3.27.92 \(\int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} (-a^5+b d-(-5 a^4+d) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5)} \, dx\) [2692]

3.27.92.1 Optimal result

Integrand size = 86, antiderivative size = 244 \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}}\right )}{d^{2/3}}+\frac {\log \left (a^2-2 a x+x^2-\sqrt [3]{d} \sqrt [3]{a b+(-a-b) x+x^2}\right )}{d^{2/3}}-\frac {\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+d^{2/3} \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{a b+(-a-b) x+x^2} \left (a^2 \sqrt [3]{d}-2 a \sqrt [3]{d} x+\sqrt [3]{d} x^2\right )\right )}{2 d^{2/3}} \]

output

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

3.27.92.2 Mathematica [A] (verified)

Time = 11.24 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.81 \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}}{2 a^2-4 a x+2 x^2+\sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}}\right )+2 \log \left (a^2-2 a x+x^2-\sqrt [3]{d} \sqrt [3]{(-a+x) (-b+x)}\right )-\log \left (a^4-4 a^3 x+6 a^2 x^2-4 a x^3+x^4+\sqrt [3]{d} (a-x)^2 \sqrt [3]{(-a+x) (-b+x)}+d^{2/3} ((-a+x) (-b+x))^{2/3}\right )}{2 d^{2/3}} \]

input

Integrate[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1 
/3)*(-a^5 + b*d - (-5*a^4 + d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5 
)),x]
 

output

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

3.27.92.3 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.

\(\displaystyle \int \frac {-x (3 a+5 b)-a (a-5 b)+4 x^2}{\sqrt [3]{(x-a) (x-b)} \left (-a^5-x \left (d-5 a^4\right )-10 a^3 x^2+10 a^2 x^3-5 a x^4+b d+x^5\right )} \, dx\)

\(\Big \downarrow \) 2048

\(\displaystyle \int \frac {-x (3 a+5 b)-a (a-5 b)+4 x^2}{\sqrt [3]{x (-a-b)+a b+x^2} \left (-a^5-x \left (d-5 a^4\right )-10 a^3 x^2+10 a^2 x^3-5 a x^4+b d+x^5\right )}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x (3 a+5 b)+a (a-5 b)-4 x^2}{\sqrt [3]{-x (a+b)+a b+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )+x \left (d-5 a^4\right )+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4 x^2}{\sqrt [3]{-x (a+b)+a b+x^2} \left (-\left (a^5 \left (1-\frac {b d}{a^5}\right )\right )+5 a^4 x \left (1-\frac {d}{5 a^4}\right )-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )}+\frac {x (3 a+5 b)}{\sqrt [3]{-x (a+b)+a b+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 x \left (1-\frac {d}{5 a^4}\right )+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}+\frac {a (a-5 b)}{\sqrt [3]{-x (a+b)+a b+x^2} \left (a^5 \left (1-\frac {b d}{a^5}\right )-5 a^4 x \left (1-\frac {d}{5 a^4}\right )+10 a^3 x^2-10 a^2 x^3+5 a x^4-x^5\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle a (a-5 b) \int \frac {1}{\sqrt [3]{x^2-(a+b) x+a b} \left (\left (1-\frac {b d}{a^5}\right ) a^5-5 \left (1-\frac {d}{5 a^4}\right ) x a^4+10 x^2 a^3-10 x^3 a^2+5 x^4 a-x^5\right )}dx+(3 a+5 b) \int \frac {x}{\sqrt [3]{x^2-(a+b) x+a b} \left (\left (1-\frac {b d}{a^5}\right ) a^5-5 \left (1-\frac {d}{5 a^4}\right ) x a^4+10 x^2 a^3-10 x^3 a^2+5 x^4 a-x^5\right )}dx+4 \int \frac {x^2}{\sqrt [3]{x^2-(a+b) x+a b} \left (-\left (\left (1-\frac {b d}{a^5}\right ) a^5\right )+5 \left (1-\frac {d}{5 a^4}\right ) x a^4-10 x^2 a^3+10 x^3 a^2-5 x^4 a+x^5\right )}dx\)

input

Int[(-(a*(a - 5*b)) - (3*a + 5*b)*x + 4*x^2)/(((-a + x)*(-b + x))^(1/3)*(- 
a^5 + b*d - (-5*a^4 + d)*x - 10*a^3*x^2 + 10*a^2*x^3 - 5*a*x^4 + x^5)),x]
 

output

$Aborted
 

3.27.92.3.1 Defintions of rubi rules used

rule 2009

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

rule 2048

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) 
, x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F 
reeQ[{a, b, c, d, e, n, p}, x]
 

rule 7292

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

rule 7293

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

3.27.92.4 Maple [F]

\[\int \frac {-a \left (a -5 b \right )-\left (3 a +5 b \right ) x +4 x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a^{5}+b d -\left (-5 a^{4}+d \right ) x -10 a^{3} x^{2}+10 a^{2} x^{3}-5 a \,x^{4}+x^{5}\right )}d x\]

input

int((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(-a^5+b*d-(-5*a^4 
+d)*x-10*a^3*x^2+10*a^2*x^3-5*a*x^4+x^5),x)
 

output

int((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(-a^5+b*d-(-5*a^4 
+d)*x-10*a^3*x^2+10*a^2*x^3-5*a*x^4+x^5),x)
 

