3.104 \(\int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx\)

Optimal. Leaf size=287 \[ \frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{\frac {1}{a x}+1}}+\frac {1003 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {1003 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {5533 x \sqrt [4]{1-\frac {1}{a x}}}{1920 a^4 \sqrt [4]{\frac {1}{a x}+1}}-\frac {1189 x^2 \sqrt [4]{1-\frac {1}{a x}}}{960 a^3 \sqrt [4]{\frac {1}{a x}+1}}+\frac {181 x^3 \sqrt [4]{1-\frac {1}{a x}}}{240 a^2 \sqrt [4]{\frac {1}{a x}+1}}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}-\frac {21 x^4 \sqrt [4]{1-\frac {1}{a x}}}{40 a \sqrt [4]{\frac {1}{a x}+1}} \]

[Out]

26111/1920*(1-1/a/x)^(1/4)/a^5/(1+1/a/x)^(1/4)+5533/1920*(1-1/a/x)^(1/4)*x/a^4/(1+1/a/x)^(1/4)-1189/960*(1-1/a
/x)^(1/4)*x^2/a^3/(1+1/a/x)^(1/4)+181/240*(1-1/a/x)^(1/4)*x^3/a^2/(1+1/a/x)^(1/4)-21/40*(1-1/a/x)^(1/4)*x^4/a/
(1+1/a/x)^(1/4)+1/5*(1-1/a/x)^(1/4)*x^5/(1+1/a/x)^(1/4)+1003/128*arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5-1
003/128*arctanh((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^5

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Rubi [A]  time = 0.16, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6171, 98, 151, 155, 12, 93, 298, 203, 206} \[ \frac {181 x^3 \sqrt [4]{1-\frac {1}{a x}}}{240 a^2 \sqrt [4]{\frac {1}{a x}+1}}-\frac {1189 x^2 \sqrt [4]{1-\frac {1}{a x}}}{960 a^3 \sqrt [4]{\frac {1}{a x}+1}}+\frac {5533 x \sqrt [4]{1-\frac {1}{a x}}}{1920 a^4 \sqrt [4]{\frac {1}{a x}+1}}+\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{\frac {1}{a x}+1}}+\frac {1003 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {1003 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {x^5 \sqrt [4]{1-\frac {1}{a x}}}{5 \sqrt [4]{\frac {1}{a x}+1}}-\frac {21 x^4 \sqrt [4]{1-\frac {1}{a x}}}{40 a \sqrt [4]{\frac {1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/E^((5*ArcCoth[a*x])/2),x]

[Out]

(26111*(1 - 1/(a*x))^(1/4))/(1920*a^5*(1 + 1/(a*x))^(1/4)) + (5533*(1 - 1/(a*x))^(1/4)*x)/(1920*a^4*(1 + 1/(a*
x))^(1/4)) - (1189*(1 - 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 + 1/(a*x))^(1/4)) + (181*(1 - 1/(a*x))^(1/4)*x^3)/(240
*a^2*(1 + 1/(a*x))^(1/4)) - (21*(1 - 1/(a*x))^(1/4)*x^4)/(40*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^5
)/(5*(1 + 1/(a*x))^(1/4)) + (1003*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5) - (1003*ArcTanh[(
1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(128*a^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 155

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m + n + p + 2, 0] && NeQ[m, -1] && (Sum
SimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1])))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 6171

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2
)), x], x, 1/x] /; FreeQ[{a, n}, x] &&  !IntegerQ[n] && IntegerQ[m]

