3.6.84 \(\int \frac {(-1+x)^2 (-10-8 x+5 x^2+5 x^3)}{(\frac {1+x}{-2+x^2})^{3/4} (-2+x^2) (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6)} \, dx\)

Optimal. Leaf size=45 \[ 2 \tan ^{-1}\left (\frac {x-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right )-2 \tanh ^{-1}\left (\frac {x-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right ) \]

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Rubi [F]  time = 14.35, 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 {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x)^2*(-10 - 8*x + 5*x^2 + 5*x^3))/(((1 + x)/(-2 + x^2))^(3/4)*(-2 + x^2)*(-3 + 7*x - 11*x^2 + 4*x^3
+ 4*x^4 - 4*x^5 + x^6)),x]

[Out]

(16*(1 + x)^(3/4)*Defer[Subst][Defer[Int][1/((-1 - 2*x^4 + x^8)^(1/4)*(16 + x^4 - 56*x^8 + 72*x^12 - 39*x^16 +
10*x^20 - x^24)), x], x, (1 + x)^(1/4)])/((-((1 + x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/4)) - (16*(1 + x)^(3/4)*
Defer[Subst][Defer[Int][1/((-1 - 2*x^4 + x^8)^(1/4)*(-16 - x^4 + 56*x^8 - 72*x^12 + 39*x^16 - 10*x^20 + x^24))
, x], x, (1 + x)^(1/4)])/((-((1 + x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/4)) - (16*(1 + x)^(3/4)*Defer[Subst][Defe
r[Int][x^4/((-1 - 2*x^4 + x^8)^(1/4)*(-16 - x^4 + 56*x^8 - 72*x^12 + 39*x^16 - 10*x^20 + x^24)), x], x, (1 + x
)^(1/4)])/((-((1 + x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/4)) - (120*(1 + x)^(3/4)*Defer[Subst][Defer[Int][x^8/((-
1 - 2*x^4 + x^8)^(1/4)*(-16 - x^4 + 56*x^8 - 72*x^12 + 39*x^16 - 10*x^20 + x^24)), x], x, (1 + x)^(1/4)])/((-(
(1 + x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/4)) + (228*(1 + x)^(3/4)*Defer[Subst][Defer[Int][x^12/((-1 - 2*x^4 + x
^8)^(1/4)*(-16 - x^4 + 56*x^8 - 72*x^12 + 39*x^16 - 10*x^20 + x^24)), x], x, (1 + x)^(1/4)])/((-((1 + x)/(2 -
x^2)))^(3/4)*(-2 + x^2)^(3/4)) - (120*(1 + x)^(3/4)*Defer[Subst][Defer[Int][x^16/((-1 - 2*x^4 + x^8)^(1/4)*(-1
6 - x^4 + 56*x^8 - 72*x^12 + 39*x^16 - 10*x^20 + x^24)), x], x, (1 + x)^(1/4)])/((-((1 + x)/(2 - x^2)))^(3/4)*
(-2 + x^2)^(3/4)) + (20*(1 + x)^(3/4)*Defer[Subst][Defer[Int][x^20/((-1 - 2*x^4 + x^8)^(1/4)*(-16 - x^4 + 56*x
^8 - 72*x^12 + 39*x^16 - 10*x^20 + x^24)), x], x, (1 + x)^(1/4)])/((-((1 + x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/
4))

Rubi steps

\begin {align*} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx &=\frac {(1+x)^{3/4} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=\frac {(1+x)^{3/4} \int \left (-\frac {10}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {12 x}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {11 x^2}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}-\frac {13 x^3}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}-\frac {5 x^4}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}+\frac {5 x^5}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )}\right ) \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=-\frac {\left (5 (1+x)^{3/4}\right ) \int \frac {x^4}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (5 (1+x)^{3/4}\right ) \int \frac {x^5}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (10 (1+x)^{3/4}\right ) \int \frac {1}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (11 (1+x)^{3/4}\right ) \int \frac {x^2}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (12 (1+x)^{3/4}\right ) \int \frac {x}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (13 (1+x)^{3/4}\right ) \int \frac {x^3}{(1+x)^{3/4} \sqrt [4]{-2+x^2} \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^5}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^2}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {-1+x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^3}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {4 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {6 x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {4 x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )}+\frac {5 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {10 x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {10 x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {5 x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^{20}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {2 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )}+\frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )}+\frac {3 x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}-\frac {3 x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}+\frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ &=\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (20 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{20}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (40 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (44 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (48 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-2 x^4+x^8} \left (16+x^4-56 x^8+72 x^{12}-39 x^{16}+10 x^{20}-x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (52 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (80 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (80 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (88 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (100 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (100 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{16}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (120 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (156 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (156 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}-\frac {\left (200 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}+\frac {\left (200 (1+x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{12}}{\sqrt [4]{-1-2 x^4+x^8} \left (-16-x^4+56 x^8-72 x^{12}+39 x^{16}-10 x^{20}+x^{24}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 2.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-1+x)^2 \left (-10-8 x+5 x^2+5 x^3\right )}{\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right ) \left (-3+7 x-11 x^2+4 x^3+4 x^4-4 x^5+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x)^2*(-10 - 8*x + 5*x^2 + 5*x^3))/(((1 + x)/(-2 + x^2))^(3/4)*(-2 + x^2)*(-3 + 7*x - 11*x^2 +
4*x^3 + 4*x^4 - 4*x^5 + x^6)),x]

