3.25.56 \(\int \frac {2 x^4-x^9}{\sqrt {-1+x^5} (a-a x^5+x^{10})} \, dx\)

Optimal. Leaf size=199 \[ -\frac {\sqrt {2} \left (a+\sqrt {a-4} \sqrt {a}-4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^5-1}}{\sqrt {-a-\sqrt {a-4} \sqrt {a}+2}}\right )}{5 \sqrt {-a-\sqrt {a-4} \sqrt {a}+2} \sqrt {a-4} \sqrt {a}}-\frac {\sqrt {2} \left (-a+\sqrt {a-4} \sqrt {a}+4\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^5-1}}{\sqrt {-a+\sqrt {a-4} \sqrt {a}+2}}\right )}{5 \sqrt {-a+\sqrt {a-4} \sqrt {a}+2} \sqrt {a-4} \sqrt {a}} \]

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Rubi [A]  time = 0.45, antiderivative size = 60, normalized size of antiderivative = 0.30, number of steps used = 6, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {1593, 6715, 826, 1164, 628} \begin {gather*} \frac {\log \left (\sqrt {a} \sqrt {x^5-1}+x^5\right )}{5 \sqrt {a}}-\frac {\log \left (x^5-\sqrt {a} \sqrt {x^5-1}\right )}{5 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^4 - x^9)/(Sqrt[-1 + x^5]*(a - a*x^5 + x^10)),x]

[Out]

-1/5*Log[x^5 - Sqrt[a]*Sqrt[-1 + x^5]]/Sqrt[a] + Log[x^5 + Sqrt[a]*Sqrt[-1 + x^5]]/(5*Sqrt[a])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {2 x^4-x^9}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx &=\int \frac {x^4 \left (2-x^5\right )}{\sqrt {-1+x^5} \left (a-a x^5+x^{10}\right )} \, dx\\ &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {2-x}{\sqrt {-1+x} \left (a-a x+x^2\right )} \, dx,x,x^5\right )\\ &=\frac {2}{5} \operatorname {Subst}\left (\int \frac {1-x^2}{1+(2-a) x^2+x^4} \, dx,x,\sqrt {-1+x^5}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a}+2 x}{-1-\sqrt {a} x-x^2} \, dx,x,\sqrt {-1+x^5}\right )}{5 \sqrt {a}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a}-2 x}{-1+\sqrt {a} x-x^2} \, dx,x,\sqrt {-1+x^5}\right )}{5 \sqrt {a}}\\ &=-\frac {\log \left (x^5-\sqrt {a} \sqrt {-1+x^5}\right )}{5 \sqrt {a}}+\frac {\log \left (x^5+\sqrt {a} \sqrt {-1+x^5}\right )}{5 \sqrt {a}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 161, normalized size = 0.81 \begin {gather*} -\frac {1}{5} \sqrt {2} \left (\frac {\left (\sqrt {\frac {a-4}{a}}+1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^5-1}}{\sqrt {-a-\sqrt {a-4} \sqrt {a}+2}}\right )}{\sqrt {-a-\sqrt {a-4} \sqrt {a}+2}}-\frac {\left (\sqrt {\frac {a-4}{a}}-1\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {x^5-1}}{\sqrt {-a+\sqrt {a-4} \sqrt {a}+2}}\right )}{\sqrt {-a+\sqrt {a-4} \sqrt {a}+2}}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(2*x^4 - x^9)/(Sqrt[-1 + x^5]*(a - a*x^5 + x^10)),x]

[Out]

-1/5*(Sqrt[2]*(((1 + Sqrt[(-4 + a)/a])*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] - a]])/Sq
rt[2 - Sqrt[-4 + a]*Sqrt[a] - a] - ((-1 + Sqrt[(-4 + a)/a])*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 + Sqrt[-4 +
a]*Sqrt[a] - a]])/Sqrt[2 + Sqrt[-4 + a]*Sqrt[a] - a]))

