3.22.97 \(\int \frac {x^5 (8-7 (1+k) x+6 k x^2)}{((1-x) x (1-k x))^{2/3} (-b+b (1+k) x-b k x^2+x^8)} \, dx\) [2197]

3.22.97.1 Optimal result

Integrand size = 57, antiderivative size = 162 \[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^3}{x^3+2 \sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (x^3-\sqrt [3]{b} \sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (x^6+\sqrt [3]{b} x^3 \sqrt [3]{x+(-1-k) x^2+k x^3}+b^{2/3} \left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]

output

3^(1/2)*arctan(3^(1/2)*x^3/(x^3+2*b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(1/3)))/b^( 
1/3)+ln(x^3-b^(1/3)*(x+(-1-k)*x^2+k*x^3)^(1/3))/b^(1/3)-1/2*ln(x^6+b^(1/3) 
*x^3*(x+(-1-k)*x^2+k*x^3)^(1/3)+b^(2/3)*(x+(-1-k)*x^2+k*x^3)^(2/3))/b^(1/3 
)
 

3.22.97.2 Mathematica [F]

\[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx \]

input

Integrate[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*( 
-b + b*(1 + k)*x - b*k*x^2 + x^8)),x]
 

output

Integrate[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*( 
-b + b*(1 + k)*x - b*k*x^2 + x^8)), x]
 

3.22.97.3 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.

\(\displaystyle \int \frac {x^5 \left (6 k x^2-7 (k+1) x+8\right )}{((1-x) x (1-k x))^{2/3} \left (-b k x^2+b (k+1) x-b+x^8\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int -\frac {x^{13/3} \left (6 k x^2-7 (k+1) x+8\right )}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}dx}{((1-x) x (1-k x))^{2/3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {x^{13/3} \left (6 k x^2-7 (k+1) x+8\right )}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}dx}{((1-x) x (1-k x))^{2/3}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \frac {x^5 \left (6 k x^2-7 (k+1) x+8\right )}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \int \left (\frac {6 k x^7}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}+\frac {7 (-k-1) x^6}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}+\frac {8 x^5}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}\right )d\sqrt [3]{x}}{((1-x) x (1-k x))^{2/3}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 x^{2/3} \left (k x^2-(k+1) x+1\right )^{2/3} \left (6 k \int \frac {x^7}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}d\sqrt [3]{x}-7 (k+1) \int \frac {x^6}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}d\sqrt [3]{x}+8 \int \frac {x^5}{\left (k x^2-(k+1) x+1\right )^{2/3} \left (-x^8+b k x^2-b (k+1) x+b\right )}d\sqrt [3]{x}\right )}{((1-x) x (1-k x))^{2/3}}\)

input

Int[(x^5*(8 - 7*(1 + k)*x + 6*k*x^2))/(((1 - x)*x*(1 - k*x))^(2/3)*(-b + b 
*(1 + k)*x - b*k*x^2 + x^8)),x]
 

output

$Aborted
 

3.22.97.3.1 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 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7293

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

3.22.97.4 Maple [F]

\[\int \frac {x^{5} \left (8-7 \left (1+k \right ) x +6 k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {2}{3}} \left (-b +b \left (1+k \right ) x -b k \,x^{2}+x^{8}\right )}d x\]

input

int(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x-b*k*x 
^2+x^8),x)
 

output

int(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x-b*k*x 
^2+x^8),x)
 

3.22.97.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\text {Timed out} \]

input

integrate(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x 
-b*k*x^2+x^8),x, algorithm="fricas")
 

output

Timed out
 

3.22.97.6 Sympy [F]

\[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int \frac {x^{5} \cdot \left (6 k x^{2} - 7 k x - 7 x + 8\right )}{\left (x \left (x - 1\right ) \left (k x - 1\right )\right )^{\frac {2}{3}} \left (- b k x^{2} + b k x + b x - b + x^{8}\right )}\, dx \]

input

integrate(x**5*(8-7*(1+k)*x+6*k*x**2)/((1-x)*x*(-k*x+1))**(2/3)/(-b+b*(1+k 
)*x-b*k*x**2+x**8),x)
 

output

Integral(x**5*(6*k*x**2 - 7*k*x - 7*x + 8)/((x*(x - 1)*(k*x - 1))**(2/3)*( 
-b*k*x**2 + b*k*x + b*x - b + x**8)), x)
 

3.22.97.7 Maxima [F]

\[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int { \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{5}}{{\left (x^{8} - b k x^{2} + b {\left (k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}}} \,d x } \]

input

integrate(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x 
-b*k*x^2+x^8),x, algorithm="maxima")
 

output

integrate((6*k*x^2 - 7*(k + 1)*x + 8)*x^5/((x^8 - b*k*x^2 + b*(k + 1)*x - 
b)*((k*x - 1)*(x - 1)*x)^(2/3)), x)
 

3.22.97.8 Giac [F]

\[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=\int { \frac {{\left (6 \, k x^{2} - 7 \, {\left (k + 1\right )} x + 8\right )} x^{5}}{{\left (x^{8} - b k x^{2} + b {\left (k + 1\right )} x - b\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {2}{3}}} \,d x } \]

input

integrate(x^5*(8-7*(1+k)*x+6*k*x^2)/((1-x)*x*(-k*x+1))^(2/3)/(-b+b*(1+k)*x 
-b*k*x^2+x^8),x, algorithm="giac")
 

output

integrate((6*k*x^2 - 7*(k + 1)*x + 8)*x^5/((x^8 - b*k*x^2 + b*(k + 1)*x - 
b)*((k*x - 1)*(x - 1)*x)^(2/3)), x)
 

3.22.97.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (8-7 (1+k) x+6 k x^2\right )}{((1-x) x (1-k x))^{2/3} \left (-b+b (1+k) x-b k x^2+x^8\right )} \, dx=-\int \frac {x^5\,\left (6\,k\,x^2-7\,x\,\left (k+1\right )+8\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{2/3}\,\left (-x^8+b\,k\,x^2-b\,\left (k+1\right )\,x+b\right )} \,d x \]

input

int(-(x^5*(6*k*x^2 - 7*x*(k + 1) + 8))/((x*(k*x - 1)*(x - 1))^(2/3)*(b - x 
^8 - b*x*(k + 1) + b*k*x^2)),x)
 

output

-int((x^5*(6*k*x^2 - 7*x*(k + 1) + 8))/((x*(k*x - 1)*(x - 1))^(2/3)*(b - x 
^8 - b*x*(k + 1) + b*k*x^2)), x)