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

Optimal. Leaf size=107 \[ \frac{2 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a (c-a c x)^{3/2}}-\frac{8 c^2 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (c-a c x)^{3/2}}+\frac{8 c \left (1-a^2 x^2\right )^{3/2}}{35 a^3 \sqrt{c-a c x}} \]

[Out]

(-8*c^2*(1 - a^2*x^2)^(3/2))/(105*a^3*(c - a*c*x)^(3/2)) + (2*c^2*x^2*(1 - a^2*x^2)^(3/2))/(7*a*(c - a*c*x)^(3
/2)) + (8*c*(1 - a^2*x^2)^(3/2))/(35*a^3*Sqrt[c - a*c*x])

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Rubi [A]  time = 0.158381, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {6128, 871, 795, 649} \[ \frac{2 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a (c-a c x)^{3/2}}-\frac{8 c^2 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (c-a c x)^{3/2}}+\frac{8 c \left (1-a^2 x^2\right )^{3/2}}{35 a^3 \sqrt{c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c^2*(1 - a^2*x^2)^(3/2))/(105*a^3*(c - a*c*x)^(3/2)) + (2*c^2*x^2*(1 - a^2*x^2)^(3/2))/(7*a*(c - a*c*x)^(3
/2)) + (8*c*(1 - a^2*x^2)^(3/2))/(35*a^3*Sqrt[c - a*c*x])

Rule 6128

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

Rule 871

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

Rule 795

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2)), Int[(d +
e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && NeQ[m + 2*p +
2, 0] && NeQ[m, 2]

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rubi steps

\begin{align*} \int e^{\tanh ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=c \int \frac{x^2 \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx\\ &=\frac{2 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a (c-a c x)^{3/2}}-\frac{(4 c) \int \frac{x \sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx}{7 a}\\ &=\frac{2 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a (c-a c x)^{3/2}}+\frac{8 c \left (1-a^2 x^2\right )^{3/2}}{35 a^3 \sqrt{c-a c x}}-\frac{(4 c) \int \frac{\sqrt{1-a^2 x^2}}{\sqrt{c-a c x}} \, dx}{35 a^2}\\ &=-\frac{8 c^2 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 (c-a c x)^{3/2}}+\frac{2 c^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{7 a (c-a c x)^{3/2}}+\frac{8 c \left (1-a^2 x^2\right )^{3/2}}{35 a^3 \sqrt{c-a c x}}\\ \end{align*}

Mathematica [A]  time = 0.0282366, size = 51, normalized size = 0.48 \[ \frac{2 (a x+1)^{3/2} \left (15 a^2 x^2-12 a x+8\right ) \sqrt{c-a c x}}{105 a^3 \sqrt{1-a x}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*(1 + a*x)^(3/2)*Sqrt[c - a*c*x]*(8 - 12*a*x + 15*a^2*x^2))/(105*a^3*Sqrt[1 - a*x])

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Maple [A]  time = 0.039, size = 48, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( ax+1 \right ) ^{2} \left ( 15\,{a}^{2}{x}^{2}-12\,ax+8 \right ) }{105\,{a}^{3}}\sqrt{-acx+c}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(a*x+1)^2*(15*a^2*x^2-12*a*x+8)*(-a*c*x+c)^(1/2)/a^3/(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.03429, size = 143, normalized size = 1.34 \begin{align*} \frac{2 \,{\left (5 \, a^{4} \sqrt{c} x^{4} - a^{3} \sqrt{c} x^{3} + 2 \, a^{2} \sqrt{c} x^{2} - 8 \, a \sqrt{c} x - 16 \, \sqrt{c}\right )}}{35 \, \sqrt{a x + 1} a^{3}} + \frac{2 \,{\left (3 \, a^{3} \sqrt{c} x^{3} - a^{2} \sqrt{c} x^{2} + 4 \, a \sqrt{c} x + 8 \, \sqrt{c}\right )}}{15 \, \sqrt{a x + 1} a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^4*sqrt(c)*x^4 - a^3*sqrt(c)*x^3 + 2*a^2*sqrt(c)*x^2 - 8*a*sqrt(c)*x - 16*sqrt(c))/(sqrt(a*x + 1)*a^3
) + 2/15*(3*a^3*sqrt(c)*x^3 - a^2*sqrt(c)*x^2 + 4*a*sqrt(c)*x + 8*sqrt(c))/(sqrt(a*x + 1)*a^3)

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Fricas [A]  time = 1.80732, size = 128, normalized size = 1.2 \begin{align*} -\frac{2 \,{\left (15 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - 4 \, a x + 8\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{105 \,{\left (a^{4} x - a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(15*a^3*x^3 + 3*a^2*x^2 - 4*a*x + 8)*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 )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.37634, size = 88, normalized size = 0.82 \begin{align*} -\frac{2 \, c{\left (\frac{22 \, \sqrt{2} \sqrt{c}}{a^{2}} - \frac{15 \,{\left (a c x + c\right )}^{\frac{7}{2}} - 42 \,{\left (a c x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{2}}{a^{2} c^{3}}\right )}}{105 \, a{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*c*(22*sqrt(2)*sqrt(c)/a^2 - (15*(a*c*x + c)^(7/2) - 42*(a*c*x + c)^(5/2)*c + 35*(a*c*x + c)^(3/2)*c^2)/
(a^2*c^3))/(a*abs(c))