3.1168 \(\int e^{3 \tanh ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\)

Optimal. Leaf size=185 \[ -\frac{c^4 (a x+1)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}+\frac{2 c^4 (a x+1)^9 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}-\frac{3 c^4 (a x+1)^8 \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{8 c^4 (a x+1)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}} \]

[Out]

(8*c^4*(1 + a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 + a*x)^8*Sqrt[c - a^2*c*x^2])/(2*a
*Sqrt[1 - a^2*x^2]) + (2*c^4*(1 + a*x)^9*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 + a*x)^10*Sqrt
[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.115192, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6143, 6140, 43} \[ -\frac{c^4 (a x+1)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}+\frac{2 c^4 (a x+1)^9 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}-\frac{3 c^4 (a x+1)^8 \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{8 c^4 (a x+1)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(9/2),x]

[Out]

(8*c^4*(1 + a*x)^7*Sqrt[c - a^2*c*x^2])/(7*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 + a*x)^8*Sqrt[c - a^2*c*x^2])/(2*a
*Sqrt[1 - a^2*x^2]) + (2*c^4*(1 + a*x)^9*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 + a*x)^10*Sqrt
[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])

Rule 6143

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

Rule 6140

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^3 (1+a x)^6 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int \left (8 (1+a x)^6-12 (1+a x)^7+6 (1+a x)^8-(1+a x)^9\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{8 c^4 (1+a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{3 c^4 (1+a x)^8 \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-a^2 x^2}}+\frac{2 c^4 (1+a x)^9 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}-\frac{c^4 (1+a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0528237, size = 68, normalized size = 0.37 \[ -\frac{c^4 (a x+1)^7 \left (21 a^3 x^3-77 a^2 x^2+98 a x-44\right ) \sqrt{c-a^2 c x^2}}{210 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(9/2),x]

[Out]

-(c^4*(1 + a*x)^7*Sqrt[c - a^2*c*x^2]*(-44 + 98*a*x - 77*a^2*x^2 + 21*a^3*x^3))/(210*a*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.03, size = 97, normalized size = 0.5 \begin{align*}{\frac{x \left ( 21\,{a}^{9}{x}^{9}+70\,{a}^{8}{x}^{8}-240\,{x}^{6}{a}^{6}-210\,{x}^{5}{a}^{5}+252\,{x}^{4}{a}^{4}+420\,{x}^{3}{a}^{3}-315\,ax-210 \right ) }{210\, \left ( ax+1 \right ) ^{3} \left ( ax-1 \right ) ^{3}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \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)*(-a^2*c*x^2+c)^(9/2),x)

[Out]

1/210*x*(21*a^9*x^9+70*a^8*x^8-240*a^6*x^6-210*a^5*x^5+252*a^4*x^4+420*a^3*x^3-315*a*x-210)*(-a^2*c*x^2+c)^(9/
2)/(a*x-1)^3/(a*x+1)^3/(-a^2*x^2+1)^(3/2)

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Maxima [B]  time = 1.1484, size = 552, normalized size = 2.98 \begin{align*} -\frac{1}{7} \, a^{6} c^{\frac{9}{2}} x^{7} + \frac{3}{5} \, a^{4} c^{\frac{9}{2}} x^{5} - a^{2} c^{\frac{9}{2}} x^{3} + c^{\frac{9}{2}} x + \frac{1}{40} \,{\left (\frac{4 \, a^{8} c^{5} x^{12}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac{19 \, a^{6} c^{5} x^{10}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac{35 \, a^{4} c^{5} x^{8}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac{30 \, a^{2} c^{5} x^{6}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac{10 \, c^{5} x^{4}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} - \frac{1}{105} \,{\left (35 \, a^{6} c^{\frac{9}{2}} x^{9} - 135 \, a^{4} c^{\frac{9}{2}} x^{7} + 189 \, a^{2} c^{\frac{9}{2}} x^{5} - 105 \, c^{\frac{9}{2}} x^{3}\right )} a^{2} + \frac{3}{8} \,{\left (\frac{a^{8} c^{5} x^{10}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac{5 \, a^{6} c^{5} x^{8}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac{10 \, a^{4} c^{5} x^{6}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac{10 \, a^{2} c^{5} x^{4}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac{4 \, c^{5}}{\sqrt{a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \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)*(-a^2*c*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

-1/7*a^6*c^(9/2)*x^7 + 3/5*a^4*c^(9/2)*x^5 - a^2*c^(9/2)*x^3 + c^(9/2)*x + 1/40*(4*a^8*c^5*x^12/sqrt(a^4*c*x^4
- 2*a^2*c*x^2 + c) - 19*a^6*c^5*x^10/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) + 35*a^4*c^5*x^8/sqrt(a^4*c*x^4 - 2*a^
2*c*x^2 + c) - 30*a^2*c^5*x^6/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) + 10*c^5*x^4/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c)
)*a^3 - 1/105*(35*a^6*c^(9/2)*x^9 - 135*a^4*c^(9/2)*x^7 + 189*a^2*c^(9/2)*x^5 - 105*c^(9/2)*x^3)*a^2 + 3/8*(a^
8*c^5*x^10/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) - 5*a^6*c^5*x^8/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) + 10*a^4*c^5*x^
6/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) - 10*a^2*c^5*x^4/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) + 4*c^5/(sqrt(a^4*c*x^4
- 2*a^2*c*x^2 + c)*a^2))*a

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Fricas [A]  time = 2.5971, size = 265, normalized size = 1.43 \begin{align*} \frac{{\left (21 \, a^{9} c^{4} x^{10} + 70 \, a^{8} c^{4} x^{9} - 240 \, a^{6} c^{4} x^{7} - 210 \, a^{5} c^{4} x^{6} + 252 \, a^{4} c^{4} x^{5} + 420 \, a^{3} c^{4} x^{4} - 315 \, a c^{4} x^{2} - 210 \, c^{4} x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{210 \,{\left (a^{2} x^{2} - 1\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)*(-a^2*c*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

1/210*(21*a^9*c^4*x^10 + 70*a^8*c^4*x^9 - 240*a^6*c^4*x^7 - 210*a^5*c^4*x^6 + 252*a^4*c^4*x^5 + 420*a^3*c^4*x^
4 - 315*a*c^4*x^2 - 210*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)*(-a**2*c*x**2+c)**(9/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}}{\left (a x + 1\right )}^{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)*(-a^2*c*x^2+c)^(9/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(9/2)*(a*x + 1)^3/(-a^2*x^2 + 1)^(3/2), x)