Rubi steps

\begin {align*} \int e^{-\frac {5}{2} \coth ^{-1}(a x)} x^4 \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/4}}{x^6 \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1}{5} \operatorname {Subst}\left (\int \frac {\frac {21}{2 a}-\frac {10 x}{a^2}}{x^5 \left (1-\frac {x}{a}\right )^{3/4} \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1}{20} \operatorname {Subst}\left (\int \frac {\frac {181}{4 a^2}-\frac {42 x}{a^3}}{x^4 \left (1-\frac {x}{a}\right )^{3/4} \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1}{60} \operatorname {Subst}\left (\int \frac {\frac {1189}{8 a^3}-\frac {543 x}{4 a^4}}{x^3 \left (1-\frac {x}{a}\right )^{3/4} \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1}{120} \operatorname {Subst}\left (\int \frac {\frac {5533}{16 a^4}-\frac {1189 x}{4 a^5}}{x^2 \left (1-\frac {x}{a}\right )^{3/4} \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1}{120} \operatorname {Subst}\left (\int \frac {\frac {15045}{32 a^5}-\frac {5533 x}{16 a^6}}{x \left (1-\frac {x}{a}\right )^{3/4} \left (1+\frac {x}{a}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1}{60} a \operatorname {Subst}\left (\int \frac {15045}{64 a^6 x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1003 \operatorname {Subst}\left (\int \frac {1}{x \left (1-\frac {x}{a}\right )^{3/4} \sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{256 a^5}\\ &=\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1003 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{64 a^5}\\ &=\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1003 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}+\frac {1003 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}\\ &=\frac {26111 \sqrt [4]{1-\frac {1}{a x}}}{1920 a^5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {5533 \sqrt [4]{1-\frac {1}{a x}} x}{1920 a^4 \sqrt [4]{1+\frac {1}{a x}}}-\frac {1189 \sqrt [4]{1-\frac {1}{a x}} x^2}{960 a^3 \sqrt [4]{1+\frac {1}{a x}}}+\frac {181 \sqrt [4]{1-\frac {1}{a x}} x^3}{240 a^2 \sqrt [4]{1+\frac {1}{a x}}}-\frac {21 \sqrt [4]{1-\frac {1}{a x}} x^4}{40 a \sqrt [4]{1+\frac {1}{a x}}}+\frac {\sqrt [4]{1-\frac {1}{a x}} x^5}{5 \sqrt [4]{1+\frac {1}{a x}}}+\frac {1003 \tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}-\frac {1003 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{128 a^5}\\ \end {align*}

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Mathematica [A]  time = 5.43, size = 198, normalized size = 0.69 \[ \frac {8 e^{-\frac {1}{2} \coth ^{-1}(a x)}-\frac {4117 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{192 \left (e^{-2 \coth ^{-1}(a x)}-1\right )}-\frac {1661 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{48 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^2}-\frac {233 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{6 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^3}-\frac {122 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^4}-\frac {32 e^{-\frac {1}{2} \coth ^{-1}(a x)}}{5 \left (e^{-2 \coth ^{-1}(a x)}-1\right )^5}+\frac {1003}{256} \log \left (1-e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )-\frac {1003}{256} \log \left (e^{-\frac {1}{2} \coth ^{-1}(a x)}+1\right )-\frac {1003}{128} \tan ^{-1}\left (e^{-\frac {1}{2} \coth ^{-1}(a x)}\right )}{a^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/E^((5*ArcCoth[a*x])/2),x]

[Out]

(8/E^(ArcCoth[a*x]/2) - 32/(5*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^5) - 122/(5*E^(ArcCoth[a*x]/2)*(-1
+ E^(-2*ArcCoth[a*x]))^4) - 233/(6*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))^3) - 1661/(48*E^(ArcCoth[a*x
]/2)*(-1 + E^(-2*ArcCoth[a*x]))^2) - 4117/(192*E^(ArcCoth[a*x]/2)*(-1 + E^(-2*ArcCoth[a*x]))) - (1003*ArcTan[E
^(-1/2*ArcCoth[a*x])])/128 + (1003*Log[1 - E^(-1/2*ArcCoth[a*x])])/256 - (1003*Log[1 + E^(-1/2*ArcCoth[a*x])])
/256)/a^5

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fricas [A]  time = 1.17, size = 119, normalized size = 0.41 \[ \frac {2 \, {\left (384 \, a^{5} x^{5} - 1008 \, a^{4} x^{4} + 1448 \, a^{3} x^{3} - 2378 \, a^{2} x^{2} + 5533 \, a x + 26111\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) - 15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) + 15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{3840 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(2*(384*a^5*x^5 - 1008*a^4*x^4 + 1448*a^3*x^3 - 2378*a^2*x^2 + 5533*a*x + 26111)*((a*x - 1)/(a*x + 1))^
(1/4) - 30090*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) + 15045*log(((a
*x - 1)/(a*x + 1))^(1/4) - 1))/a^5

