\(\int \frac {(1+2 x+x^2) (-2-(-1+k) (1+k) x+2 k^2 x^2)}{((1-x^2) (1-k^2 x^2))^{3/4} (1-d-(1+3 d) x-(3 d+k^2) x^2+(-d+k^2) x^3)} \, dx\) [1266]

Optimal result

Integrand size = 88, antiderivative size = 92 \[ \int \frac {\left (1+2 x+x^2\right ) \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{d}+\sqrt [4]{d} x}{\sqrt [4]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{d^{3/4}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(Sqrt[2]*k^(3/2)*Sqrt[-1 + k^2]*Sqrt[(-1 - k^2 + 2*k^2*x^2)^2]*Sqrt[(1 + k^2*(1 - 2*x^2))^2/((1 - k^2)^2*(1 -
(2*k*Sqrt[(1 - x^2)*(1 - k^2*x^2)])/(1 - k^2))^2)]*(1 - (2*k*Sqrt[(1 - x^2)*(1 - k^2*x^2)])/(1 - k^2))*Ellipti
cF[2*ArcTan[(Sqrt[2]*Sqrt[k]*((1 - x^2)*(1 - k^2*x^2))^(1/4))/Sqrt[-1 + k^2]], 1/2])/((d - k^2)*(1 + k^2 - 2*k
^2*x^2)*Sqrt[(-1 - k^2*(1 - 2*x^2))^2]) + ((k^2 + 5*k^4 - d*(1 - 3*k^2))*x*((1 - x^2)/(1 - k^2*x^2))^(3/4)*(1
- k^2*x^2)*Hypergeometric2F1[1/2, 3/4, 3/2, ((1 - k^2)*x^2)/(1 - k^2*x^2)])/((d - k^2)^2*(1 - (1 + k^2)*x^2 +
k^2*x^4)^(3/4)) - ((k^2 + 7*k^4 + 3*d^2*(1 - k^2) - d*(1 + 2*k^2 + 5*k^4))*Defer[Int][1/((1 - d - (1 + 3*d)*x
- (3*d + k^2)*x^2 - (d - k^2)*x^3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(3/4)), x])/(d - k^2)^2 + ((k^2 - k^6 - 6*d^
2*(1 - k^2) - d*(1 - 14*k^2 - 19*k^4))*Defer[Int][x/((1 - d - (1 + 3*d)*x - (3*d + k^2)*x^2 - (d - k^2)*x^3)*(
1 + (-1 - k^2)*x^2 + k^2*x^4)^(3/4)), x])/(d - k^2)^2 + ((24*d*k^4 - 3*d^2*(1 - k^2) + k^4*(3 + 5*k^2))*Defer[
Int][x^2/((1 - d - (1 + 3*d)*x - (3*d + k^2)*x^2 - (d - k^2)*x^3)*(1 + (-1 - k^2)*x^2 + k^2*x^4)^(3/4)), x])/(
d - k^2)^2

Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+x)^2 \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (\left (1-x^2\right ) \left (1-k^2 x^2\right )\right )^{3/4} \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right )} \, dx \\ & = \int \frac {(1+x)^2 \left (-2-(-1+k) (1+k) x+2 k^2 x^2\right )}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2+\left (-d+k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx \\ & = \int \frac {(1+x)^2 \left (-2+(1-k) (1+k) x+2 k^2 x^2\right )}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx \\ & = \int \left (\frac {k^2+5 k^4-d \left (1-3 k^2\right )}{\left (d-k^2\right )^2 \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}}-\frac {2 k^2 x}{\left (d-k^2\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}}+\frac {-3 d^2 \left (1-k^2\right )-k^2 \left (1+7 k^2\right )+d \left (1+2 k^2+5 k^4\right )+\left (k^2-k^6-6 d^2 \left (1-k^2\right )-d \left (1-14 k^2-19 k^4\right )\right ) x+\left (24 d k^4-3 d^2 \left (1-k^2\right )+k^4 \left (3+5 k^2\right )\right ) x^2}{\left (d-k^2\right )^2 \left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}}\right ) \, dx \\ & = \frac {\int \frac {-3 d^2 \left (1-k^2\right )-k^2 \left (1+7 k^2\right )+d \left (1+2 k^2+5 k^4\right )+\left (k^2-k^6-6 d^2 \left (1-k^2\right )-d \left (1-14 k^2-19 k^4\right )\right ) x+\left (24 d k^4-3 d^2 \left (1-k^2\right )+k^4 \left (3+5 k^2\right )\right ) x^2}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2}-\frac {\left (2 k^2\right ) \int \frac {x}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{d-k^2}+\frac {\left (k^2+5 k^4-d \left (1-3 k^2\right )\right ) \int \frac {1}{\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2} \\ & = \frac {\int \left (\frac {k^2 \left (-1-7 k^2\right ) \left (1-\frac {d \left (1+2 k^2+5 k^4+3 d \left (-1+k^2\right )\right )}{k^2+7 k^4}\right )}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}}+\frac {\left (-6 d^2 \left (1-k^2\right )-d \left (1-14 k^2-19 k^4\right )+k^2 \left (1-k^4\right )\right ) x}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}}+\frac {\left (24 d k^4-3 d^2 \left (1-k^2\right )+k^4 \left (3+5 k^2\right )\right ) x^2}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}}\right ) \, dx}{\left (d-k^2\right )^2}-\frac {k^2 \text {Subst}\left (\int \frac {1}{\left (1+\left (-1-k^2\right ) x+k^2 x^2\right )^{3/4}} \, dx,x,x^2\right )}{d-k^2}+\frac {\left (\left (k^2+5 k^4-d \left (1-3 k^2\right )\right ) \left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}\right ) \int \frac {1}{\left (1-x^2\right )^{3/4} \left (1-k^2 x^2\right )^{3/4}} \, dx}{\left (d-k^2\right )^2 \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \\ & = \frac {\left (k^2+5 k^4-d \left (1-3 k^2\right )\right ) x \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {\left (1-k^2\right ) x^2}{1-k^2 x^2}\right )}{\left (d-k^2\right )^2 \left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{3/4}}+\frac {\left (24 d k^4-3 d^2 \left (1-k^2\right )+k^4 \left (3+5 k^2\right )\right ) \int \frac {x^2}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2}+\frac {\left (k^2-k^6-6 d^2 \left (1-k^2\right )-d \left (1-14 k^2-19 k^4\right )\right ) \int \frac {x}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2}+\frac {\left (-3 d^2 \left (1-k^2\right )-k^2 \left (1+7 k^2\right )+d \left (1+2 k^2+5 k^4\right )\right ) \int \frac {1}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2}-\frac {\left (4 k^2 \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-4 k^2+\left (-1-k^2\right )^2+4 k^2 x^4}} \, dx,x,\sqrt [4]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}\right )}{\left (d-k^2\right ) \left (-1-k^2+2 k^2 x^2\right )} \\ & = \frac {\sqrt {2} k^{3/2} \sqrt {-1+k^2} \sqrt {\left (-1-k^2+2 k^2 x^2\right )^2} \sqrt {\frac {\left (1+k^2 \left (1-2 x^2\right )\right )^2}{\left (1-k^2\right )^2 \left (1-\frac {2 k \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{1-k^2}\right )^2}} \left (1-\frac {2 k \sqrt {\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{1-k^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt {k} \sqrt [4]{\left (1-x^2\right ) \left (1-k^2 x^2\right )}}{\sqrt {-1+k^2}}\right ),\frac {1}{2}\right )}{\left (d-k^2\right ) \left (1+k^2-2 k^2 x^2\right ) \sqrt {\left (-1-k^2 \left (1-2 x^2\right )\right )^2}}+\frac {\left (k^2+5 k^4-d \left (1-3 k^2\right )\right ) x \left (\frac {1-x^2}{1-k^2 x^2}\right )^{3/4} \left (1-k^2 x^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\frac {\left (1-k^2\right ) x^2}{1-k^2 x^2}\right )}{\left (d-k^2\right )^2 \left (1-\left (1+k^2\right ) x^2+k^2 x^4\right )^{3/4}}+\frac {\left (24 d k^4-3 d^2 \left (1-k^2\right )+k^4 \left (3+5 k^2\right )\right ) \int \frac {x^2}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2}+\frac {\left (k^2-k^6-6 d^2 \left (1-k^2\right )-d \left (1-14 k^2-19 k^4\right )\right ) \int \frac {x}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2}+\frac {\left (-3 d^2 \left (1-k^2\right )-k^2 \left (1+7 k^2\right )+d \left (1+2 k^2+5 k^4\right )\right ) \int \frac {1}{\left (1-d-(1+3 d) x-\left (3 d+k^2\right ) x^2-\left (d-k^2\right ) x^3\right ) \left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{3/4}} \, dx}{\left (d-k^2\right )^2} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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