3.27.92.5 Fricas [F(-1)]

Timed out. \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\text {Timed out} \]

input

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(-a^5+b*d-( 
-5*a^4+d)*x-10*a^3*x^2+10*a^2*x^3-5*a*x^4+x^5),x, algorithm="fricas")
 

output

Timed out
 

3.27.92.6 Sympy [F]

\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int \frac {\left (- a + x\right ) \left (a - 5 b + 4 x\right )}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )} \left (- a^{5} + 5 a^{4} x - 10 a^{3} x^{2} + 10 a^{2} x^{3} - 5 a x^{4} + b d - d x + x^{5}\right )}\, dx \]

input

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x**2)/((-a+x)*(-b+x))**(1/3)/(-a**5+b* 
d-(-5*a**4+d)*x-10*a**3*x**2+10*a**2*x**3-5*a*x**4+x**5),x)
 

output

Integral((-a + x)*(a - 5*b + 4*x)/(((-a + x)*(-b + x))**(1/3)*(-a**5 + 5*a 
**4*x - 10*a**3*x**2 + 10*a**2*x**3 - 5*a*x**4 + b*d - d*x + x**5)), x)
 

3.27.92.7 Maxima [F]

\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int { \frac {{\left (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}} \,d x } \]

input

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(-a^5+b*d-( 
-5*a^4+d)*x-10*a^3*x^2+10*a^2*x^3-5*a*x^4+x^5),x, algorithm="maxima")
 

output

integrate(((a - 5*b)*a + (3*a + 5*b)*x - 4*x^2)/((a^5 + 10*a^3*x^2 - 10*a^ 
2*x^3 + 5*a*x^4 - x^5 - b*d - (5*a^4 - d)*x)*((a - x)*(b - x))^(1/3)), x)
 

3.27.92.8 Giac [F]

\[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int { \frac {{\left (a - 5 \, b\right )} a + {\left (3 \, a + 5 \, b\right )} x - 4 \, x^{2}}{{\left (a^{5} + 10 \, a^{3} x^{2} - 10 \, a^{2} x^{3} + 5 \, a x^{4} - x^{5} - b d - {\left (5 \, a^{4} - d\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\right )^{\frac {1}{3}}} \,d x } \]

input

integrate((-a*(a-5*b)-(3*a+5*b)*x+4*x^2)/((-a+x)*(-b+x))^(1/3)/(-a^5+b*d-( 
-5*a^4+d)*x-10*a^3*x^2+10*a^2*x^3-5*a*x^4+x^5),x, algorithm="giac")
 

output

integrate(((a - 5*b)*a + (3*a + 5*b)*x - 4*x^2)/((a^5 + 10*a^3*x^2 - 10*a^ 
2*x^3 + 5*a*x^4 - x^5 - b*d - (5*a^4 - d)*x)*((a - x)*(b - x))^(1/3)), x)
 

3.27.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-a (a-5 b)-(3 a+5 b) x+4 x^2}{\sqrt [3]{(-a+x) (-b+x)} \left (-a^5+b d-\left (-5 a^4+d\right ) x-10 a^3 x^2+10 a^2 x^3-5 a x^4+x^5\right )} \, dx=\int \frac {-4\,x^2+\left (3\,a+5\,b\right )\,x+a\,\left (a-5\,b\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (5\,a\,x^4-b\,d+x\,\left (d-5\,a^4\right )+a^5-x^5-10\,a^2\,x^3+10\,a^3\,x^2\right )} \,d x \]

input

int((a*(a - 5*b) + x*(3*a + 5*b) - 4*x^2)/(((a - x)*(b - x))^(1/3)*(5*a*x^ 
4 - b*d + x*(d - 5*a^4) + a^5 - x^5 - 10*a^2*x^3 + 10*a^3*x^2)),x)
 

output

int((a*(a - 5*b) + x*(3*a + 5*b) - 4*x^2)/(((a - x)*(b - x))^(1/3)*(5*a*x^ 
4 - b*d + x*(d - 5*a^4) + a^5 - x^5 - 10*a^2*x^3 + 10*a^3*x^2)), x)