3.7.49 \(\int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} (-2+x^2) (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9)} \, dx\)

Optimal. Leaf size=51 \[ 2 \tanh ^{-1}\left (\frac {x^2-1}{\sqrt [4]{\frac {x+1}{x^2-2}}}\right )-2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {x+1}{x^2-2}}}{x^2-1}\right ) \]

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Rubi [F]  time = 9.30, 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 {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)*(-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5
+ 5*x^6 - 5*x^7 - x^8 + x^9)),x]

[Out]

(-24*(-((1 + x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/4)*Defer[Subst][Defer[Int][x^2/((-1 - 2*x^4 + x^8)^(3/4)*(-1 -
16*x^12 + 56*x^20 - 72*x^24 + 39*x^28 - 10*x^32 + x^36)), x], x, (1 + x)^(1/4)])/(1 + x)^(3/4) - (36*(-((1 +
x)/(2 - x^2)))^(3/4)*(-2 + x^2)^(3/4)*Defer[Subst][Defer[Int][x^6/((-1 - 2*x^4 + x^8)^(3/4)*(-1 - 16*x^12 + 56
*x^20 - 72*x^24 + 39*x^28 - 10*x^32 + x^36)), x], x, (1 + x)^(1/4)])/(1 + x)^(3/4) + (104*(-((1 + x)/(2 - x^2)
))^(3/4)*(-2 + x^2)^(3/4)*Defer[Subst][Defer[Int][x^10/((-1 - 2*x^4 + x^8)^(3/4)*(-1 - 16*x^12 + 56*x^20 - 72*
x^24 + 39*x^28 - 10*x^32 + x^36)), x], x, (1 + x)^(1/4)])/(1 + x)^(3/4) - (36*(-((1 + x)/(2 - x^2)))^(3/4)*(-2
+ x^2)^(3/4)*Defer[Subst][Defer[Int][x^14/((-1 - 2*x^4 + x^8)^(3/4)*(-1 - 16*x^12 + 56*x^20 - 72*x^24 + 39*x^
28 - 10*x^32 + x^36)), x], x, (1 + x)^(1/4)])/(1 + x)^(3/4)

Rubi steps

\begin {align*} \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx &=\int \frac {\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2-16 x+x^2+9 x^3\right )}{3+x-9 x^2+16 x^4-14 x^6+6 x^8-x^{10}} \, dx\\ &=\frac {\left (\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \frac {(1+x)^{3/4} \left (-2-16 x+x^2+9 x^3\right )}{\left (-2+x^2\right )^{3/4} \left (3+x-9 x^2+16 x^4-14 x^6+6 x^8-x^{10}\right )} \, dx}{(1+x)^{3/4}}\\ &=\frac {\left (\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \frac {-2-16 x+x^2+9 x^3}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (3-2 x-7 x^2+7 x^3+9 x^4-9 x^5-5 x^6+5 x^7+x^8-x^9\right )} \, dx}{(1+x)^{3/4}}\\ &=\frac {\left (\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \left (\frac {2}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )}+\frac {16 x}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )}-\frac {x^2}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )}-\frac {9 x^3}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )}\right ) \, dx}{(1+x)^{3/4}}\\ &=-\frac {\left (\left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \frac {x^2}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx}{(1+x)^{3/4}}+\frac {\left (2 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \frac {1}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx}{(1+x)^{3/4}}-\frac {\left (9 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \frac {x^3}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx}{(1+x)^{3/4}}+\frac {\left (16 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \int \frac {x}{\sqrt [4]{1+x} \left (-2+x^2\right )^{3/4} \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx}{(1+x)^{3/4}}\\ &=-\frac {\left (4 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (8 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}-\frac {\left (36 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )^3}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (64 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}\\ &=-\frac {\left (4 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}-\frac {2 x^6}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}+\frac {x^{10}}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (8 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}-\frac {\left (36 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \left (-\frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}+\frac {3 x^6}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}-\frac {3 x^{10}}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}+\frac {x^{14}}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (64 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \left (-\frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}+\frac {x^6}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )}\right ) \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}\\ &=-\frac {\left (4 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}-\frac {\left (4 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (8 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (8 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (36 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}-\frac {\left (36 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}-\frac {\left (64 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (64 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}-\frac {\left (108 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}+\frac {\left (108 \left (\frac {1+x}{-2+x^2}\right )^{3/4} \left (-2+x^2\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (-1-2 x^4+x^8\right )^{3/4} \left (-1-16 x^{12}+56 x^{20}-72 x^{24}+39 x^{28}-10 x^{32}+x^{36}\right )} \, dx,x,\sqrt [4]{1+x}\right )}{(1+x)^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 2.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+16 x-x^2-9 x^3}{\sqrt [4]{\frac {1+x}{-2+x^2}} \left (-2+x^2\right ) \left (-3+2 x+7 x^2-7 x^3-9 x^4+9 x^5+5 x^6-5 x^7-x^8+x^9\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)*(-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 +
9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)),x]

