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

Optimal. Leaf size=179 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {11 x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{24 a^2}+\frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {5 x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{12 a} \]

[Out]

11/24*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x/a^2+5/12*(1-1/a/x)^(3/4)*(1+1/a/x)^(1/4)*x^2/a+1/3*(1-1/a/x)^(3/4)*(1+
1/a/x)^(1/4)*x^3+3/8*arctan((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/a^3+3/8*arctanh((1+1/a/x)^(1/4)/(1-1/a/x)^(1/4))/
a^3

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6171, 99, 151, 12, 93, 212, 206, 203} \[ \frac {11 x \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{24 a^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{a x}+1}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {1}{3} x^3 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}+\frac {5 x^2 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{\frac {1}{a x}+1}}{12 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(24*a^2) + (5*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(12*
a) + ((1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/3 + (3*ArcTan[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*
a^3) + (3*ArcTanh[(1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)])/(8*a^3)

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 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (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 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 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 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 {1}{2} \coth ^{-1}(a x)} x^2 \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt [4]{1+\frac {x}{a}}}{x^4 \sqrt [4]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\frac {5}{2 a}+\frac {2 x}{a^2}}{x^3 \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{12 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3+\frac {1}{6} \operatorname {Subst}\left (\int \frac {-\frac {11}{4 a^2}-\frac {5 x}{2 a^3}}{x^2 \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{24 a^2}+\frac {5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{12 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3-\frac {1}{6} \operatorname {Subst}\left (\int \frac {9}{8 a^3 x \sqrt [4]{1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{24 a^2}+\frac {5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{12 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3-\frac {3 \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 )}{16 a^3}\\ &=\frac {11 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{24 a^2}+\frac {5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{12 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3-\frac {3 \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 )}{4 a^3}\\ &=\frac {11 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{24 a^2}+\frac {5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{12 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3+\frac {3 \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 )}{8 a^3}+\frac {3 \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 )}{8 a^3}\\ &=\frac {11 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x}{24 a^2}+\frac {5 \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^2}{12 a}+\frac {1}{3} \left (1-\frac {1}{a x}\right )^{3/4} \sqrt [4]{1+\frac {1}{a x}} x^3+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+\frac {1}{a x}}}{\sqrt [4]{1-\frac {1}{a x}}}\right )}{8 a^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 9.06, size = 399, normalized size = 2.23 \[ -\frac {e^{-\frac {7}{2} \coth ^{-1}(a x)} \left (1280 e^{8 \coth ^{-1}(a x)} \left (1346 e^{2 \coth ^{-1}(a x)}+557 e^{4 \coth ^{-1}(a x)}+821\right ) \, _4F_3\left (2,2,2,\frac {9}{4};1,1,\frac {21}{4};e^{2 \coth ^{-1}(a x)}\right )+10240 e^{8 \coth ^{-1}(a x)} \left (42 e^{2 \coth ^{-1}(a x)}+19 e^{4 \coth ^{-1}(a x)}+23\right ) \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {21}{4};e^{2 \coth ^{-1}(a x)}\right )+20480 e^{8 \coth ^{-1}(a x)} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{2 \coth ^{-1}(a x)}\right )+40960 e^{10 \coth ^{-1}(a x)} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{2 \coth ^{-1}(a x)}\right )+20480 e^{12 \coth ^{-1}(a x)} \, _6F_5\left (2,2,2,2,2,\frac {9}{4};1,1,1,1,\frac {21}{4};e^{2 \coth ^{-1}(a x)}\right )[phone] e^{2 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 \coth ^{-1}(a x)}\right )-635067810 e^{4 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 \coth ^{-1}(a x)}\right )-384831720 e^{6 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 \coth ^{-1}(a x)}\right )[phone] e^{8 \coth ^{-1}(a x)} \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 \coth ^{-1}(a x)}\right )[phone] \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 \coth ^{-1}(a x)}\right )-946471617 e^{2 \coth ^{-1}(a x)}[phone] e^{4 \coth ^{-1}(a x)}[phone] e^{6 \coth ^{-1}(a x)}-23818496 e^{8 \coth ^{-1}(a x)}-1070609085\right )}{1909440 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/1909440*(-1070609085 - 946471617*E^(2*ArcCoth[a*x]) + 369641285*E^(4*ArcCoth[a*x]) + 351173641*E^(6*ArcCoth
[a*x]) - 23818496*E^(8*ArcCoth[a*x]) + 1070609085*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 7323498
00*E^(2*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] - 635067810*E^(4*ArcCoth[a*x])*Hyperg
eometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] - 384831720*E^(6*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(
2*ArcCoth[a*x])] + 60913125*E^(8*ArcCoth[a*x])*Hypergeometric2F1[1/4, 1, 5/4, E^(2*ArcCoth[a*x])] + 1280*E^(8*
ArcCoth[a*x])*(821 + 1346*E^(2*ArcCoth[a*x]) + 557*E^(4*ArcCoth[a*x]))*HypergeometricPFQ[{2, 2, 2, 9/4}, {1, 1
, 21/4}, E^(2*ArcCoth[a*x])] + 10240*E^(8*ArcCoth[a*x])*(23 + 42*E^(2*ArcCoth[a*x]) + 19*E^(4*ArcCoth[a*x]))*H
ypergeometricPFQ[{2, 2, 2, 2, 9/4}, {1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 20480*E^(8*ArcCoth[a*x])*Hypergeome
tricPFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 40960*E^(10*ArcCoth[a*x])*Hypergeometr
icPFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])] + 20480*E^(12*ArcCoth[a*x])*Hypergeometric
PFQ[{2, 2, 2, 2, 2, 9/4}, {1, 1, 1, 1, 21/4}, E^(2*ArcCoth[a*x])])/(a^3*E^((7*ArcCoth[a*x])/2))

