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

3.30.38.1 Optimal result

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

output

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)
 

3.30.38.2 Mathematica [F]

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

input

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]
 

output

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]
 

3.30.38.3 Rubi [F]

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.

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

\(\Big \downarrow \) 7270

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (-9 a \text {Subst}\left (\int \frac {x \sqrt [3]{x^3+a-b}}{d x^{12}+4 a \left (1-\frac {b}{a}\right ) d x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) d x^6-\left (1-4 (a-b)^3 d\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right ) d}dx,x,\sqrt [3]{x-a}\right )-9 \text {Subst}\left (\int \frac {x^4 \sqrt [3]{x^3+a-b}}{d x^{12}+4 a \left (1-\frac {b}{a}\right ) d x^9+6 a^2 \left (\frac {b (b-2 a)}{a^2}+1\right ) d x^6-\left (1-4 (a-b)^3 d\right ) x^3+a^4 \left (\frac {b \left (-4 a^3+6 b a^2-4 b^2 a+b^3\right )}{a^4}+1\right ) d}dx,x,\sqrt [3]{x-a}\right )+3 (4 a-b) \text {Subst}\left (\int \frac {x \sqrt [3]{x^3+a-b}}{d \left (x^3+a\right )^4-4 b d \left (x^3+a\right )^3+6 b^2 d \left (x^3+a\right )^2-\left (4 d b^3+1\right ) \left (x^3+a\right )+a \left (\frac {d b^4}{a}+1\right )}dx,x,\sqrt [3]{x-a}\right )\right )}{\sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\)

input

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]
 

output

$Aborted
 

3.30.38.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7270

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]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.30.38.4 Maple [F]

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

input

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)
 

output

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)
 

3.30.38.5 Fricas [A] (verification not implemented)

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 ] \]

input

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, algorithm="fricas")
 

output

[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]
 

3.30.38.6 Sympy [F(-1)]

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

input

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)
 

output

Timed out
 

3.30.38.7 Maxima [F]

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

input

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, algorithm="maxima")
 

output

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)
 

3.30.38.8 Giac [F]

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

input

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, algorithm="giac")
 

output

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)
 

3.30.38.9 Mupad [F(-1)]

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

input

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)
 

output

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)