[Out]

Integrate[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)*(-3 + 2*x + 7*x^2 - 7*x^3 - 9*x^4 +
9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)), x]

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IntegrateAlgebraic [A]  time = 0.20, size = 51, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1+x}{-2+x^2}}}{-1+x^2}\right )+2 \tanh ^{-1}\left (\frac {-1+x^2}{\sqrt [4]{\frac {1+x}{-2+x^2}}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + 16*x - x^2 - 9*x^3)/(((1 + x)/(-2 + x^2))^(1/4)*(-2 + x^2)*(-3 + 2*x + 7*x^2 - 7*x^3 -
9*x^4 + 9*x^5 + 5*x^6 - 5*x^7 - x^8 + x^9)),x]

[Out]

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

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fricas [B]  time = 100.27, size = 331, normalized size = 6.49 \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^{7} - x^{6} - 4 \, x^{5} + 4 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}\right )}}{x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3}\right ) + \log \left (-\frac {x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, 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^{5} - x^{4} - 3 \, x^{3} + 3 \, x^{2} + 2 \, x - 2\right )} \sqrt {\frac {x + 1}{x^{2} - 2}} + 2 \, {\left (x^{7} - x^{6} - 4 \, x^{5} + 4 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} - 2 \, x + 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}} + 2 \, x - 1}{x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3
),x, algorithm="fricas")

[Out]

-arctan(2*((x^3 - x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(3/4) + (x^7 - x^6 - 4*x^5 + 4*x^4 + 5*x^3 - 5*x^2 - 2*x
+ 2)*((x + 1)/(x^2 - 2))^(1/4))/(x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3)) + log(-
(x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*(x^3 - x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(3/4)
+ 2*(x^5 - x^4 - 3*x^3 + 3*x^2 + 2*x - 2)*sqrt((x + 1)/(x^2 - 2)) + 2*(x^7 - x^6 - 4*x^5 + 4*x^4 + 5*x^3 - 5*
x^2 - 2*x + 2)*((x + 1)/(x^2 - 2))^(1/4) + 2*x - 1)/(x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2
+ 2*x - 3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {9 \, x^{3} + x^{2} - 16 \, x - 2}{{\left (x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3
),x, algorithm="giac")

[Out]

integrate(-(9*x^3 + x^2 - 16*x - 2)/((x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3)*(x^
2 - 2)*((x + 1)/(x^2 - 2))^(1/4)), x)

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maple [C]  time = 13.18, size = 1012, normalized size = 19.84

method	result	size
trager	\(\ln \left (-\frac {-1+2 x +x^{9}-5 x^{7}+9 x^{5}+5 x^{6}+7 x^{2}-7 x^{3}-x^{8}-9 x^{4}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}+2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{5}-2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{4}-6 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{3}+6 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x^{2}+4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, x +2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{7}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{6}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}-8 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}+8 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x +10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}-10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x}{x^{9}-x^{8}-5 x^{7}+5 x^{6}+9 x^{5}-9 x^{4}-7 x^{3}+7 x^{2}+2 x -3}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right )+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{9}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{8}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{7}-9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}+9 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+7 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-7 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-6 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+6 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}-2 \sqrt {-\frac {-1-x}{x^{2}-2}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{7}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{6}-2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{3}-8 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{5}+2 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x^{2}+8 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{4}+4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {3}{4}} x +10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{3}-10 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x^{2}-4 \left (-\frac {-1-x}{x^{2}-2}\right )^{\frac {1}{4}} x}{x^{9}-x^{8}-5 x^{7}+5 x^{6}+9 x^{5}-9 x^{4}-7 x^{3}+7 x^{2}+2 x -3}\right )\)	\(1012\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3),x,me
thod=_RETURNVERBOSE)

[Out]

