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

Optimal. Leaf size=274 \[ \frac{c^2 x^{m+1} \sqrt{c-a^2 c x^2}}{(m+1) \sqrt{1-a^2 x^2}}+\frac{a c^2 x^{m+2} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-a^2 x^2}}-\frac{2 a^2 c^2 x^{m+3} \sqrt{c-a^2 c x^2}}{(m+3) \sqrt{1-a^2 x^2}}-\frac{2 a^3 c^2 x^{m+4} \sqrt{c-a^2 c x^2}}{(m+4) \sqrt{1-a^2 x^2}}+\frac{a^4 c^2 x^{m+5} \sqrt{c-a^2 c x^2}}{(m+5) \sqrt{1-a^2 x^2}}+\frac{a^5 c^2 x^{m+6} \sqrt{c-a^2 c x^2}}{(m+6) \sqrt{1-a^2 x^2}} \]

[Out]

(c^2*x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2]) + (a*c^2*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 +
m)*Sqrt[1 - a^2*x^2]) - (2*a^2*c^2*x^(3 + m)*Sqrt[c - a^2*c*x^2])/((3 + m)*Sqrt[1 - a^2*x^2]) - (2*a^3*c^2*x^(
4 + m)*Sqrt[c - a^2*c*x^2])/((4 + m)*Sqrt[1 - a^2*x^2]) + (a^4*c^2*x^(5 + m)*Sqrt[c - a^2*c*x^2])/((5 + m)*Sqr
t[1 - a^2*x^2]) + (a^5*c^2*x^(6 + m)*Sqrt[c - a^2*c*x^2])/((6 + m)*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.229839, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 88} \[ \frac{c^2 x^{m+1} \sqrt{c-a^2 c x^2}}{(m+1) \sqrt{1-a^2 x^2}}+\frac{a c^2 x^{m+2} \sqrt{c-a^2 c x^2}}{(m+2) \sqrt{1-a^2 x^2}}-\frac{2 a^2 c^2 x^{m+3} \sqrt{c-a^2 c x^2}}{(m+3) \sqrt{1-a^2 x^2}}-\frac{2 a^3 c^2 x^{m+4} \sqrt{c-a^2 c x^2}}{(m+4) \sqrt{1-a^2 x^2}}+\frac{a^4 c^2 x^{m+5} \sqrt{c-a^2 c x^2}}{(m+5) \sqrt{1-a^2 x^2}}+\frac{a^5 c^2 x^{m+6} \sqrt{c-a^2 c x^2}}{(m+6) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*x^(1 + m)*Sqrt[c - a^2*c*x^2])/((1 + m)*Sqrt[1 - a^2*x^2]) + (a*c^2*x^(2 + m)*Sqrt[c - a^2*c*x^2])/((2 +
m)*Sqrt[1 - a^2*x^2]) - (2*a^2*c^2*x^(3 + m)*Sqrt[c - a^2*c*x^2])/((3 + m)*Sqrt[1 - a^2*x^2]) - (2*a^3*c^2*x^(
4 + m)*Sqrt[c - a^2*c*x^2])/((4 + m)*Sqrt[1 - a^2*x^2]) + (a^4*c^2*x^(5 + m)*Sqrt[c - a^2*c*x^2])/((5 + m)*Sqr
t[1 - a^2*x^2]) + (a^5*c^2*x^(6 + m)*Sqrt[c - a^2*c*x^2])/((6 + m)*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^{\tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^{5/2} \, dx &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int e^{\tanh ^{-1}(a x)} x^m \left (1-a^2 x^2\right )^{5/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int x^m (1-a x)^2 (1+a x)^3 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^2 \sqrt{c-a^2 c x^2}\right ) \int \left (x^m+a x^{1+m}-2 a^2 x^{2+m}-2 a^3 x^{3+m}+a^4 x^{4+m}+a^5 x^{5+m}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{c^2 x^{1+m} \sqrt{c-a^2 c x^2}}{(1+m) \sqrt{1-a^2 x^2}}+\frac{a c^2 x^{2+m} \sqrt{c-a^2 c x^2}}{(2+m) \sqrt{1-a^2 x^2}}-\frac{2 a^2 c^2 x^{3+m} \sqrt{c-a^2 c x^2}}{(3+m) \sqrt{1-a^2 x^2}}-\frac{2 a^3 c^2 x^{4+m} \sqrt{c-a^2 c x^2}}{(4+m) \sqrt{1-a^2 x^2}}+\frac{a^4 c^2 x^{5+m} \sqrt{c-a^2 c x^2}}{(5+m) \sqrt{1-a^2 x^2}}+\frac{a^5 c^2 x^{6+m} \sqrt{c-a^2 c x^2}}{(6+m) \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.093042, size = 102, normalized size = 0.37 \[ \frac{c^2 x^{m+1} \sqrt{c-a^2 c x^2} \left (\frac{a^5 x^5}{m+6}+\frac{a^4 x^4}{m+5}-\frac{2 a^3 x^3}{m+4}-\frac{2 a^2 x^2}{m+3}+\frac{a x}{m+2}+\frac{1}{m+1}\right )}{\sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.033, size = 377, normalized size = 1.4 \begin{align*}{\frac{{x}^{1+m} \left ({a}^{5}{m}^{5}{x}^{5}+15\,{a}^{5}{m}^{4}{x}^{5}+85\,{a}^{5}{m}^{3}{x}^{5}+{a}^{4}{m}^{5}{x}^{4}+225\,{a}^{5}{m}^{2}{x}^{5}+16\,{a}^{4}{m}^{4}{x}^{4}+274\,{a}^{5}m{x}^{5}+95\,{a}^{4}{m}^{3}{x}^{4}-2\,{a}^{3}{m}^{5}{x}^{3}+120\,{x}^{5}{a}^{5}+260\,{a}^{4}{m}^{2}{x}^{4}-34\,{a}^{3}{m}^{4}{x}^{3}+324\,{a}^{4}m{x}^{4}-214\,{a}^{3}{m}^{3}{x}^{3}-2\,{a}^{2}{m}^{5}{x}^{2}+144\,{x}^{4}{a}^{4}-614\,{a}^{3}{m}^{2}{x}^{3}-36\,{a}^{2}{m}^{4}{x}^{2}-792\,{a}^{3}m{x}^{3}-242\,{a}^{2}{m}^{3}{x}^{2}+a{m}^{5}x-360\,{x}^{3}{a}^{3}-744\,{a}^{2}{m}^{2}{x}^{2}+19\,a{m}^{4}x-1016\,{a}^{2}m{x}^{2}+137\,a{m}^{3}x+{m}^{5}-480\,{a}^{2}{x}^{2}+461\,a{m}^{2}x+20\,{m}^{4}+702\,amx+155\,{m}^{3}+360\,ax+580\,{m}^{2}+1044\,m+720 \right ) }{ \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{2} \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) } \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{5}{2}}}{\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^m*(-a^2*c*x^2+c)^(5/2),x)

