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

Optimal. Leaf size=89 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{d} x-\sqrt [4]{d}}{\sqrt [4]{k x^3+(-k-1) x^2+x}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{d} x-\sqrt [4]{d}}{\sqrt [4]{k x^3+(-k-1) x^2+x}}\right )}{d^{3/4}} \]

________________________________________________________________________________________

Rubi [F]  time = 13.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(1 - x)^(3/4)*x^(3/4)*(1 - k*x)^(3/4)*Defer[Subst][Defer[Int][(1 - x^4)^(5/4)/((1 - k*x^4)^(3/4)*(-x^4 + k*
x^8 - d*(-1 + x^4)^3)), x], x, x^(1/4)])/((1 - x)*x*(1 - k*x))^(3/4) + (8*(1 - k)*(1 - x)^(3/4)*x^(3/4)*(1 - k
*x)^(3/4)*Defer[Subst][Defer[Int][(x^4*(1 - x^4)^(5/4))/((1 - k*x^4)^(3/4)*(-x^4 + k*x^8 - d*(-1 + x^4)^3)), x
], x, x^(1/4)])/((1 - x)*x*(1 - k*x))^(3/4) + (4*k*(1 - x)^(3/4)*x^(3/4)*(1 - k*x)^(3/4)*Defer[Subst][Defer[In
t][(x^8*(1 - x^4)^(5/4))/((1 - k*x^4)^(3/4)*(x^4 - k*x^8 + d*(-1 + x^4)^3)), x], x, x^(1/4)])/((1 - x)*x*(1 -
k*x))^(3/4)

Rubi steps

\begin {align*} \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx &=\int \frac {(-1+x)^2 \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx\\ &=\frac {\left ((1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \int \frac {(-1+x)^2 \left (-1+2 (-1+k) x+k x^2\right )}{(1-x)^{3/4} x^{3/4} (1-k x)^{3/4} \left (-d+(1+3 d) x-(3 d+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 {(1-x)^{5/4} \left (-1+2 (-1+k) x+k x^2\right )}{x^{3/4} (1-k x)^{3/4} \left (-d+(1+3 d) x-(3 d+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 ) \operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4} \left (-1+2 (-1+k) x^4+k x^8\right )}{\left (1-k x^4\right )^{3/4} \left (-d+(1+3 d) x^4-(3 d+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 ) \operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4} \left (-1+2 (-1+k) x^4+k x^8\right )}{\left (1-k x^4\right )^{3/4} \left (x^4-k x^8+d \left (-1+x^4\right )^3\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 ) \operatorname {Subst}\left (\int \left (\frac {\left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )}+\frac {2 (1-k) x^4 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )}+\frac {k x^8 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-d+(1+3 d) x^4-3 d \left (1+\frac {k}{3 d}\right ) x^8+d x^{12}\right )}\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 ) \operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (8 (1-k) (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (d-(1+3 d) x^4+3 d \left (1+\frac {k}{3 d}\right ) x^8-d x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 k (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-d+(1+3 d) x^4-3 d \left (1+\frac {k}{3 d}\right ) 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 ) \operatorname {Subst}\left (\int \frac {\left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-x^4+k x^8-d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (8 (1-k) (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (-x^4+k x^8-d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}+\frac {\left (4 k (1-x)^{3/4} x^{3/4} (1-k x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8 \left (1-x^4\right )^{5/4}}{\left (1-k x^4\right )^{3/4} \left (x^4-k x^8+d \left (-1+x^4\right )^3\right )} \, dx,x,\sqrt [4]{x}\right )}{((1-x) x (1-k x))^{3/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 1.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-2 x+x^2\right ) \left (-1+2 (-1+k) x+k x^2\right )}{((1-x) x (1-k x))^{3/4} \left (-d+(1+3 d) x-(3 d+k) x^2+d x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.27, size = 89, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right )}{d^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {-\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{x+(-1-k) x^2+k x^3}}\right )}{d^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (k x^{2} + 2 \, {\left (k - 1\right )} x - 1\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left (d x^{3} - {\left (3 \, d + k\right )} x^{2} + {\left (3 \, d + 1\right )} x - d\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-2 x +1\right ) \left (-1+2 \left (-1+k \right ) x +k \,x^{2}\right )}{\left (\left (1-x \right ) x \left (-k x +1\right )\right )^{\frac {3}{4}} \left (-d +\left (1+3 d \right ) x -\left (3 d +k \right ) x^{2}+d \,x^{3}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (k x^{2} + 2 \, {\left (k - 1\right )} x - 1\right )} {\left (x^{2} - 2 \, x + 1\right )}}{{\left (d x^{3} - {\left (3 \, d + k\right )} x^{2} + {\left (3 \, d + 1\right )} x - d\right )} \left ({\left (k x - 1\right )} {\left (x - 1\right )} x\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^2-2\,x+1\right )\,\left (2\,x\,\left (k-1\right )+k\,x^2-1\right )}{{\left (x\,\left (k\,x-1\right )\,\left (x-1\right )\right )}^{3/4}\,\left (-d\,x^3+\left (3\,d+k\right )\,x^2+\left (-3\,d-1\right )\,x+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________