3.12.37 \(\int \frac {(-1-2 (-1+k) x+k x^2) (1-2 k x+k^2 x^2)}{((1-x) x (1-k x))^{3/4} (-d+(1+3 d k) x-(1+3 d k^2) x^2+d k^3 x^3)} \, dx\) [1137]

3.12.37.1 Optimal result

Integrand size = 79, antiderivative size = 85 \[ \int \frac {\left (-1-2 (-1+k) x+k x^2\right ) \left (1-2 k x+k^2 x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d k) x-\left (1+3 d k^2\right ) x^2+d k^3 x^3\right )} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{d} \left (x+(-1-k) x^2+k x^3\right )^{3/4}}{(-1+x) x}\right )}{d^{3/4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{d} \left (x+(-1-k) x^2+k x^3\right )^{3/4}}{(-1+x) x}\right )}{d^{3/4}} \]

output

2*arctan(d^(1/4)*(x+(-1-k)*x^2+k*x^3)^(3/4)/(-1+x)/x)/d^(3/4)-2*arctanh(d^ 
(1/4)*(x+(-1-k)*x^2+k*x^3)^(3/4)/(-1+x)/x)/d^(3/4)
 

3.12.37.2 Mathematica [F]

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

input

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

output

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

3.12.37.3 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.

\(\displaystyle \int \frac {\left (k x^2-2 (k-1) x-1\right ) \left (k^2 x^2-2 k x+1\right )}{((1-x) x (1-k x))^{3/4} \left (d k^3 x^3-x^2 \left (3 d k^2+1\right )+x (3 d k+1)-d\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{3/4} \left (k x^2-(k+1) x+1\right )^{3/4} \int \frac {\left (-k x^2-2 (1-k) x+1\right ) \left (k^2 x^2-2 k x+1\right )}{x^{3/4} \left (k x^2-(k+1) x+1\right )^{3/4} \left (-d k^3 x^3+\left (3 d k^2+1\right ) x^2-(3 d k+1) x+d\right )}dx}{((1-x) x (1-k x))^{3/4}}\)

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 1380

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

$Aborted
 

3.12.37.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1380

Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S 
imp[1/c^p   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
 

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

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] 
]
 

3.12.37.4 Maple [F]

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

input

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

output

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

3.12.37.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.12.37.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.12.37.7 Maxima [F]

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

input

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

output

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

3.12.37.8 Giac [F]

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

input

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

output

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

3.12.37.9 Mupad [F(-1)]

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

input

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

output

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