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IntegrateAlgebraic [A]  time = 0.38, size = 199, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {2} \left (-4+\sqrt {-4+a} \sqrt {a}+a\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x^5}}{\sqrt {2-\sqrt {-4+a} \sqrt {a}-a}}\right )}{5 \sqrt {2-\sqrt {-4+a} \sqrt {a}-a} \sqrt {-4+a} \sqrt {a}}-\frac {\sqrt {2} \left (4+\sqrt {-4+a} \sqrt {a}-a\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x^5}}{\sqrt {2+\sqrt {-4+a} \sqrt {a}-a}}\right )}{5 \sqrt {2+\sqrt {-4+a} \sqrt {a}-a} \sqrt {-4+a} \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2*x^4 - x^9)/(Sqrt[-1 + x^5]*(a - a*x^5 + x^10)),x]

[Out]

-1/5*(Sqrt[2]*(-4 + Sqrt[-4 + a]*Sqrt[a] + a)*ArcTan[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] -
a]])/(Sqrt[2 - Sqrt[-4 + a]*Sqrt[a] - a]*Sqrt[-4 + a]*Sqrt[a]) - (Sqrt[2]*(4 + Sqrt[-4 + a]*Sqrt[a] - a)*ArcTa
n[(Sqrt[2]*Sqrt[-1 + x^5])/Sqrt[2 + Sqrt[-4 + a]*Sqrt[a] - a]])/(5*Sqrt[2 + Sqrt[-4 + a]*Sqrt[a] - a]*Sqrt[-4
+ a]*Sqrt[a])

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fricas [A]  time = 0.49, size = 86, normalized size = 0.43 \begin {gather*} \left [\frac {\log \left (\frac {x^{10} + 2 \, \sqrt {x^{5} - 1} \sqrt {a} x^{5} + a x^{5} - a}{x^{10} - a x^{5} + a}\right )}{5 \, \sqrt {a}}, -\frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {x^{5} - 1} \sqrt {-a} x^{5}}{a x^{5} - a}\right )}{5 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^9+2*x^4)/(x^5-1)^(1/2)/(x^10-a*x^5+a),x, algorithm="fricas")

[Out]

[1/5*log((x^10 + 2*sqrt(x^5 - 1)*sqrt(a)*x^5 + a*x^5 - a)/(x^10 - a*x^5 + a))/sqrt(a), -2/5*sqrt(-a)*arctan(sq
rt(x^5 - 1)*sqrt(-a)*x^5/(a*x^5 - a))/a]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^9+2*x^4)/(x^5-1)^(1/2)/(x^10-a*x^5+a),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a]=[88]Warning, need to choose a branch for the root of a polynomial with parameters. This migh
t be wrong.The choice was done assuming [a]=[-46]Undef/Unsigned Inf encountered in limitLimit: Max order reach
ed or unable to make series expansion Error: Bad Argument Value

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {-x^{9}+2 x^{4}}{\sqrt {x^{5}-1}\, \left (x^{10}-a \,x^{5}+a \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^9+2*x^4)/(x^5-1)^(1/2)/(x^10-a*x^5+a),x)

[Out]

int((-x^9+2*x^4)/(x^5-1)^(1/2)/(x^10-a*x^5+a),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {x^{9} - 2 \, x^{4}}{{\left (x^{10} - a x^{5} + a\right )} \sqrt {x^{5} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^9+2*x^4)/(x^5-1)^(1/2)/(x^10-a*x^5+a),x, algorithm="maxima")

[Out]

-integrate((x^9 - 2*x^4)/((x^10 - a*x^5 + a)*sqrt(x^5 - 1)), x)

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mupad [B]  time = 2.24, size = 47, normalized size = 0.24 \begin {gather*} \frac {\ln \left (\frac {a\,x^5-a+x^{10}+2\,\sqrt {a}\,x^5\,\sqrt {x^5-1}}{x^{10}-a\,x^5+a}\right )}{5\,\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4 - x^9)/((x^5 - 1)^(1/2)*(a - a*x^5 + x^10)),x)

[Out]

log((a*x^5 - a + x^10 + 2*a^(1/2)*x^5*(x^5 - 1)^(1/2))/(a - a*x^5 + x^10))/(5*a^(1/2))

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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**9+2*x**4)/(x**5-1)**(1/2)/(x**10-a*x**5+a),x)

[Out]

Timed out

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