3.78 \(\int e^{\frac{5}{2} \coth ^{-1}(a x)} x^3 \, dx\)

Optimal. Leaf size=250 \[ \frac{113 x^2 \sqrt [4]{\frac{1}{a x}+1}}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 x \sqrt [4]{\frac{1}{a x}+1}}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}-\frac{2467 \sqrt [4]{\frac{1}{a x}+1}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{475 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{475 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{x^4 \sqrt [4]{\frac{1}{a x}+1}}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 x^3 \sqrt [4]{\frac{1}{a x}+1}}{24 a \sqrt [4]{1-\frac{1}{a x}}} \]

[Out]

(-2467*(1 + 1/(a*x))^(1/4))/(192*a^4*(1 - 1/(a*x))^(1/4)) + (521*(1 + 1/(a*x))^(1/4)*x)/(192*a^3*(1 - 1/(a*x))
^(1/4)) + (113*(1 + 1/(a*x))^(1/4)*x^2)/(96*a^2*(1 - 1/(a*x))^(1/4)) + (17*(1 + 1/(a*x))^(1/4)*x^3)/(24*a*(1 -
1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^4)/(4*(1 - 1/(a*x))^(1/4)) + (475*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/
(a*x))^(1/4)])/(64*a^4) + (475*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)

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Rubi [A]  time = 0.135991, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 11, 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, 212, 206, 203} \[ \frac{113 x^2 \sqrt [4]{\frac{1}{a x}+1}}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 x \sqrt [4]{\frac{1}{a x}+1}}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}-\frac{2467 \sqrt [4]{\frac{1}{a x}+1}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{475 \tan ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{475 \tanh ^{-1}\left (\frac{\sqrt [4]{\frac{1}{a x}+1}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{x^4 \sqrt [4]{\frac{1}{a x}+1}}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 x^3 \sqrt [4]{\frac{1}{a x}+1}}{24 a \sqrt [4]{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[E^((5*ArcCoth[a*x])/2)*x^3,x]

[Out]

(-2467*(1 + 1/(a*x))^(1/4))/(192*a^4*(1 - 1/(a*x))^(1/4)) + (521*(1 + 1/(a*x))^(1/4)*x)/(192*a^3*(1 - 1/(a*x))
^(1/4)) + (113*(1 + 1/(a*x))^(1/4)*x^2)/(96*a^2*(1 - 1/(a*x))^(1/4)) + (17*(1 + 1/(a*x))^(1/4)*x^3)/(24*a*(1 -
1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^4)/(4*(1 - 1/(a*x))^(1/4)) + (475*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/
(a*x))^(1/4)])/(64*a^4) + (475*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(64*a^4)

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]

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 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 212

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

Rubi steps

\begin{align*} \int e^{\frac{5}{2} \coth ^{-1}(a x)} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/4}}{x^5 \left (1-\frac{x}{a}\right )^{5/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-\frac{17}{2 a}-\frac{8 x}{a^2}}{x^4 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{12} \operatorname{Subst}\left (\int \frac{\frac{113}{4 a^2}+\frac{51 x}{2 a^3}}{x^3 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{24} \operatorname{Subst}\left (\int \frac{-\frac{521}{8 a^3}-\frac{113 x}{2 a^4}}{x^2 \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{521 \sqrt [4]{1+\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{\frac{1425}{16 a^4}+\frac{521 x}{8 a^5}}{x \left (1-\frac{x}{a}\right )^{5/4} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2467 \sqrt [4]{1+\frac{1}{a x}}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 \sqrt [4]{1+\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{1}{12} a \operatorname{Subst}\left (\int -\frac{1425}{32 a^5 x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2467 \sqrt [4]{1+\frac{1}{a x}}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 \sqrt [4]{1+\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}-\frac{475 \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/4}} \, dx,x,\frac{1}{x}\right )}{128 a^4}\\ &=-\frac{2467 \sqrt [4]{1+\frac{1}{a x}}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 \sqrt [4]{1+\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}-\frac{475 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{32 a^4}\\ &=-\frac{2467 \sqrt [4]{1+\frac{1}{a x}}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 \sqrt [4]{1+\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{475 \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 )}{64 a^4}+\frac{475 \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 )}{64 a^4}\\ &=-\frac{2467 \sqrt [4]{1+\frac{1}{a x}}}{192 a^4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{521 \sqrt [4]{1+\frac{1}{a x}} x}{192 a^3 \sqrt [4]{1-\frac{1}{a x}}}+\frac{113 \sqrt [4]{1+\frac{1}{a x}} x^2}{96 a^2 \sqrt [4]{1-\frac{1}{a x}}}+\frac{17 \sqrt [4]{1+\frac{1}{a x}} x^3}{24 a \sqrt [4]{1-\frac{1}{a x}}}+\frac{\sqrt [4]{1+\frac{1}{a x}} x^4}{4 \sqrt [4]{1-\frac{1}{a x}}}+\frac{475 \tan ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}+\frac{475 \tanh ^{-1}\left (\frac{\sqrt [4]{1+\frac{1}{a x}}}{\sqrt [4]{1-\frac{1}{a x}}}\right )}{64 a^4}\\ \end{align*}