ln(-(-1+2*x+x^9-5*x^7+9*x^5+5*x^6+7*x^2-7*x^3-x^8-9*x^4+4*(-(-1-x)/(x^2-2))^(1/4)-4*(-(-1-x)/(x^2-2))^(1/2)+4*
(-(-1-x)/(x^2-2))^(3/4)+2*(-(-1-x)/(x^2-2))^(1/2)*x^5-2*(-(-1-x)/(x^2-2))^(1/2)*x^4-6*(-(-1-x)/(x^2-2))^(1/2)*
x^3+6*(-(-1-x)/(x^2-2))^(1/2)*x^2+4*(-(-1-x)/(x^2-2))^(1/2)*x+2*(-(-1-x)/(x^2-2))^(1/4)*x^7-2*(-(-1-x)/(x^2-2)
)^(1/4)*x^6+2*(-(-1-x)/(x^2-2))^(3/4)*x^3-8*(-(-1-x)/(x^2-2))^(1/4)*x^5-2*(-(-1-x)/(x^2-2))^(3/4)*x^2+8*(-(-1-
x)/(x^2-2))^(1/4)*x^4-4*(-(-1-x)/(x^2-2))^(3/4)*x+10*(-(-1-x)/(x^2-2))^(1/4)*x^3-10*(-(-1-x)/(x^2-2))^(1/4)*x^
2-4*(-(-1-x)/(x^2-2))^(1/4)*x)/(x^9-x^8-5*x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3))+RootOf(_Z^2+1)*ln((-2*Root
Of(_Z^2+1)*x-7*RootOf(_Z^2+1)*x^2-RootOf(_Z^2+1)*x^9+RootOf(_Z^2+1)*x^8+5*RootOf(_Z^2+1)*x^7-5*RootOf(_Z^2+1)*
x^6-9*RootOf(_Z^2+1)*x^5+9*RootOf(_Z^2+1)*x^4+7*RootOf(_Z^2+1)*x^3-4*(-(-1-x)/(x^2-2))^(1/2)*RootOf(_Z^2+1)+Ro
otOf(_Z^2+1)+4*(-(-1-x)/(x^2-2))^(1/4)-4*(-(-1-x)/(x^2-2))^(3/4)-6*(-(-1-x)/(x^2-2))^(1/2)*RootOf(_Z^2+1)*x^3+
6*(-(-1-x)/(x^2-2))^(1/2)*RootOf(_Z^2+1)*x^2+4*(-(-1-x)/(x^2-2))^(1/2)*RootOf(_Z^2+1)*x+2*(-(-1-x)/(x^2-2))^(1
/2)*RootOf(_Z^2+1)*x^5-2*(-(-1-x)/(x^2-2))^(1/2)*RootOf(_Z^2+1)*x^4+2*(-(-1-x)/(x^2-2))^(1/4)*x^7-2*(-(-1-x)/(
x^2-2))^(1/4)*x^6-2*(-(-1-x)/(x^2-2))^(3/4)*x^3-8*(-(-1-x)/(x^2-2))^(1/4)*x^5+2*(-(-1-x)/(x^2-2))^(3/4)*x^2+8*
(-(-1-x)/(x^2-2))^(1/4)*x^4+4*(-(-1-x)/(x^2-2))^(3/4)*x+10*(-(-1-x)/(x^2-2))^(1/4)*x^3-10*(-(-1-x)/(x^2-2))^(1
/4)*x^2-4*(-(-1-x)/(x^2-2))^(1/4)*x)/(x^9-x^8-5*x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {9 \, x^{3} + x^{2} - 16 \, x - 2}{{\left (x^{9} - x^{8} - 5 \, x^{7} + 5 \, x^{6} + 9 \, x^{5} - 9 \, x^{4} - 7 \, x^{3} + 7 \, x^{2} + 2 \, x - 3\right )} {\left (x^{2} - 2\right )} \left (\frac {x + 1}{x^{2} - 2}\right )^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-9*x^3-x^2+16*x+2)/((1+x)/(x^2-2))^(1/4)/(x^2-2)/(x^9-x^8-5*x^7+5*x^6+9*x^5-9*x^4-7*x^3+7*x^2+2*x-3
),x, algorithm="maxima")

[Out]

-integrate((9*x^3 + x^2 - 16*x - 2)/((x^9 - x^8 - 5*x^7 + 5*x^6 + 9*x^5 - 9*x^4 - 7*x^3 + 7*x^2 + 2*x - 3)*(x^
2 - 2)*((x + 1)/(x^2 - 2))^(1/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {-9\,x^3-x^2+16\,x+2}{\left (x^2-2\right )\,{\left (\frac {x+1}{x^2-2}\right )}^{1/4}\,\left (x^9-x^8-5\,x^7+5\,x^6+9\,x^5-9\,x^4-7\,x^3+7\,x^2+2\,x-3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x - x^2 - 9*x^3 + 2)/((x^2 - 2)*((x + 1)/(x^2 - 2))^(1/4)*(2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6
- 5*x^7 - x^8 + x^9 - 3)),x)

[Out]

int((16*x - x^2 - 9*x^3 + 2)/((x^2 - 2)*((x + 1)/(x^2 - 2))^(1/4)*(2*x + 7*x^2 - 7*x^3 - 9*x^4 + 9*x^5 + 5*x^6
- 5*x^7 - x^8 + x^9 - 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((-9*x**3-x**2+16*x+2)/((1+x)/(x**2-2))**(1/4)/(x**2-2)/(x**9-x**8-5*x**7+5*x**6+9*x**5-9*x**4-7*x**3
+7*x**2+2*x-3),x)

[Out]

Timed out

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