3.1262 \(\int e^{-3 \tanh ^{-1}(a x)} x^3 \sqrt{c-a^2 c x^2} \, dx\)

Optimal. Leaf size=225 \[ \frac{a x^5 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-a^2 x^2}}-\frac{3 x^4 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-a^2 x^2}}+\frac{4 x^3 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-a^2 x^2}}+\frac{4 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^4 \sqrt{1-a^2 x^2}} \]

[Out]

(4*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - a^2*x^2]) - (2*x^2*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - a^2*x^2]) + (4*x
^3*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (3*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2]) + (a*x^5*S
qrt[c - a^2*c*x^2])/(5*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^4*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.224854, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 88} \[ \frac{a x^5 \sqrt{c-a^2 c x^2}}{5 \sqrt{1-a^2 x^2}}-\frac{3 x^4 \sqrt{c-a^2 c x^2}}{4 \sqrt{1-a^2 x^2}}+\frac{4 x^3 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{a^2 \sqrt{1-a^2 x^2}}+\frac{4 x \sqrt{c-a^2 c x^2}}{a^3 \sqrt{1-a^2 x^2}}-\frac{4 \sqrt{c-a^2 c x^2} \log (a x+1)}{a^4 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x*Sqrt[c - a^2*c*x^2])/(a^3*Sqrt[1 - a^2*x^2]) - (2*x^2*Sqrt[c - a^2*c*x^2])/(a^2*Sqrt[1 - a^2*x^2]) + (4*x
^3*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (3*x^4*Sqrt[c - a^2*c*x^2])/(4*Sqrt[1 - a^2*x^2]) + (a*x^5*S
qrt[c - a^2*c*x^2])/(5*Sqrt[1 - a^2*x^2]) - (4*Sqrt[c - a^2*c*x^2]*Log[1 + a*x])/(a^4*Sqrt[1 - a^2*x^2])

Rule 6153

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

Rule 6150

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

Rubi steps

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

Mathematica [A]  time = 0.0573029, size = 81, normalized size = 0.36 \[ \frac{\sqrt{c-a^2 c x^2} \left (-\frac{2 x^2}{a^2}+\frac{4 x}{a^3}-\frac{4 \log (a x+1)}{a^4}+\frac{a x^5}{5}+\frac{4 x^3}{3 a}-\frac{3 x^4}{4}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - a^2*c*x^2]*((4*x)/a^3 - (2*x^2)/a^2 + (4*x^3)/(3*a) - (3*x^4)/4 + (a*x^5)/5 - (4*Log[1 + a*x])/a^4))
/Sqrt[1 - a^2*x^2]

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Maple [A]  time = 0.086, size = 88, normalized size = 0.4 \begin{align*}{\frac{-12\,{x}^{5}{a}^{5}+45\,{x}^{4}{a}^{4}-80\,{x}^{3}{a}^{3}+120\,{a}^{2}{x}^{2}-240\,ax+240\,\ln \left ( ax+1 \right ) }{ \left ( 60\,{a}^{2}{x}^{2}-60 \right ){a}^{4}}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)*(-12*x^5*a^5+45*x^4*a^4-80*x^3*a^3+120*a^2*x^2-240*a*x+240*ln(a
*x+1))/(a^2*x^2-1)/a^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.56111, size = 869, normalized size = 3.86 \begin{align*} \left [\frac{120 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \log \left (\frac{a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x +{\left (a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) -{\left (12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{60 \,{\left (a^{6} x^{2} - a^{4}\right )}}, -\frac{240 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right ) +{\left (12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{60 \,{\left (a^{6} x^{2} - a^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(120*(a^2*x^2 - 1)*sqrt(c)*log((a^6*c*x^6 + 4*a^5*c*x^5 + 5*a^4*c*x^4 - 4*a^2*c*x^2 - 4*a*c*x + (a^4*x^4
+ 4*a^3*x^3 + 6*a^2*x^2 + 4*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - 2*c)/(a^4*x^4 + 2*a^3*x^3
- 2*a*x - 1)) - (12*a^5*x^5 - 45*a^4*x^4 + 80*a^3*x^3 - 120*a^2*x^2 + 240*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*
x^2 + 1))/(a^6*x^2 - a^4), -1/60*(240*(a^2*x^2 - 1)*sqrt(-c)*arctan(sqrt(-a^2*c*x^2 + c)*(a^2*x^2 + 2*a*x + 2)
*sqrt(-a^2*x^2 + 1)*sqrt(-c)/(a^4*c*x^4 + 2*a^3*c*x^3 - a^2*c*x^2 - 2*a*c*x)) + (12*a^5*x^5 - 45*a^4*x^4 + 80*
a^3*x^3 - 120*a^2*x^2 + 240*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^6*x^2 - a^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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