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giac [A]  time = 0.21, size = 254, normalized size = 0.89 \[ -\frac {1}{3840} \, a {\left (\frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {15045 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{6}} - \frac {30720 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{6}} - \frac {4 \, {\left (\frac {33816 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a x + 1} - \frac {61130 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{2}} + \frac {49120 \, {\left (a x - 1\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{3}} - \frac {20585 \, {\left (a x - 1\right )}^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{{\left (a x + 1\right )}^{4}} - 7365 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{a^{6} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*a*(30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 15045*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^6 - 15
045*log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^6 - 30720*((a*x - 1)/(a*x + 1))^(1/4)/a^6 - 4*(33816*(a*x - 1)
*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1) - 61130*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^2 + 49120*(a*
x - 1)^3*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^3 - 20585*(a*x - 1)^4*((a*x - 1)/(a*x + 1))^(1/4)/(a*x + 1)^4 -
7365*((a*x - 1)/(a*x + 1))^(1/4))/(a^6*((a*x - 1)/(a*x + 1) - 1)^5))

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maple [F]  time = 0.36, size = 0, normalized size = 0.00 \[ \int x^{4} \left (\frac {a x -1}{a x +1}\right )^{\frac {5}{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.41, size = 279, normalized size = 0.97 \[ -\frac {1}{3840} \, a {\left (\frac {4 \, {\left (20585 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{4}} - 49120 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{4}} + 61130 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{4}} - 33816 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{4}} + 7365 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{6}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{6}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{6}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{6}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{6}}{{\left (a x + 1\right )}^{5}} - a^{6}} + \frac {30090 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{6}} + \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{6}} - \frac {15045 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{6}} - \frac {30720 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}}{a^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3840*a*(4*(20585*((a*x - 1)/(a*x + 1))^(17/4) - 49120*((a*x - 1)/(a*x + 1))^(13/4) + 61130*((a*x - 1)/(a*x
+ 1))^(9/4) - 33816*((a*x - 1)/(a*x + 1))^(5/4) + 7365*((a*x - 1)/(a*x + 1))^(1/4))/(5*(a*x - 1)*a^6/(a*x + 1)
- 10*(a*x - 1)^2*a^6/(a*x + 1)^2 + 10*(a*x - 1)^3*a^6/(a*x + 1)^3 - 5*(a*x - 1)^4*a^6/(a*x + 1)^4 + (a*x - 1)
^5*a^6/(a*x + 1)^5 - a^6) + 30090*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^6 + 15045*log(((a*x - 1)/(a*x + 1))^(1
/4) + 1)/a^6 - 15045*log(((a*x - 1)/(a*x + 1))^(1/4) - 1)/a^6 - 30720*((a*x - 1)/(a*x + 1))^(1/4)/a^6)

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mupad [B]  time = 0.08, size = 253, normalized size = 0.88 \[ \frac {8\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{a^5}+\frac {\frac {491\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}}{64}-\frac {1409\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/4}}{40}+\frac {6113\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/4}}{96}-\frac {307\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/4}}{6}+\frac {4117\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/4}}{192}}{a^5+\frac {10\,a^5\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a^5\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a^5\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a^5\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {5\,a^5\,\left (a\,x-1\right )}{a\,x+1}}-\frac {1003\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{128\,a^5}+\frac {\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\,1{}\mathrm {i}\right )\,1003{}\mathrm {i}}{128\,a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((a*x - 1)/(a*x + 1))^(1/4)*1i)*1003i)/(128*a^5) + (8*((a*x - 1)/(a*x + 1))^(1/4))/a^5 + ((491*((a*x - 1
)/(a*x + 1))^(1/4))/64 - (1409*((a*x - 1)/(a*x + 1))^(5/4))/40 + (6113*((a*x - 1)/(a*x + 1))^(9/4))/96 - (307*
((a*x - 1)/(a*x + 1))^(13/4))/6 + (4117*((a*x - 1)/(a*x + 1))^(17/4))/192)/(a^5 + (10*a^5*(a*x - 1)^2)/(a*x +
1)^2 - (10*a^5*(a*x - 1)^3)/(a*x + 1)^3 + (5*a^5*(a*x - 1)^4)/(a*x + 1)^4 - (a^5*(a*x - 1)^5)/(a*x + 1)^5 - (5
*a^5*(a*x - 1))/(a*x + 1)) - (1003*atan(((a*x - 1)/(a*x + 1))^(1/4)))/(128*a^5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*((a*x-1)/(a*x+1))**(5/4),x)

[Out]

Timed out

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