3.5.3 \(\int e^{3 \tanh ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\) [403]

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

[Out]

-2/3*(a*x+1)^(3/2)*(-a*c*x+c)^(3/2)/a^3/c/(-a*x+1)^(3/2)-2/7*(a*x+1)^(7/2)*(-a*c*x+c)^(3/2)/a^3/c/(-a*x+1)^(3/
2)+4*(-a*c*x+c)^(3/2)*arctanh(1/2*(a*x+1)^(1/2)*2^(1/2))*2^(1/2)/a^3/c/(-a*x+1)^(3/2)-4*(-a*c*x+c)^(3/2)*(a*x+
1)^(1/2)/a^3/c/(-a*x+1)^(3/2)

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Rubi [A]
time = 0.11, antiderivative size = 169, normalized size of antiderivative = 1.00, 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 = {6265, 23, 90, 52, 65, 214} \begin {gather*} -\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}} \end {gather*}

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 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 52

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 65

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 90

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 214

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

Rule 6265

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]
)

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 ) \text {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*}

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Mathematica [A]
time = 0.05, size = 84, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x} \left (52+16 a x+9 a^2 x^2+3 a^3 x^3\right )-42 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{21 a^3 \sqrt {1-a x}} \end {gather*}

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 = 1.19, size = 129, normalized size = 0.76

method	result	size
default	\(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (-3 a^{3} x^{3} \sqrt {\left (a x +1\right ) c}-9 a^{2} x^{2} \sqrt {\left (a x +1\right ) c}+42 \sqrt {c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right )-16 \sqrt {\left (a x +1\right ) c}\, a x -52 \sqrt {\left (a x +1\right ) c}\right )}{21 \left (a x -1\right ) \sqrt {\left (a x +1\right ) c}\, a^{3}}\)	\(129\)
risch	\(\frac {2 \left (3 a^{3} x^{3}+9 a^{2} x^{2}+16 a x +52\right ) \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{21 a^{3} \sqrt {\left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctanh \left (\frac {\sqrt {c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a^{3} \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\)	\(170\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.36, size = 258, normalized size = 1.53 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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 = 0.43, size = 130, normalized size = 0.77 \begin {gather*} \frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} {\left (21 \, c \arctan \left (\frac {\sqrt {c}}{\sqrt {-c}}\right ) + 40 \, \sqrt {-c} \sqrt {c}\right )}}{a^{2} \sqrt {-c} c} - \frac {\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}}{a^{2} c^{4}}\right )}}{21 \, a {\left | c \right |}} \end {gather*}

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]

2/21*c^2*(2*sqrt(2)*(21*c*arctan(sqrt(c)/sqrt(-c)) + 40*sqrt(-c)*sqrt(c))/(a^2*sqrt(-c)*c) - (42*sqrt(2)*c^4*a
rctan(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^2*c^4))/(a*abs(c))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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