3.24.17 \(\int \frac {5 x-4 (1+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} (-1+(1+k) x-k x^2+b x^5)} \, dx\) [2317]

3.24.17.1 Optimal result

Integrand size = 56, antiderivative size = 179 \[ \int \frac {5 x-4 (1+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^5\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x+(-1-k) x^2+k x^3}}{2 \sqrt [3]{b} x^2+\sqrt [3]{x+(-1-k) x^2+k x^3}}\right )}{\sqrt [3]{b}}+\frac {\log \left (-\sqrt [3]{b} x^2+\sqrt [3]{x+(-1-k) x^2+k x^3}\right )}{\sqrt [3]{b}}-\frac {\log \left (b^{2/3} x^4+\sqrt [3]{b} x^2 \sqrt [3]{x+(-1-k) x^2+k x^3}+\left (x+(-1-k) x^2+k x^3\right )^{2/3}\right )}{2 \sqrt [3]{b}} \]

output

3^(1/2)*arctan(3^(1/2)*(x+(-1-k)*x^2+k*x^3)^(1/3)/(2*b^(1/3)*x^2+(x+(-1-k) 
*x^2+k*x^3)^(1/3)))/b^(1/3)+ln(-b^(1/3)*x^2+(x+(-1-k)*x^2+k*x^3)^(1/3))/b^ 
(1/3)-1/2*ln(b^(2/3)*x^4+b^(1/3)*x^2*(x+(-1-k)*x^2+k*x^3)^(1/3)+(x+(-1-k)* 
x^2+k*x^3)^(2/3))/b^(1/3)
 

3.24.17.2 Mathematica [A] (verified)

Time = 11.71 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.80 \[ \int \frac {5 x-4 (1+k) x^2+3 k x^3}{\sqrt [3]{(1-x) x (1-k x)} \left (-1+(1+k) x-k x^2+b x^5\right )} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{(-1+x) x (-1+k x)}}{2 \sqrt [3]{b} x^2+\sqrt [3]{(-1+x) x (-1+k x)}}\right )+2 \log \left (-\sqrt [3]{b} x^2+\sqrt [3]{(-1+x) x (-1+k x)}\right )-\log \left (b^{2/3} x^4+\sqrt [3]{b} x^2 \sqrt [3]{(-1+x) x (-1+k x)}+((-1+x) x (-1+k x))^{2/3}\right )}{2 \sqrt [3]{b}} \]

input

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

output

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

3.24.17.3 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.

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

\(\Big \downarrow \) 2028

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \int \left (\frac {3 k x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+k x^2-(k+1) x+1\right )}+\frac {4 (-k-1) x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+k x^2-(k+1) x+1\right )}+\frac {5 x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+k x^2-(k+1) x+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{(1-x) x (1-k x)}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 \sqrt [3]{x} \sqrt [3]{k x^2-(k+1) x+1} \left (5 \int \frac {x^{4/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+k x^2-(k+1) x+1\right )}d\sqrt [3]{x}-4 (k+1) \int \frac {x^{7/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+k x^2-(k+1) x+1\right )}d\sqrt [3]{x}+3 k \int \frac {x^{10/3}}{\sqrt [3]{k x^2-(k+1) x+1} \left (-b x^5+k x^2-(k+1) x+1\right )}d\sqrt [3]{x}\right )}{\sqrt [3]{(1-x) x (1-k x)}}\)

input

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

output

$Aborted
 

3.24.17.3.1 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 2028

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)*(x_)^(t_.))^(p_.), 
x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r))^p*Fx, x] /; FreeQ[ 
{a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] &&  !(E 
qQ[p, 1] && EqQ[u, 1])
 

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] 
]
 

3.24.17.4 Maple [F]

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

input

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

output

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

3.24.17.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.24.17.6 Sympy [F]

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

input

integrate((5*x-4*(1+k)*x**2+3*k*x**3)/((1-x)*x*(-k*x+1))**(1/3)/(-1+(1+k)* 
x-k*x**2+b*x**5),x)
 

output

Integral(x*(3*k*x**2 - 4*k*x - 4*x + 5)/((x*(x - 1)*(k*x - 1))**(1/3)*(b*x 
**5 - k*x**2 + k*x + x - 1)), x)
 

3.24.17.7 Maxima [F]

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

input

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

output

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

3.24.17.8 Giac [F]

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

input

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

output

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

3.24.17.9 Mupad [F(-1)]

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

input

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

output

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