3.1378 \(\int \frac{e^{-5 \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^{27/2}} \, dx\)

Optimal. Leaf size=60 \[ \frac{(5 a x+1) \sqrt{1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (a x+1)^{15} \sqrt{c-a^2 c x^2}} \]

[Out]

((1 + 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(1 - a*x)^10*(1 + a*x)^15*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.237762, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6153, 6150, 81} \[ \frac{(5 a x+1) \sqrt{1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (a x+1)^{15} \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(E^(5*ArcTanh[a*x])*(c - a^2*c*x^2)^(27/2)),x]

[Out]

((1 + 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(1 - a*x)^10*(1 + a*x)^15*Sqrt[c - a^2*c*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 81

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*
x)^(n + 1)*(e + f*x)^(p + 1)*(2*a*d*f*(n + p + 3) - b*(d*e*(n + 2) + c*f*(p + 2)) + b*d*f*(n + p + 2)*x))/(d^2
*f^2*(n + p + 2)*(n + p + 3)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && NeQ[n + p + 3,
0] && EqQ[d*f*(n + p + 2)*(a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1)))) - b*(d*e*(n + 1)
+ c*f*(p + 1))*(a*d*f*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2))), 0]

Rubi steps

\begin{align*} \int \frac{e^{-5 \tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{-5 \tanh ^{-1}(a x)} x^2}{\left (1-a^2 x^2\right )^{27/2}} \, dx}{c^{13} \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^2}{(1-a x)^{11} (1+a x)^{16}} \, dx}{c^{13} \sqrt{c-a^2 c x^2}}\\ &=\frac{(1+5 a x) \sqrt{1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{10} (1+a x)^{15} \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.466298, size = 59, normalized size = 0.98 \[ \frac{(5 a x+1) \sqrt{1-a^2 x^2}}{120 a^3 c^{13} (a x-1)^{10} (a x+1)^{15} \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(E^(5*ArcTanh[a*x])*(c - a^2*c*x^2)^(27/2)),x]

[Out]

((1 + 5*a*x)*Sqrt[1 - a^2*x^2])/(120*a^3*c^13*(-1 + a*x)^10*(1 + a*x)^15*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.037, size = 49, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 5\,ax+1 \right ) \left ( ax-1 \right ) }{120\,{a}^{3} \left ( ax+1 \right ) ^{4}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{27}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a*x+1)^5*(-a^2*x^2+1)^(5/2)/(-a^2*c*x^2+c)^(27/2),x)

[Out]

-1/120*(a*x-1)*(5*a*x+1)*(-a^2*x^2+1)^(5/2)/(a*x+1)^4/a^3/(-a^2*c*x^2+c)^(27/2)

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Maxima [B]  time = 3.73154, size = 369, normalized size = 6.15 \begin{align*} \frac{5 \, a \sqrt{c} x + \sqrt{c}}{120 \,{\left (a^{28} c^{14} x^{25} + 5 \, a^{27} c^{14} x^{24} - 40 \, a^{25} c^{14} x^{22} - 50 \, a^{24} c^{14} x^{21} + 126 \, a^{23} c^{14} x^{20} + 280 \, a^{22} c^{14} x^{19} - 160 \, a^{21} c^{14} x^{18} - 765 \, a^{20} c^{14} x^{17} - 105 \, a^{19} c^{14} x^{16} + 1248 \, a^{18} c^{14} x^{15} + 720 \, a^{17} c^{14} x^{14} - 1260 \, a^{16} c^{14} x^{13} - 1260 \, a^{15} c^{14} x^{12} + 720 \, a^{14} c^{14} x^{11} + 1248 \, a^{13} c^{14} x^{10} - 105 \, a^{12} c^{14} x^{9} - 765 \, a^{11} c^{14} x^{8} - 160 \, a^{10} c^{14} x^{7} + 280 \, a^{9} c^{14} x^{6} + 126 \, a^{8} c^{14} x^{5} - 50 \, a^{7} c^{14} x^{4} - 40 \, a^{6} c^{14} x^{3} + 5 \, a^{4} c^{14} x + a^{3} c^{14}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x+1)^5*(-a^2*x^2+1)^(5/2)/(-a^2*c*x^2+c)^(27/2),x, algorithm="maxima")

[Out]

1/120*(5*a*sqrt(c)*x + sqrt(c))/(a^28*c^14*x^25 + 5*a^27*c^14*x^24 - 40*a^25*c^14*x^22 - 50*a^24*c^14*x^21 + 1
26*a^23*c^14*x^20 + 280*a^22*c^14*x^19 - 160*a^21*c^14*x^18 - 765*a^20*c^14*x^17 - 105*a^19*c^14*x^16 + 1248*a
^18*c^14*x^15 + 720*a^17*c^14*x^14 - 1260*a^16*c^14*x^13 - 1260*a^15*c^14*x^12 + 720*a^14*c^14*x^11 + 1248*a^1
3*c^14*x^10 - 105*a^12*c^14*x^9 - 765*a^11*c^14*x^8 - 160*a^10*c^14*x^7 + 280*a^9*c^14*x^6 + 126*a^8*c^14*x^5
- 50*a^7*c^14*x^4 - 40*a^6*c^14*x^3 + 5*a^4*c^14*x + a^3*c^14)

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Fricas [B]  time = 3.0069, size = 1226, normalized size = 20.43 \begin{align*} \frac{{\left (a^{22} x^{25} + 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} + 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} - 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} - 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} - 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} - 50 \, a x^{4} - 40 \, x^{3}\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{120 \,{\left (a^{27} c^{14} x^{27} + 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} - 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} + 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} - 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} + 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} + 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} - 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} + 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} - 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} + 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} - 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} + 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} + a^{2} c^{14} x^{2} - 5 \, a c^{14} x - c^{14}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x+1)^5*(-a^2*x^2+1)^(5/2)/(-a^2*c*x^2+c)^(27/2),x, algorithm="fricas")

