3.10.60 \(\int \frac {(3+x^4) \sqrt {x+x^4-x^5}}{(-1+x^4) (-1+x^3+x^4)} \, dx\)

Optimal. Leaf size=73 \[ 2 \tanh ^{-1}\left (\frac {x \sqrt {-x^5+x^4+x}}{x^4-x^3-1}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {-x^5+x^4+x}}{x^4-x^3-1}\right ) \]

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Rubi [F]  time = 5.68, 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 (3+x^4\right ) \sqrt {x+x^4-x^5}}{\left (-1+x^4\right ) \left (-1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((3 + x^4)*Sqrt[x + x^4 - x^5])/((-1 + x^4)*(-1 + x^3 + x^4)),x]

[Out]

((-I)*Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][Sqrt[1 + x^6 - x^8]/(I - x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[
1 + x^3 - x^4]) + ((-1)^(1/4)*Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][Sqrt[1 + x^6 - x^8]/((-1)^(1/4) - x)
, x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]) - ((-1)^(3/4)*Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][Sqr
t[1 + x^6 - x^8]/(-(-1)^(3/4) - x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]) + (Sqrt[x + x^4 - x^5]*Defe
r[Subst][Defer[Int][Sqrt[1 + x^6 - x^8]/(-1 + x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]) - (I*Sqrt[x +
x^4 - x^5]*Defer[Subst][Defer[Int][Sqrt[1 + x^6 - x^8]/(I + x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]
) - (Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][Sqrt[1 + x^6 - x^8]/(1 + x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1
+ x^3 - x^4]) + ((-1)^(1/4)*Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][Sqrt[1 + x^6 - x^8]/((-1)^(1/4) + x),
x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]) - ((-1)^(3/4)*Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][Sqrt
[1 + x^6 - x^8]/(-(-1)^(3/4) + x), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]) - (6*Sqrt[x + x^4 - x^5]*Def
er[Subst][Defer[Int][(x^2*Sqrt[1 + x^6 - x^8])/(-1 + x^6 + x^8), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + x^3 - x^4]
) - (8*Sqrt[x + x^4 - x^5]*Defer[Subst][Defer[Int][(x^4*Sqrt[1 + x^6 - x^8])/(-1 + x^6 + x^8), x], x, Sqrt[x]]
)/(Sqrt[x]*Sqrt[1 + x^3 - x^4])

Rubi steps

\begin {align*} \int \frac {\left (3+x^4\right ) \sqrt {x+x^4-x^5}}{\left (-1+x^4\right ) \left (-1+x^3+x^4\right )} \, dx &=\frac {\sqrt {x+x^4-x^5} \int \frac {\sqrt {x} \sqrt {1+x^3-x^4} \left (3+x^4\right )}{\left (-1+x^4\right ) \left (-1+x^3+x^4\right )} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=\frac {\sqrt {x+x^4-x^5} \int \left (\frac {\sqrt {x} \sqrt {1+x^3-x^4}}{-1+x}+\frac {\sqrt {x} \sqrt {1+x^3-x^4}}{1+x}-\frac {2 x^{3/2} \sqrt {1+x^3-x^4}}{1+x^2}+\frac {(-3-4 x) \sqrt {x} \sqrt {1+x^3-x^4}}{-1+x^3+x^4}\right ) \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=\frac {\sqrt {x+x^4-x^5} \int \frac {\sqrt {x} \sqrt {1+x^3-x^4}}{-1+x} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\sqrt {x+x^4-x^5} \int \frac {\sqrt {x} \sqrt {1+x^3-x^4}}{1+x} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\sqrt {x+x^4-x^5} \int \frac {(-3-4 x) \sqrt {x} \sqrt {1+x^3-x^4}}{-1+x^3+x^4} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \int \frac {x^{3/2} \sqrt {1+x^3-x^4}}{1+x^2} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \int \left (\frac {i x^{3/2} \sqrt {1+x^3-x^4}}{2 (i-x)}+\frac {i x^{3/2} \sqrt {1+x^3-x^4}}{2 (i+x)}\right ) \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{-1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3-4 x^2\right ) \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \int \frac {x^{3/2} \sqrt {1+x^3-x^4}}{i-x} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \int \frac {x^{3/2} \sqrt {1+x^3-x^4}}{i+x} \, dx}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (\sqrt {1+x^6-x^8}+\frac {\sqrt {1+x^6-x^8}}{-1+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (\sqrt {1+x^6-x^8}-\frac {\sqrt {1+x^6-x^8}}{1+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^2 \sqrt {1+x^6-x^8}}{-1+x^6+x^8}-\frac {4 x^4 \sqrt {1+x^6-x^8}}{-1+x^6+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{i-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{i+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+2 \frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \sqrt {1+x^6-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{-1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (6 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (8 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (-i \sqrt {1+x^6-x^8}-x^2 \sqrt {1+x^6-x^8}-\frac {\sqrt {1+x^6-x^8}}{i-x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (-i \sqrt {1+x^6-x^8}+x^2 \sqrt {1+x^6-x^8}-\frac {\sqrt {1+x^6-x^8}}{i+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+2 \frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \sqrt {1+x^6-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {1+x^6-x^8}}{2 (i-x)}+\frac {i \sqrt {1+x^6-x^8}}{2 (i+x)}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {1+x^6-x^8}}{2 (-1+x)}-\frac {\sqrt {1+x^6-x^8}}{2 (1+x)}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (6 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (8 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i-x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\sqrt {x+x^4-x^5} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{-1+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\sqrt {x+x^4-x^5} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{1+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (6 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (8 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {(-1)^{3/4} \sqrt {1+x^6-x^8}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {1+x^6-x^8}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (2 i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {\sqrt [4]{-1} \sqrt {1+x^6-x^8}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {1+x^6-x^8}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\sqrt {x+x^4-x^5} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{-1+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\sqrt {x+x^4-x^5} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{1+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (6 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (8 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ &=-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (i \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\sqrt {x+x^4-x^5} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{-1+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\sqrt {x+x^4-x^5} \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{1+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (6 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left (8 \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+x^6-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (\sqrt [4]{-1} \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{\sqrt [4]{-1}-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}+\frac {\left (\sqrt [4]{-1} \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{\sqrt [4]{-1}+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left ((-1)^{3/4} \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{-(-1)^{3/4}-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}-\frac {\left ((-1)^{3/4} \sqrt {x+x^4-x^5}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+x^6-x^8}}{-(-1)^{3/4}+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x^3-x^4}}\\ \end {align*}