Mathematica [A]  time = 5.23914, size = 161, normalized size = 0.64 \[ \frac{-3072 e^{\frac{1}{2} \coth ^{-1}(a x)}+\frac{6292 e^{\frac{1}{2} \coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}+\frac{7376 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^2}+\frac{5248 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^3}+\frac{1536 e^{\frac{1}{2} \coth ^{-1}(a x)}}{\left (e^{2 \coth ^{-1}(a x)}-1\right )^4}-1425 \log \left (1-e^{\frac{1}{2} \coth ^{-1}(a x)}\right )+1425 \log \left (e^{\frac{1}{2} \coth ^{-1}(a x)}+1\right )+2850 \tan ^{-1}\left (e^{\frac{1}{2} \coth ^{-1}(a x)}\right )}{384 a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((5*ArcCoth[a*x])/2)*x^3,x]

[Out]

(-3072*E^(ArcCoth[a*x]/2) + (1536*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^4 + (5248*E^(ArcCoth[a*x]/2))/
(-1 + E^(2*ArcCoth[a*x]))^3 + (7376*E^(ArcCoth[a*x]/2))/(-1 + E^(2*ArcCoth[a*x]))^2 + (6292*E^(ArcCoth[a*x]/2)
)/(-1 + E^(2*ArcCoth[a*x])) + 2850*ArcTan[E^(ArcCoth[a*x]/2)] - 1425*Log[1 - E^(ArcCoth[a*x]/2)] + 1425*Log[1
+ E^(ArcCoth[a*x]/2)])/(384*a^4)

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Maple [F]  time = 0.325, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.4941, size = 321, normalized size = 1.28 \begin{align*} \frac{1}{384} \, a{\left (\frac{4 \,{\left (\frac{4645 \,{\left (a x - 1\right )}}{a x + 1} - \frac{7483 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{5415 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{1425 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} - 768\right )}}{a^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{17}{4}} - 4 \, a^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{13}{4}} + 6 \, a^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{4}} - 4 \, a^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}} + a^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} - \frac{2850 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{5}} + \frac{1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{5}} - \frac{1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*a*(4*(4645*(a*x - 1)/(a*x + 1) - 7483*(a*x - 1)^2/(a*x + 1)^2 + 5415*(a*x - 1)^3/(a*x + 1)^3 - 1425*(a*x
- 1)^4/(a*x + 1)^4 - 768)/(a^5*((a*x - 1)/(a*x + 1))^(17/4) - 4*a^5*((a*x - 1)/(a*x + 1))^(13/4) + 6*a^5*((a*
x - 1)/(a*x + 1))^(9/4) - 4*a^5*((a*x - 1)/(a*x + 1))^(5/4) + a^5*((a*x - 1)/(a*x + 1))^(1/4)) - 2850*arctan((
(a*x - 1)/(a*x + 1))^(1/4))/a^5 + 1425*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^5 - 1425*log(((a*x - 1)/(a*x + 1
))^(1/4) - 1)/a^5)

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Fricas [A]  time = 1.70379, size = 389, normalized size = 1.56 \begin{align*} -\frac{2850 \,{\left (a x - 1\right )} \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right ) - 1425 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right ) + 1425 \,{\left (a x - 1\right )} \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1\right ) - 2 \,{\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 362 \, a^{3} x^{3} + 747 \, a^{2} x^{2} - 1946 \, a x - 2467\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{384 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(2850*(a*x - 1)*arctan(((a*x - 1)/(a*x + 1))^(1/4)) - 1425*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) +
1) + 1425*(a*x - 1)*log(((a*x - 1)/(a*x + 1))^(1/4) - 1) - 2*(48*a^5*x^5 + 184*a^4*x^4 + 362*a^3*x^3 + 747*a^2
*x^2 - 1946*a*x - 2467)*((a*x - 1)/(a*x + 1))^(3/4))/(a^5*x - a^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20687, size = 301, normalized size = 1.2 \begin{align*} -\frac{1}{384} \, a{\left (\frac{2850 \, \arctan \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}\right )}{a^{5}} - \frac{1425 \, \log \left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} + 1\right )}{a^{5}} + \frac{1425 \, \log \left ({\left | \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}} - 1 \right |}\right )}{a^{5}} + \frac{3072}{a^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{4}}} + \frac{4 \,{\left (\frac{2875 \,{\left (a x - 1\right )} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{a x + 1} - \frac{2343 \,{\left (a x - 1\right )}^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{2}} + \frac{657 \,{\left (a x - 1\right )}^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}}{{\left (a x + 1\right )}^{3}} - 1573 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{4}}\right )}}{a^{5}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*a*(2850*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^5 - 1425*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^5 + 1425*
log(abs(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^5 + 3072/(a^5*((a*x - 1)/(a*x + 1))^(1/4)) + 4*(2875*(a*x - 1)*((a
*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 2343*(a*x - 1)^2*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 + 657*(a*x - 1)^
3*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^3 - 1573*((a*x - 1)/(a*x + 1))^(3/4))/(a^5*((a*x - 1)/(a*x + 1) - 1)^4
))