[Out]

1/120*(a^22*x^25 + 5*a^21*x^24 - 40*a^19*x^22 - 50*a^18*x^21 + 126*a^17*x^20 + 280*a^16*x^19 - 160*a^15*x^18 -
765*a^14*x^17 - 105*a^13*x^16 + 1248*a^12*x^15 + 720*a^11*x^14 - 1260*a^10*x^13 - 1260*a^9*x^12 + 720*a^8*x^1
1 + 1248*a^7*x^10 - 105*a^6*x^9 - 765*a^5*x^8 - 160*a^4*x^7 + 280*a^3*x^6 + 126*a^2*x^5 - 50*a*x^4 - 40*x^3)*s
qrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^27*c^14*x^27 + 5*a^26*c^14*x^26 - a^25*c^14*x^25 - 45*a^24*c^14*x^24
- 50*a^23*c^14*x^23 + 166*a^22*c^14*x^22 + 330*a^21*c^14*x^21 - 286*a^20*c^14*x^20 - 1045*a^19*c^14*x^19 + 55
*a^18*c^14*x^18 + 2013*a^17*c^14*x^17 + 825*a^16*c^14*x^16 - 2508*a^15*c^14*x^15 - 1980*a^14*c^14*x^14 + 1980*
a^13*c^14*x^13 + 2508*a^12*c^14*x^12 - 825*a^11*c^14*x^11 - 2013*a^10*c^14*x^10 - 55*a^9*c^14*x^9 + 1045*a^8*c
^14*x^8 + 286*a^7*c^14*x^7 - 330*a^6*c^14*x^6 - 166*a^5*c^14*x^5 + 50*a^4*c^14*x^4 + 45*a^3*c^14*x^3 + a^2*c^1
4*x^2 - 5*a*c^14*x - c^14)

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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(x**2/(a*x+1)**5*(-a**2*x**2+1)**(5/2)/(-a**2*c*x**2+c)**(27/2),x)

[Out]

Timed out

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Giac [B]  time = 29.0573, size = 664, normalized size = 11.07 \begin{align*} -\frac{1}{2013265920} \,{\left (\frac{2451570 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{9} + 1514205 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{8} + 769120 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{7} + 318780 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{6} + 106260 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{5} + 27830 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{4} + 5520 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{3} + 780 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )}^{2} + 70 \, \sqrt{c}{\left (\frac{2}{a x + 1} - 1\right )} + 3 \, \sqrt{c}}{a^{4} c^{14}{\left (\frac{2}{a x + 1} - 1\right )}^{10}} - \frac{2 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{15} + 45 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{14} + 480 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{13} + 3220 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{12} + 15180 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{11} + 53130 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{10} + 141680 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{9} + 288420 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{8} + 432630 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{7} + 408595 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{6} - 891480 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{4} - 2080120 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{3} - 3120180 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}^{2} - 3565920 \, a^{56} c^{\frac{393}{2}}{\left (\frac{2}{a x + 1} - 1\right )}}{a^{60} c^{210}}\right )}{\left | a \right |} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a*x+1)^5*(-a^2*x^2+1)^(5/2)/(-a^2*c*x^2+c)^(27/2),x, algorithm="giac")

[Out]

-1/2013265920*((2451570*sqrt(c)*(2/(a*x + 1) - 1)^9 + 1514205*sqrt(c)*(2/(a*x + 1) - 1)^8 + 769120*sqrt(c)*(2/
(a*x + 1) - 1)^7 + 318780*sqrt(c)*(2/(a*x + 1) - 1)^6 + 106260*sqrt(c)*(2/(a*x + 1) - 1)^5 + 27830*sqrt(c)*(2/
(a*x + 1) - 1)^4 + 5520*sqrt(c)*(2/(a*x + 1) - 1)^3 + 780*sqrt(c)*(2/(a*x + 1) - 1)^2 + 70*sqrt(c)*(2/(a*x + 1
) - 1) + 3*sqrt(c))/(a^4*c^14*(2/(a*x + 1) - 1)^10) - (2*a^56*c^(393/2)*(2/(a*x + 1) - 1)^15 + 45*a^56*c^(393/
2)*(2/(a*x + 1) - 1)^14 + 480*a^56*c^(393/2)*(2/(a*x + 1) - 1)^13 + 3220*a^56*c^(393/2)*(2/(a*x + 1) - 1)^12 +
15180*a^56*c^(393/2)*(2/(a*x + 1) - 1)^11 + 53130*a^56*c^(393/2)*(2/(a*x + 1) - 1)^10 + 141680*a^56*c^(393/2)
*(2/(a*x + 1) - 1)^9 + 288420*a^56*c^(393/2)*(2/(a*x + 1) - 1)^8 + 432630*a^56*c^(393/2)*(2/(a*x + 1) - 1)^7 +
408595*a^56*c^(393/2)*(2/(a*x + 1) - 1)^6 - 891480*a^56*c^(393/2)*(2/(a*x + 1) - 1)^4 - 2080120*a^56*c^(393/2
)*(2/(a*x + 1) - 1)^3 - 3120180*a^56*c^(393/2)*(2/(a*x + 1) - 1)^2 - 3565920*a^56*c^(393/2)*(2/(a*x + 1) - 1))
/(a^60*c^210))*abs(a)