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

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

[Out]

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

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Rubi [A]  time = 0.19981, 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 = {6192, 6193, 43} \[ \frac{(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{2 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{2 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6192

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

Rule 6193

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

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

Mathematica [A]  time = 0.0575649, size = 71, normalized size = 0.38 \[ \frac{c^4 (a x+1)^7 \left (21 a^3 x^3-77 a^2 x^2+98 a x-44\right ) \sqrt{c-a^2 c x^2}}{210 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcCoth[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^2*Sqrt[1 - 1/(a^2*x^2)]*
x)

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Maple [A]  time = 0.128, size = 100, normalized size = 0.5 \begin{align*}{\frac{x \left ( 21\,{a}^{9}{x}^{9}+70\,{x}^{8}{a}^{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 ) ^{6}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+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)^6/((a*x-1)/(a*x+1))^(3/2)

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Maxima [A]  time = 1.07743, size = 275, normalized size = 1.49 \begin{align*} \frac{{\left (21 \, a^{11} \sqrt{-c} c^{4} x^{11} + 49 \, a^{10} \sqrt{-c} c^{4} x^{10} - 70 \, a^{9} \sqrt{-c} c^{4} x^{9} - 240 \, a^{8} \sqrt{-c} c^{4} x^{8} + 30 \, a^{7} \sqrt{-c} c^{4} x^{7} + 462 \, a^{6} \sqrt{-c} c^{4} x^{6} + 168 \, a^{5} \sqrt{-c} c^{4} x^{5} - 420 \, a^{4} \sqrt{-c} c^{4} x^{4} - 315 \, a^{3} \sqrt{-c} c^{4} x^{3} + 105 \, a^{2} \sqrt{-c} c^{4} x^{2} + 210 \, \sqrt{-c} c^{4}\right )}{\left (a x + 1\right )}^{2}}{210 \,{\left (a^{3} x^{2} + 2 \, a^{2} x + a\right )}{\left (a x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/210*(21*a^11*sqrt(-c)*c^4*x^11 + 49*a^10*sqrt(-c)*c^4*x^10 - 70*a^9*sqrt(-c)*c^4*x^9 - 240*a^8*sqrt(-c)*c^4*
x^8 + 30*a^7*sqrt(-c)*c^4*x^7 + 462*a^6*sqrt(-c)*c^4*x^6 + 168*a^5*sqrt(-c)*c^4*x^5 - 420*a^4*sqrt(-c)*c^4*x^4
 - 315*a^3*sqrt(-c)*c^4*x^3 + 105*a^2*sqrt(-c)*c^4*x^2 + 210*sqrt(-c)*c^4)*(a*x + 1)^2/((a^3*x^2 + 2*a^2*x + a
)*(a*x - 1))

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Fricas [A]  time = 1.50789, size = 212, normalized size = 1.15 \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}}{210 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+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)/a

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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(1/((a*x-1)/(a*x+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 (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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