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 103, normalized size = 0.58 \[ \frac {2 \, {\left (8 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 21 \, a x + 11\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}} - 18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right ) + 9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right ) - 9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{48 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(2*(8*a^3*x^3 + 18*a^2*x^2 + 21*a*x + 11)*((a*x - 1)/(a*x + 1))^(3/4) - 18*arctan(((a*x - 1)/(a*x + 1))^(
1/4)) + 9*log(((a*x - 1)/(a*x + 1))^(1/4) + 1) - 9*log(((a*x - 1)/(a*x + 1))^(1/4) - 1))/a^3

________________________________________________________________________________________

giac [A]  time = 0.24, size = 172, normalized size = 0.96 \[ -\frac {1}{48} \, a {\left (\frac {18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{4}} - \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {9 \, \log \left ({\left | \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1 \right |}\right )}{a^{4}} - \frac {4 \, {\left (\frac {6 \, {\left (a x - 1\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{a x + 1} - \frac {9 \, {\left (a x - 1\right )}^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}}{{\left (a x + 1\right )}^{2}} - 29 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{a^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*a*(18*arctan(((a*x - 1)/(a*x + 1))^(1/4))/a^4 - 9*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 + 9*log(abs((
(a*x - 1)/(a*x + 1))^(1/4) - 1))/a^4 - 4*(6*(a*x - 1)*((a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1) - 9*(a*x - 1)^2*((
a*x - 1)/(a*x + 1))^(3/4)/(a*x + 1)^2 - 29*((a*x - 1)/(a*x + 1))^(3/4))/(a^4*((a*x - 1)/(a*x + 1) - 1)^3))

________________________________________________________________________________________

maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 187, normalized size = 1.04 \[ -\frac {1}{48} \, a {\left (\frac {4 \, {\left (9 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{4}} - 6 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{4}} + 29 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{4}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{4}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{4}}{{\left (a x + 1\right )}^{3}} - a^{4}} + \frac {18 \, \arctan \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}}\right )}{a^{4}} - \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} + 1\right )}{a^{4}} + \frac {9 \, \log \left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{4}} - 1\right )}{a^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*a*(4*(9*((a*x - 1)/(a*x + 1))^(11/4) - 6*((a*x - 1)/(a*x + 1))^(7/4) + 29*((a*x - 1)/(a*x + 1))^(3/4))/(
3*(a*x - 1)*a^4/(a*x + 1) - 3*(a*x - 1)^2*a^4/(a*x + 1)^2 + (a*x - 1)^3*a^4/(a*x + 1)^3 - a^4) + 18*arctan(((a
*x - 1)/(a*x + 1))^(1/4))/a^4 - 9*log(((a*x - 1)/(a*x + 1))^(1/4) + 1)/a^4 + 9*log(((a*x - 1)/(a*x + 1))^(1/4)
- 1)/a^4)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 157, normalized size = 0.88 \[ \frac {\frac {29\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4}}{12}-\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/4}}{2}+\frac {3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/4}}{4}}{a^3+\frac {3\,a^3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a^3\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {3\,a^3\,\left (a\,x-1\right )}{a\,x+1}}-\frac {3\,\mathrm {atan}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3}+\frac {3\,\mathrm {atanh}\left ({\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4}\right )}{8\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((29*((a*x - 1)/(a*x + 1))^(3/4))/12 - ((a*x - 1)/(a*x + 1))^(7/4)/2 + (3*((a*x - 1)/(a*x + 1))^(11/4))/4)/(a^
3 + (3*a^3*(a*x - 1)^2)/(a*x + 1)^2 - (a^3*(a*x - 1)^3)/(a*x + 1)^3 - (3*a^3*(a*x - 1))/(a*x + 1)) - (3*atan((
(a*x - 1)/(a*x + 1))^(1/4)))/(8*a^3) + (3*atanh(((a*x - 1)/(a*x + 1))^(1/4)))/(8*a^3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt [4]{\frac {a x - 1}{a x + 1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))**(1/4)*x**2,x)

[Out]

Integral(x**2/((a*x - 1)/(a*x + 1))**(1/4), x)

________________________________________________________________________________________