+x)^2} \left (b^4+a d-\left (4 b^3+d\right ) x+6 b^2 x^2-4 b x^3+x^4\right )} \, dx=\int -\frac {\left (b-x\right )\,\left (b-4\,a+3\,x\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (a\,d-4\,b\,x^3-x\,\left (4\,b^3+d\right )+b^4+x^4+6\,b^2\,x^2\right )} \,d x \]
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)