\(\int \frac {(-1+x) (-1+k x) (3-2 (1+k) x+k x^2)}{x ((1-x) x (1-k x))^{3/4} (-1+(1+k) x-k x^2+d x^3)} \, dx\) [1329]

Optimal result

Integrand size = 62, antiderivative size = 96 \[ \int \frac {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x ((1-x) x (1-k x))^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx=\frac {4 \sqrt [4]{x+(-1-k) x^2+k x^3}}{x}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(4*(1 - x)^(3/4)*(1 - k*x)^(3/4)*AppellF1[-3/4, -1/4, -1/4, 1/4, x, k*x])/((1 - x)*x*(1 - k*x))^(3/4) + (8*k*(
1 - x)^(3/4)*x^(3/4)*(1 - k*x)^(3/4)*Defer[Subst][Defer[Int][(x^4*(1 - x^4)^(1/4)*(1 - k*x^4)^(1/4))/(1 - (1 +
k)*x^4 + k*x^8 - d*x^12), x], x, x^(1/4)])/((1 - x)*x*(1 - k*x))^(3/4) + (4*(1 + k)*(1 - x)^(3/4)*x^(3/4)*(1
- k*x)^(3/4)*Defer[Subst][Defer[Int][((1 - x^4)^(1/4)*(1 - k*x^4)^(1/4))/(-1 + (1 + k)*x^4 - k*x^8 + d*x^12),
x], x, x^(1/4)])/((1 - x)*x*(1 - k*x))^(3/4) + (12*d*(1 - x)^(3/4)*x^(3/4)*(1 - k*x)^(3/4)*Defer[Subst][Defer[
Int][(x^8*(1 - x^4)^(1/4)*(1 - k*x^4)^(1/4))/(-1 + (1 + k)*x^4 - k*x^8 + d*x^12), x], x, x^(1/4)])/((1 - x)*x*
(1 - k*x))^(3/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{(1-x)^{3/4} x^{7/4} (1-k x)^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}} \\ & = -\frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {\sqrt [4]{1-x} (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x^{7/4} (1-k x)^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {\sqrt [4]{1-x} \sqrt [4]{1-k x} \left (3-2 (1+k) x+k x^2\right )}{x^{7/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4} \left (3-2 (1+k) x^4+k x^8\right )}{x^4 \left (-1+(1+k) x^4-k x^8+d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \left (-\frac {3 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{x^4}+\frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4} \left (-1-k+2 k x^4-3 d x^8\right )}{1-(1+k) x^4+k x^8-d x^{12}}\right ) \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4} \left (-1-k+2 k x^4-3 d x^8\right )}{1-(1+k) x^4+k x^8-d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}-\frac {\left (12 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{x^4} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {4 (1-x)^{3/4} (1-k x)^{3/4} \operatorname {AppellF1}\left (-\frac {3}{4},-\frac {1}{4},-\frac {1}{4},\frac {1}{4},x,k x\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \left (\frac {2 k x^4 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{1-(1+k) x^4+k x^8-d x^{12}}+\frac {(1+k) \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}}+\frac {3 d x^8 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}}\right ) \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ & = \frac {4 (1-x)^{3/4} (1-k x)^{3/4} \operatorname {AppellF1}\left (-\frac {3}{4},-\frac {1}{4},-\frac {1}{4},\frac {1}{4},x,k x\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (12 d (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^8 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (8 k (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^4 \sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{1-(1+k) x^4+k x^8-d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 (1+k) (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \text {Subst}\left (\int \frac {\sqrt [4]{1-x^4} \sqrt [4]{1-k x^4}}{-1+(1+k) x^4-k x^8+d x^{12}} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int \frac {(-1+x) (-1+k x) \left (3-2 (1+k) x+k x^2\right )}{x ((1-x) x (1-k x))^{3/4} \left (-1+(1+k) x-k x^2+d x^3\right )} \, dx=\frac {4 \sqrt [4]{(-1+x) x (-1+k x)}}{x}+2 \sqrt [4]{d} \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{(-1+x) x (-1+k x)}}\right )-2 \sqrt [4]{d} \text {arctanh}\left (\frac {\sqrt [4]{d} x}{\sqrt [4]{(-1+x) x (-1+k x)}}\right ) \]

[In]

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

[Out]

(4*((-1 + x)*x*(-1 + k*x))^(1/4))/x + 2*d^(1/4)*ArcTan[(d^(1/4)*x)/((-1 + x)*x*(-1 + k*x))^(1/4)] - 2*d^(1/4)*
ArcTanh[(d^(1/4)*x)/((-1 + x)*x*(-1 + k*x))^(1/4)]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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