[Out]

x^(1+m)*(a^5*m^5*x^5+15*a^5*m^4*x^5+85*a^5*m^3*x^5+a^4*m^5*x^4+225*a^5*m^2*x^5+16*a^4*m^4*x^4+274*a^5*m*x^5+95
*a^4*m^3*x^4-2*a^3*m^5*x^3+120*a^5*x^5+260*a^4*m^2*x^4-34*a^3*m^4*x^3+324*a^4*m*x^4-214*a^3*m^3*x^3-2*a^2*m^5*
x^2+144*a^4*x^4-614*a^3*m^2*x^3-36*a^2*m^4*x^2-792*a^3*m*x^3-242*a^2*m^3*x^2+a*m^5*x-360*a^3*x^3-744*a^2*m^2*x
^2+19*a*m^4*x-1016*a^2*m*x^2+137*a*m^3*x+m^5-480*a^2*x^2+461*a*m^2*x+20*m^4+702*a*m*x+155*m^3+360*a*x+580*m^2+
1044*m+720)*(-a^2*c*x^2+c)^(5/2)/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)/(a*x-1)^2/(a*x+1)^2/(-a^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.01227, size = 194, normalized size = 0.71 \begin{align*} \frac{{\left ({\left (m^{2} + 6 \, m + 8\right )} a^{4} c^{\frac{5}{2}} x^{6} - 2 \,{\left (m^{2} + 8 \, m + 12\right )} a^{2} c^{\frac{5}{2}} x^{4} +{\left (m^{2} + 10 \, m + 24\right )} c^{\frac{5}{2}} x^{2}\right )} a x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} + \frac{{\left ({\left (m^{2} + 4 \, m + 3\right )} a^{4} c^{\frac{5}{2}} x^{5} - 2 \,{\left (m^{2} + 6 \, m + 5\right )} a^{2} c^{\frac{5}{2}} x^{3} +{\left (m^{2} + 8 \, m + 15\right )} c^{\frac{5}{2}} x\right )} x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \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^m*(-a^2*c*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

