3.1373 \(\int \frac{e^{5 \tanh ^{-1}(a x)} x^2}{(c-a^2 c x^2)^{27/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.230833, 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} \[ -\frac{(1-5 a x) \sqrt{1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{15} (a x+1)^{10} \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(5*ArcTanh[a*x])*x^2)/(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)^15*(1 + a*x)^10*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)^{16} (1+a x)^{11}} \, 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)^{15} (1+a x)^{10} \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.447027, size = 59, normalized size = 0.98 \[ \frac{(1-5 a x) \sqrt{1-a^2 x^2}}{120 a^3 c^{13} (a x-1)^{15} (a x+1)^{10} \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(5*ArcTanh[a*x])*x^2)/(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)^15*(1 + a*x)^10*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.04, size = 49, normalized size = 0.8 \begin{align*} -{\frac{ \left ( ax-1 \right ) \left ( ax+1 \right ) ^{6} \left ( 5\,ax-1 \right ) }{120\,{a}^{3}} \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((a*x+1)^5/(-a^2*x^2+1)^(5/2)*x^2/(-a^2*c*x^2+c)^(27/2),x)

[Out]

-1/120*(a*x-1)*(a*x+1)^6*(5*a*x-1)/a^3/(-a^2*x^2+1)^(5/2)/(-a^2*c*x^2+c)^(27/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{5} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{27}{2}}{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.90849, size = 1227, normalized size = 20.45 \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((a*x+1)^5/(-a^2*x^2+1)^(5/2)*x^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^
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)*
sqrt(-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^2
4 - 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 - 5
5*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)

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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)**5/(-a**2*x**2+1)**(5/2)*x**2/(-a**2*c*x**2+c)**(27/2),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 )}^{5} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{27}{2}}{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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