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Mathematica [F]  time = 1.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+x^4\right ) \sqrt {x+x^4-x^5}}{\left (-1+x^4\right ) \left (-1+x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((3 + x^4)*Sqrt[x + x^4 - x^5])/((-1 + x^4)*(-1 + x^3 + x^4)),x]

[Out]

Integrate[((3 + x^4)*Sqrt[x + x^4 - x^5])/((-1 + x^4)*(-1 + x^3 + x^4)), x]

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IntegrateAlgebraic [A]  time = 0.22, size = 73, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x \sqrt {x+x^4-x^5}}{-1-x^3+x^4}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x+x^4-x^5}}{-1-x^3+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((3 + x^4)*Sqrt[x + x^4 - x^5])/((-1 + x^4)*(-1 + x^3 + x^4)),x]

[Out]

2*ArcTanh[(x*Sqrt[x + x^4 - x^5])/(-1 - x^3 + x^4)] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*Sqrt[x + x^4 - x^5])/(-1 -
x^3 + x^4)]

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fricas [A]  time = 0.68, size = 122, normalized size = 1.67 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{8} - 14 \, x^{7} + 17 \, x^{6} - 2 \, x^{4} + 14 \, x^{3} - 4 \, \sqrt {2} {\left (x^{5} - 3 \, x^{4} - x\right )} \sqrt {-x^{5} + x^{4} + x} + 1}{x^{8} + 2 \, x^{7} + x^{6} - 2 \, x^{4} - 2 \, x^{3} + 1}\right ) + \log \left (-\frac {x^{4} - 2 \, x^{3} + 2 \, \sqrt {-x^{5} + x^{4} + x} x - 1}{x^{4} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(-x^5+x^4+x)^(1/2)/(x^4-1)/(x^4+x^3-1),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*log((x^8 - 14*x^7 + 17*x^6 - 2*x^4 + 14*x^3 - 4*sqrt(2)*(x^5 - 3*x^4 - x)*sqrt(-x^5 + x^4 + x) + 1
)/(x^8 + 2*x^7 + x^6 - 2*x^4 - 2*x^3 + 1)) + log(-(x^4 - 2*x^3 + 2*sqrt(-x^5 + x^4 + x)*x - 1)/(x^4 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(-x^5+x^4+x)^(1/2)/(x^4-1)/(x^4+x^3-1),x, algorithm="giac")

[Out]

integrate(sqrt(-x^5 + x^4 + x)*(x^4 + 3)/((x^4 + x^3 - 1)*(x^4 - 1)), x)

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maple [C]  time = 0.60, size = 111, normalized size = 1.52

method	result	size
trager	\(\ln \left (\frac {x^{4}-2 x^{3}+2 x \sqrt {-x^{5}+x^{4}+x}-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}-3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}-4 x \sqrt {-x^{5}+x^{4}+x}-\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{4}+x^{3}-1}\right )\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+3)*(-x^5+x^4+x)^(1/2)/(x^4-1)/(x^4+x^3-1),x,method=_RETURNVERBOSE)

[Out]

ln((x^4-2*x^3+2*x*(-x^5+x^4+x)^(1/2)-1)/(-1+x)/(1+x)/(x^2+1))+RootOf(_Z^2-2)*ln(-(RootOf(_Z^2-2)*x^4-3*RootOf(
_Z^2-2)*x^3-4*x*(-x^5+x^4+x)^(1/2)-RootOf(_Z^2-2))/(x^4+x^3-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(-x^5+x^4+x)^(1/2)/(x^4-1)/(x^4+x^3-1),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^5 + x^4 + x)*(x^4 + 3)/((x^4 + x^3 - 1)*(x^4 - 1)), x)

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mupad [B]  time = 3.47, size = 81, normalized size = 1.11 \begin {gather*} \ln \left (\frac {2\,x\,\sqrt {-x^5+x^4+x}-2\,x^3+x^4-1}{x^4-1}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^3-x^4+2\,\sqrt {2}\,x\,\sqrt {-x^5+x^4+x}+1}{x^4+x^3-1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 3)*(x + x^4 - x^5)^(1/2))/((x^4 - 1)*(x^3 + x^4 - 1)),x)

[Out]

log((2*x*(x + x^4 - x^5)^(1/2) - 2*x^3 + x^4 - 1)/(x^4 - 1)) + 2^(1/2)*log((3*x^3 - x^4 + 2*2^(1/2)*x*(x + x^4
- x^5)^(1/2) + 1)/(x^3 + x^4 - 1))

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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((x**4+3)*(-x**5+x**4+x)**(1/2)/(x**4-1)/(x**4+x**3-1),x)

[Out]

Timed out

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