((m^2 + 6*m + 8)*a^4*c^(5/2)*x^6 - 2*(m^2 + 8*m + 12)*a^2*c^(5/2)*x^4 + (m^2 + 10*m + 24)*c^(5/2)*x^2)*a*x^m/(
m^3 + 12*m^2 + 44*m + 48) + ((m^2 + 4*m + 3)*a^4*c^(5/2)*x^5 - 2*(m^2 + 6*m + 5)*a^2*c^(5/2)*x^3 + (m^2 + 8*m
+ 15)*c^(5/2)*x)*x^m/(m^3 + 9*m^2 + 23*m + 15)

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Fricas [A]  time = 2.26833, size = 1058, normalized size = 3.86 \begin{align*} \frac{{\left ({\left (a^{5} c^{2} m^{5} + 15 \, a^{5} c^{2} m^{4} + 85 \, a^{5} c^{2} m^{3} + 225 \, a^{5} c^{2} m^{2} + 274 \, a^{5} c^{2} m + 120 \, a^{5} c^{2}\right )} x^{6} +{\left (a^{4} c^{2} m^{5} + 16 \, a^{4} c^{2} m^{4} + 95 \, a^{4} c^{2} m^{3} + 260 \, a^{4} c^{2} m^{2} + 324 \, a^{4} c^{2} m + 144 \, a^{4} c^{2}\right )} x^{5} - 2 \,{\left (a^{3} c^{2} m^{5} + 17 \, a^{3} c^{2} m^{4} + 107 \, a^{3} c^{2} m^{3} + 307 \, a^{3} c^{2} m^{2} + 396 \, a^{3} c^{2} m + 180 \, a^{3} c^{2}\right )} x^{4} - 2 \,{\left (a^{2} c^{2} m^{5} + 18 \, a^{2} c^{2} m^{4} + 121 \, a^{2} c^{2} m^{3} + 372 \, a^{2} c^{2} m^{2} + 508 \, a^{2} c^{2} m + 240 \, a^{2} c^{2}\right )} x^{3} +{\left (a c^{2} m^{5} + 19 \, a c^{2} m^{4} + 137 \, a c^{2} m^{3} + 461 \, a c^{2} m^{2} + 702 \, a c^{2} m + 360 \, a c^{2}\right )} x^{2} +{\left (c^{2} m^{5} + 20 \, c^{2} m^{4} + 155 \, c^{2} m^{3} + 580 \, c^{2} m^{2} + 1044 \, c^{2} m + 720 \, c^{2}\right )} x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} x^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} -{\left (a^{2} m^{6} + 21 \, a^{2} m^{5} + 175 \, a^{2} m^{4} + 735 \, a^{2} m^{3} + 1624 \, a^{2} m^{2} + 1764 \, a^{2} m + 720 \, a^{2}\right )} x^{2} + 1624 \, m^{2} + 1764 \, m + 720} \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^m*(-a^2*c*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

((a^5*c^2*m^5 + 15*a^5*c^2*m^4 + 85*a^5*c^2*m^3 + 225*a^5*c^2*m^2 + 274*a^5*c^2*m + 120*a^5*c^2)*x^6 + (a^4*c^
2*m^5 + 16*a^4*c^2*m^4 + 95*a^4*c^2*m^3 + 260*a^4*c^2*m^2 + 324*a^4*c^2*m + 144*a^4*c^2)*x^5 - 2*(a^3*c^2*m^5
+ 17*a^3*c^2*m^4 + 107*a^3*c^2*m^3 + 307*a^3*c^2*m^2 + 396*a^3*c^2*m + 180*a^3*c^2)*x^4 - 2*(a^2*c^2*m^5 + 18*
a^2*c^2*m^4 + 121*a^2*c^2*m^3 + 372*a^2*c^2*m^2 + 508*a^2*c^2*m + 240*a^2*c^2)*x^3 + (a*c^2*m^5 + 19*a*c^2*m^4
+ 137*a*c^2*m^3 + 461*a*c^2*m^2 + 702*a*c^2*m + 360*a*c^2)*x^2 + (c^2*m^5 + 20*c^2*m^4 + 155*c^2*m^3 + 580*c^
2*m^2 + 1044*c^2*m + 720*c^2)*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3
- (a^2*m^6 + 21*a^2*m^5 + 175*a^2*m^4 + 735*a^2*m^3 + 1624*a^2*m^2 + 1764*a^2*m + 720*a^2)*x^2 + 1624*m^2 + 1
764*m + 720)

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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)*x**m*(-a**2*c*x**2+c)**(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^{2} c x^{2} + c\right )}^{\frac{5}{2}}{\left (a x + 1\right )} x^{m}}{\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)*x^m*(-a^2*c*x^2+c)^(5/2),x, algorithm="giac")

[Out]

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