3.688 \(\int e^{\tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{5/2} \, dx\)

Optimal. Leaf size=219 \[ \frac{a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac{2 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^4 x^5 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \]

[Out]

-((c - c/(a^2*x^2))^(5/2)*x)/(4*(1 - a^2*x^2)^(5/2)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(3*(1 - a^2*x^2)^(5/2))
+ (a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(1 - a^2*x^2)^(5/2) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(1 - a^2*x^2)^(
5/2) + (a^5*(c - c/(a^2*x^2))^(5/2)*x^6)/(1 - a^2*x^2)^(5/2) + (a^4*(c - c/(a^2*x^2))^(5/2)*x^5*Log[x])/(1 - a
^2*x^2)^(5/2)

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Rubi [A]  time = 0.178567, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6160, 6150, 88} \[ \frac{a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac{2 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{3 \left (1-a^2 x^2\right )^{5/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^4 x^5 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{5/2}}{\left (1-a^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(5/2),x]

[Out]

-((c - c/(a^2*x^2))^(5/2)*x)/(4*(1 - a^2*x^2)^(5/2)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(3*(1 - a^2*x^2)^(5/2))
+ (a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(1 - a^2*x^2)^(5/2) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(1 - a^2*x^2)^(
5/2) + (a^5*(c - c/(a^2*x^2))^(5/2)*x^6)/(1 - a^2*x^2)^(5/2) + (a^4*(c - c/(a^2*x^2))^(5/2)*x^5*Log[x])/(1 - a
^2*x^2)^(5/2)

Rule 6160

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/
(1 + (c*x^2)/d)^p, Int[(u*(1 + (c*x^2)/d)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] &
& EqQ[c + a^2*d, 0] &&  !IntegerQ[p] &&  !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^{\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{5/2} \, dx &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2}}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \frac{(1-a x)^2 (1+a x)^3}{x^5} \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5\right ) \int \left (a^5+\frac{1}{x^5}+\frac{a}{x^4}-\frac{2 a^2}{x^3}-\frac{2 a^3}{x^2}+\frac{a^4}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{5/2}}\\ &=-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} x}{4 \left (1-a^2 x^2\right )^{5/2}}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^2}{3 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^3}{\left (1-a^2 x^2\right )^{5/2}}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^4}{\left (1-a^2 x^2\right )^{5/2}}+\frac{a^5 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^6}{\left (1-a^2 x^2\right )^{5/2}}+\frac{a^4 \left (c-\frac{c}{a^2 x^2}\right )^{5/2} x^5 \log (x)}{\left (1-a^2 x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0462039, size = 82, normalized size = 0.37 \[ \frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}} \left (12 a^5 x^5+24 a^3 x^3+12 a^2 x^2+12 a^4 x^4 \log (x)-4 a x-3\right )}{12 a^4 x^3 \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]*(c - c/(a^2*x^2))^(5/2),x]

[Out]

(c^2*Sqrt[c - c/(a^2*x^2)]*(-3 - 4*a*x + 12*a^2*x^2 + 24*a^3*x^3 + 12*a^5*x^5 + 12*a^4*x^4*Log[x]))/(12*a^4*x^
3*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.153, size = 86, normalized size = 0.4 \begin{align*} -{\frac{x \left ( 12\,{x}^{5}{a}^{5}+12\,{a}^{4}\ln \left ( x \right ){x}^{4}+24\,{x}^{3}{a}^{3}+12\,{a}^{2}{x}^{2}-4\,ax-3 \right ) }{12\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{5}{2}}}\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)*(c-c/a^2/x^2)^(5/2),x)

[Out]

-1/12*(c*(a^2*x^2-1)/a^2/x^2)^(5/2)*x/(a^2*x^2-1)^3*(-a^2*x^2+1)^(1/2)*(12*x^5*a^5+12*a^4*ln(x)*x^4+24*x^3*a^3
+12*a^2*x^2-4*a*x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)*(c - c/(a^2*x^2))^(5/2)/sqrt(-a^2*x^2 + 1), x)

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Fricas [A]  time = 2.63352, size = 984, normalized size = 4.49 \begin{align*} \left [\frac{6 \,{\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} +{\left (a x^{5} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) -{\left (12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} -{\left (12 \, a^{5} + 24 \, a^{3} + 12 \, a^{2} - 4 \, a - 3\right )} c^{2} x^{4} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \,{\left (a^{6} x^{5} - a^{4} x^{3}\right )}}, -\frac{12 \,{\left (a^{5} c^{2} x^{5} - a^{3} c^{2} x^{3}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + a x\right )} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) +{\left (12 \, a^{5} c^{2} x^{5} + 24 \, a^{3} c^{2} x^{3} -{\left (12 \, a^{5} + 24 \, a^{3} + 12 \, a^{2} - 4 \, a - 3\right )} c^{2} x^{4} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{12 \,{\left (a^{6} x^{5} - a^{4} x^{3}\right )}}\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)*(c-c/a^2/x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/12*(6*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*x^2
+ 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) - (12*a^5*c^2*x^5 + 24*a^3*c^2*x^3 - (12*
a^5 + 24*a^3 + 12*a^2 - 4*a - 3)*c^2*x^4 + 12*a^2*c^2*x^2 - 4*a*c^2*x - 3*c^2)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*
x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3), -1/12*(12*(a^5*c^2*x^5 - a^3*c^2*x^3)*sqrt(c)*arctan(sqrt(-a^2*x^2 +
1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)*c*x^2 + c)) + (12*a^5*c^2*x^5
+ 24*a^3*c^2*x^3 - (12*a^5 + 24*a^3 + 12*a^2 - 4*a - 3)*c^2*x^4 + 12*a^2*c^2*x^2 - 4*a*c^2*x - 3*c^2)*sqrt(-a
^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^6*x^5 - a^4*x^3)]

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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)/(-a**2*x**2+1)**(1/2)*(c-c/a**2/x**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{2}}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(c-c/a^2/x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((a*x + 1)*(c - c/(a^2*x^2))^(5/2)/sqrt(-a^2*x^2 + 1), x)