3.582 \(\int e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx\)

Optimal. Leaf size=292 \[ -\frac{11 x^2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{24 a^2 \sqrt{1-a x}}-\frac{21 x (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{32 a^3 \sqrt{1-a x}}-\frac{107 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{64 a^3 \sqrt{1-a x}}-\frac{363 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{64 a^{7/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{7/2} \sqrt{1-a x}}-\frac{x^3 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]

[Out]

(-107*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(64*a^3*Sqrt[1 - a*x]) - (21*Sqrt[c - c/(a*x)]*x*(1 + a*x)^(3/2))/(32
*a^3*Sqrt[1 - a*x]) - (11*Sqrt[c - c/(a*x)]*x^2*(1 + a*x)^(3/2))/(24*a^2*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x
^3*(1 + a*x)^(3/2))/(4*a*Sqrt[1 - a*x]) - (363*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(64*a^(7/2)
*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(7
/2)*Sqrt[1 - a*x])

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Rubi [A]  time = 0.313426, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6134, 6129, 101, 154, 157, 54, 215, 93, 206} \[ -\frac{11 x^2 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{24 a^2 \sqrt{1-a x}}-\frac{21 x (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{32 a^3 \sqrt{1-a x}}-\frac{107 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{64 a^3 \sqrt{1-a x}}-\frac{363 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{64 a^{7/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{7/2} \sqrt{1-a x}}-\frac{x^3 (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^3,x]

[Out]

(-107*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(64*a^3*Sqrt[1 - a*x]) - (21*Sqrt[c - c/(a*x)]*x*(1 + a*x)^(3/2))/(32
*a^3*Sqrt[1 - a*x]) - (11*Sqrt[c - c/(a*x)]*x^2*(1 + a*x)^(3/2))/(24*a^2*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x
^3*(1 + a*x)^(3/2))/(4*a*Sqrt[1 - a*x]) - (363*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(64*a^(7/2)
*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(7
/2)*Sqrt[1 - a*x])

Rule 6134

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Dist[(x^p*(c + d/x)^p)/(1 + (c*
x)/d)^p, Int[(u*(1 + (c*x)/d)^p*E^(n*ArcTanh[a*x]))/x^p, x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*
d^2, 0] &&  !IntegerQ[p]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 154

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

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

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 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])

Rubi steps

\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^3 \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{3 \tanh ^{-1}(a x)} x^{5/2} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{5/2} (1+a x)^{3/2}}{1-a x} \, dx}{\sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2} \sqrt{1+a x} \left (\frac{5}{2}+\frac{11 a x}{2}\right )}{1-a x} \, dx}{4 a \sqrt{1-a x}}\\ &=-\frac{11 \sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}-\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x} \sqrt{1+a x} \left (-\frac{33 a}{4}-\frac{63 a^2 x}{4}\right )}{1-a x} \, dx}{12 a^3 \sqrt{1-a x}}\\ &=-\frac{21 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt{1-a x}}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1+a x} \left (\frac{63 a^2}{8}+\frac{321 a^3 x}{8}\right )}{\sqrt{x} (1-a x)} \, dx}{24 a^5 \sqrt{1-a x}}\\ &=-\frac{107 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{64 a^3 \sqrt{1-a x}}-\frac{21 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt{1-a x}}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}-\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-\frac{447 a^3}{16}-\frac{1089 a^4 x}{16}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{24 a^6 \sqrt{1-a x}}\\ &=-\frac{107 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{64 a^3 \sqrt{1-a x}}-\frac{21 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt{1-a x}}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}-\frac{\left (363 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{128 a^3 \sqrt{1-a x}}+\frac{\left (4 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a^3 \sqrt{1-a x}}\\ &=-\frac{107 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{64 a^3 \sqrt{1-a x}}-\frac{21 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt{1-a x}}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}-\frac{\left (363 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{64 a^3 \sqrt{1-a x}}+\frac{\left (8 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a^3 \sqrt{1-a x}}\\ &=-\frac{107 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{64 a^3 \sqrt{1-a x}}-\frac{21 \sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt{1-a x}}-\frac{11 \sqrt{c-\frac{c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt{1-a x}}-\frac{363 \sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{64 a^{7/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{a^{7/2} \sqrt{1-a x}}\\ \end{align*}

Mathematica [A]  time = 0.0990608, size = 130, normalized size = 0.45 \[ -\frac{\sqrt{c-\frac{c}{a x}} \left (\sqrt{a} x \sqrt{a x+1} \left (48 a^3 x^3+136 a^2 x^2+214 a x+447\right )+1089 \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-768 \sqrt{2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{192 a^{7/2} \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])*Sqrt[c - c/(a*x)]*x^3,x]

[Out]

-(Sqrt[c - c/(a*x)]*(Sqrt[a]*x*Sqrt[1 + a*x]*(447 + 214*a*x + 136*a^2*x^2 + 48*a^3*x^3) + 1089*Sqrt[x]*ArcSinh
[Sqrt[a]*Sqrt[x]] - 768*Sqrt[2]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(192*a^(7/2)*Sqrt[1
- a*x])

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Maple [A]  time = 0.149, size = 247, normalized size = 0.9 \begin{align*}{\frac{x\sqrt{2}}{768\,ax-768}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 96\,{a}^{9/2}\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}{x}^{3}+272\,\sqrt{- \left ( ax+1 \right ) x}{a}^{7/2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}+428\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x+894\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-1089\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+1536\,\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ) \sqrt{a} \right ){a}^{-{\frac{9}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{-{a}^{-1}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*x^3*(c-c/a/x)^(1/2),x)

[Out]

1/768*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(96*a^(9/2)*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*x^3+272*(
-(a*x+1)*x)^(1/2)*a^(7/2)*2^(1/2)*(-1/a)^(1/2)*x^2+428*(-(a*x+1)*x)^(1/2)*a^(5/2)*2^(1/2)*(-1/a)^(1/2)*x+894*(
-(a*x+1)*x)^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)-1089*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)
*(-1/a)^(1/2)+1536*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a-3*a*x-1)/(a*x-1))*a^(1/2))*2^(1/2)/a^(9/2)/
(a*x-1)/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3} \sqrt{c - \frac{c}{a x}} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x + 1)^3*sqrt(c - c/(a*x))*x^3/(-a^2*x^2 + 1)^(3/2), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*x**3*(c-c/a/x)**(1/2),x)

[Out]

Integral(x**3*sqrt(-c*(-1 + 1/(a*x)))*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3} \sqrt{c - \frac{c}{a x}} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*x + 1)^3*sqrt(c - c/(a*x))*x^3/(-a^2*x^2 + 1)^(3/2), x)