[Out]

Integrate[((-1 + x)^2*(-10 - 8*x + 5*x^2 + 5*x^3))/(((1 + x)/(-2 + x^2))^(3/4)*(-2 + x^2)*(-3 + 7*x - 11*x^2 +
4*x^3 + 4*x^4 - 4*x^5 + x^6)), x]

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IntegrateAlgebraic [A]  time = 4.45, size = 45, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {-1+x}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right )-2 \tanh ^{-1}\left (\frac {-1+x}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x)^2*(-10 - 8*x + 5*x^2 + 5*x^3))/(((1 + x)/(-2 + x^2))^(3/4)*(-2 + x^2)*(-3 + 7*x -
11*x^2 + 4*x^3 + 4*x^4 - 4*x^5 + x^6)),x]

[Out]

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

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fricas [B]  time = 48.31, size = 256, normalized size = 5.69 \begin {gather*} -\arctan \left (\frac {2 \, {\left ({\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + {\left (x^{5} - 3 \, x^{4} + x^{3} + 5 \, x^{2} - 6 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}\right )}}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3}\right ) + \log \left (\frac {x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} - 2 \, {\left (x^{3} - x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}} + 2 \, {\left (x^{4} - 2 \, x^{3} - x^{2} + 4 \, x - 2\right )} \sqrt {\frac {x + 1}{x^{2} - 2}} - 2 \, {\left (x^{5} - 3 \, x^{4} + x^{3} + 5 \, x^{2} - 6 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}} + 9 \, x - 1}{x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^2*(5*x^3+5*x^2-8*x-10)/((1+x)/(x^2-2))^(3/4)/(x^2-2)/(x^6-4*x^5+4*x^4+4*x^3-11*x^2+7*x-3),x,
algorithm="fricas")

[Out]

-arctan(2*((x^3 - x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(3/4) + (x^5 - 3*x^4 + x^3 + 5*x^2 - 6*x + 2)*((x + 1)/(x
^2 - 2))^(1/4))/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 11*x^2 + 7*x - 3)) + log((x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 11*x^2
- 2*(x^3 - x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(3/4) + 2*(x^4 - 2*x^3 - x^2 + 4*x - 2)*sqrt((x + 1)/(x^2 - 2))
- 2*(x^5 - 3*x^4 + x^3 + 5*x^2 - 6*x + 2)*((x + 1)/(x^2 - 2))^(1/4) + 9*x - 1)/(x^6 - 4*x^5 + 4*x^4 + 4*x^3 -
11*x^2 + 7*x - 3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{3} + 5 \, x^{2} - 8 \, x - 10\right )} {\left (x - 1\right )}^{2}}{{\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^2*(5*x^3+5*x^2-8*x-10)/((1+x)/(x^2-2))^(3/4)/(x^2-2)/(x^6-4*x^5+4*x^4+4*x^3-11*x^2+7*x-3),x,
algorithm="giac")

[Out]

integrate((5*x^3 + 5*x^2 - 8*x - 10)*(x - 1)^2/((x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 11*x^2 + 7*x - 3)*(x^2 - 2)*((x
+ 1)/(x^2 - 2))^(3/4)), x)

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maple [C]  time = 4.75, size = 807, normalized size = 17.93

method	result	size
trager	\(-\ln \left (-\frac {2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}+2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{4}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}+x^{6}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{3}-6 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}-4 x^{5}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x -2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{2}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}+4 x^{4}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}+8 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x +10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}+4 x^{3}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}-12 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x -11 x^{2}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}+9 x -1}{x^{6}-4 x^{5}+4 x^{4}+4 x^{3}-11 x^{2}+7 x -3}\right )-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{3}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{2}+6 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}\, x -2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1-x}{x^{2}-2}}-10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}+11 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+12 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x -9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}-4 x^{5}+4 x^{4}+4 x^{3}-11 x^{2}+7 x -3}\right )\)	\(807\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)^2*(5*x^3+5*x^2-8*x-10)/((1+x)/(x^2-2))^(3/4)/(x^2-2)/(x^6-4*x^5+4*x^4+4*x^3-11*x^2+7*x-3),x,method=
_RETURNVERBOSE)

