3.403 \(\int e^{3 \tanh ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx\)

Optimal. Leaf size=169 \[ -\frac{2 (a x+1)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}-\frac{2 (a x+1)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac{4 \sqrt{a x+1} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{a^3 c (1-a x)^{3/2}} \]

[Out]

(-4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(a^3*c*(1 - a*x)^(3/2)) - (2*(1 + a*x)^(3/2)*(c - a*c*x)^(3/2))/(3*a^3*c*
(1 - a*x)^(3/2)) - (2*(1 + a*x)^(7/2)*(c - a*c*x)^(3/2))/(7*a^3*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3
/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(a^3*c*(1 - a*x)^(3/2))

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Rubi [A]  time = 0.159788, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {6130, 23, 88, 50, 63, 208} \[ -\frac{2 (a x+1)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}-\frac{2 (a x+1)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac{4 \sqrt{a x+1} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{a^3 c (1-a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*x^2*Sqrt[c - a*c*x],x]

[Out]

(-4*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(a^3*c*(1 - a*x)^(3/2)) - (2*(1 + a*x)^(3/2)*(c - a*c*x)^(3/2))/(3*a^3*c*
(1 - a*x)^(3/2)) - (2*(1 + a*x)^(7/2)*(c - a*c*x)^(3/2))/(7*a^3*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*(c - a*c*x)^(3
/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(a^3*c*(1 - a*x)^(3/2))

Rule 6130

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 88

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int e^{3 \tanh ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=\int \frac{x^2 (1+a x)^{3/2} \sqrt{c-a c x}}{(1-a x)^{3/2}} \, dx\\ &=\frac{(c-a c x)^{3/2} \int \frac{x^2 (1+a x)^{3/2}}{c-a c x} \, dx}{(1-a x)^{3/2}}\\ &=\frac{(c-a c x)^{3/2} \int \left (-\frac{(1+a x)^{5/2}}{a^2 c}+\frac{(1+a x)^{3/2}}{a^2 (c-a c x)}\right ) \, dx}{(1-a x)^{3/2}}\\ &=-\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac{(c-a c x)^{3/2} \int \frac{(1+a x)^{3/2}}{c-a c x} \, dx}{a^2 (1-a x)^{3/2}}\\ &=-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac{\left (2 (c-a c x)^{3/2}\right ) \int \frac{\sqrt{1+a x}}{c-a c x} \, dx}{a^2 (1-a x)^{3/2}}\\ &=-\frac{4 \sqrt{1+a x} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac{\left (4 (c-a c x)^{3/2}\right ) \int \frac{1}{\sqrt{1+a x} (c-a c x)} \, dx}{a^2 (1-a x)^{3/2}}\\ &=-\frac{4 \sqrt{1+a x} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac{\left (8 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-c x^2} \, dx,x,\sqrt{1+a x}\right )}{a^3 (1-a x)^{3/2}}\\ &=-\frac{4 \sqrt{1+a x} (c-a c x)^{3/2}}{a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{3/2} (c-a c x)^{3/2}}{3 a^3 c (1-a x)^{3/2}}-\frac{2 (1+a x)^{7/2} (c-a c x)^{3/2}}{7 a^3 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+a x}}{\sqrt{2}}\right )}{a^3 c (1-a x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0637418, size = 84, normalized size = 0.5 \[ -\frac{2 \sqrt{c-a c x} \left (\sqrt{a x+1} \left (3 a^3 x^3+9 a^2 x^2+16 a x+52\right )-42 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{21 a^3 \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])*x^2*Sqrt[c - a*c*x],x]

[Out]

(-2*Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(52 + 16*a*x + 9*a^2*x^2 + 3*a^3*x^3) - 42*Sqrt[2]*ArcTanh[Sqrt[1 + a*x]/Sq
rt[2]]))/(21*a^3*Sqrt[1 - a*x])

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Maple [A]  time = 0.13, size = 129, normalized size = 0.8 \begin{align*} -{\frac{2}{ \left ( 21\,ax-21 \right ){a}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( -3\,{x}^{3}{a}^{3}\sqrt{c \left ( ax+1 \right ) }-9\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }+42\,\sqrt{c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -16\,xa\sqrt{c \left ( ax+1 \right ) }-52\,\sqrt{c \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}} \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^2*(-a*c*x+c)^(1/2),x)

[Out]

-2/21*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(-3*x^3*a^3*(c*(a*x+1))^(1/2)-9*x^2*a^2*(c*(a*x+1))^(1/2)+42*c^(1/
2)*2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-16*x*a*(c*(a*x+1))^(1/2)-52*(c*(a*x+1))^(1/2))/(a*x-
1)/(c*(a*x+1))^(1/2)/a^3

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

[Out]

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

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Fricas [A]  time = 1.84437, size = 605, normalized size = 3.58 \begin{align*} \left [\frac{2 \,{\left (21 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (3 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 16 \, a x + 52\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right )}}{21 \,{\left (a^{4} x - a^{3}\right )}}, \frac{2 \,{\left (42 \, \sqrt{2}{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (3 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 16 \, a x + 52\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}\right )}}{21 \,{\left (a^{4} x - a^{3}\right )}}\right ] \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^2*(-a*c*x+c)^(1/2),x, algorithm="fricas")

[Out]

[2/21*(21*sqrt(2)*(a*x - 1)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*
sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + (3*a^3*x^3 + 9*a^2*x^2 + 16*a*x + 52)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x +
c))/(a^4*x - a^3), 2/21*(42*sqrt(2)*(a*x - 1)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqr
t(-c)/(a^2*c*x^2 - c)) + (3*a^3*x^3 + 9*a^2*x^2 + 16*a*x + 52)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a^4*x - a
^3)]

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

[Out]

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

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Giac [A]  time = 1.36461, size = 169, normalized size = 1. \begin{align*} \frac{4 \, \sqrt{2}{\left (21 \, c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) + 40 \, \sqrt{-c} c^{\frac{3}{2}}\right )}}{21 \, a^{3} \sqrt{-c}{\left | c \right |}} - \frac{2 \,{\left (\frac{42 \, \sqrt{2} c^{4} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c}} + 3 \,{\left (a c x + c\right )}^{\frac{7}{2}} + 7 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{2} + 42 \, \sqrt{a c x + c} c^{3}\right )}}{21 \, a^{3} c^{2}{\left | c \right |}} \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^2*(-a*c*x+c)^(1/2),x, algorithm="giac")

[Out]

4/21*sqrt(2)*(21*c^2*arctan(sqrt(c)/sqrt(-c)) + 40*sqrt(-c)*c^(3/2))/(a^3*sqrt(-c)*abs(c)) - 2/21*(42*sqrt(2)*
c^4*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/sqrt(-c) + 3*(a*c*x + c)^(7/2) + 7*(a*c*x + c)^(3/2)*c^2 + 42
*sqrt(a*c*x + c)*c^3)/(a^3*c^2*abs(c))