[Out]

-ln(-(2*(-(-1-x)/(x^2-2))^(3/4)*x^3+2*(-(-1-x)/(x^2-2))^(1/2)*x^4+2*(-(-1-x)/(x^2-2))^(1/4)*x^5+x^6-2*(-(-1-x)
/(x^2-2))^(3/4)*x^2-4*(-(-1-x)/(x^2-2))^(1/2)*x^3-6*(-(-1-x)/(x^2-2))^(1/4)*x^4-4*x^5-4*(-(-1-x)/(x^2-2))^(3/4
)*x-2*(-(-1-x)/(x^2-2))^(1/2)*x^2+2*(-(-1-x)/(x^2-2))^(1/4)*x^3+4*x^4+4*(-(-1-x)/(x^2-2))^(3/4)+8*(-(-1-x)/(x^
2-2))^(1/2)*x+10*(-(-1-x)/(x^2-2))^(1/4)*x^2+4*x^3-4*(-(-1-x)/(x^2-2))^(1/2)-12*(-(-1-x)/(x^2-2))^(1/4)*x-11*x
^2+4*(-(-1-x)/(x^2-2))^(1/4)+9*x-1)/(x^6-4*x^5+4*x^4+4*x^3-11*x^2+7*x-3))-RootOf(_Z^2+1)*ln(-(2*RootOf(_Z^2+1)
*(-(-1-x)/(x^2-2))^(1/2)*x^4-RootOf(_Z^2+1)*x^6+2*(-(-1-x)/(x^2-2))^(3/4)*x^3-4*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2
))^(1/2)*x^3-2*(-(-1-x)/(x^2-2))^(1/4)*x^5+4*RootOf(_Z^2+1)*x^5-2*(-(-1-x)/(x^2-2))^(3/4)*x^2-2*RootOf(_Z^2+1)
*(-(-1-x)/(x^2-2))^(1/2)*x^2+6*(-(-1-x)/(x^2-2))^(1/4)*x^4-4*RootOf(_Z^2+1)*x^4-4*(-(-1-x)/(x^2-2))^(3/4)*x+8*
RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)*x-2*(-(-1-x)/(x^2-2))^(1/4)*x^3-4*RootOf(_Z^2+1)*x^3+4*(-(-1-x)/(x^2-2)
)^(3/4)-4*RootOf(_Z^2+1)*(-(-1-x)/(x^2-2))^(1/2)-10*(-(-1-x)/(x^2-2))^(1/4)*x^2+11*RootOf(_Z^2+1)*x^2+12*(-(-1
-x)/(x^2-2))^(1/4)*x-9*RootOf(_Z^2+1)*x-4*(-(-1-x)/(x^2-2))^(1/4)+RootOf(_Z^2+1))/(x^6-4*x^5+4*x^4+4*x^3-11*x^
2+7*x-3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{3} + 5 \, x^{2} - 8 \, x - 10\right )} {\left (x - 1\right )}^{2}}{{\left (x^{6} - 4 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} + 7 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)^2*(5*x^3+5*x^2-8*x-10)/((1+x)/(x^2-2))^(3/4)/(x^2-2)/(x^6-4*x^5+4*x^4+4*x^3-11*x^2+7*x-3),x,
algorithm="maxima")

[Out]

integrate((5*x^3 + 5*x^2 - 8*x - 10)*(x - 1)^2/((x^6 - 4*x^5 + 4*x^4 + 4*x^3 - 11*x^2 + 7*x - 3)*(x^2 - 2)*((x
+ 1)/(x^2 - 2))^(3/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {{\left (x-1\right )}^2\,\left (-5\,x^3-5\,x^2+8\,x+10\right )}{\left (x^2-2\right )\,{\left (\frac {x+1}{x^2-2}\right )}^{3/4}\,\left (x^6-4\,x^5+4\,x^4+4\,x^3-11\,x^2+7\,x-3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - 1)^2*(8*x - 5*x^2 - 5*x^3 + 10))/((x^2 - 2)*((x + 1)/(x^2 - 2))^(3/4)*(7*x - 11*x^2 + 4*x^3 + 4*x^4
- 4*x^5 + x^6 - 3)),x)

[Out]

int(-((x - 1)^2*(8*x - 5*x^2 - 5*x^3 + 10))/((x^2 - 2)*((x + 1)/(x^2 - 2))^(3/4)*(7*x - 11*x^2 + 4*x^3 + 4*x^4
- 4*x^5 + x^6 - 3)), x)

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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((-1+x)**2*(5*x**3+5*x**2-8*x-10)/((1+x)/(x**2-2))**(3/4)/(x**2-2)/(x**6-4*x**5+4*x**4+4*x**3-11*x**2
+7*x-3),x)

[Out]

Timed out

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