\(\int \frac {e^{5 \text {arctanh}(a x)} x^2}{(c-a^2 c x^2)^{27/2}} \, dx\) [1396]

Optimal result

Integrand size = 27, antiderivative size = 60 \[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=-\frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{15} (1+a x)^{10} \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.98 \[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (-1+a x)^{15} (1+a x)^{10} \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6703, 6700, 91}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2 e^{5 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{27/2}} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (1-a^2 x^2\right )^{27/2}}dx}{c^{13} \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 6700

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {x^2}{(1-a x)^{16} (a x+1)^{11}}dx}{c^{13} \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 91

\(\displaystyle -\frac {(1-5 a x) \sqrt {1-a^2 x^2}}{120 a^3 c^{13} (1-a x)^{15} (a x+1)^{10} \sqrt {c-a^2 c x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 91

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> 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]
 

rule 6700

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[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 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[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]) &&  !I 
ntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.82

method	result	size
gosper	\(-\frac {\left (a x -1\right ) \left (a x +1\right )^{6} \left (5 a x -1\right )}{120 a^{3} \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (-a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\)	\(49\)
default	\(-\frac {\left (5 a x -1\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{120 \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right )^{10} \left (a x -1\right )^{15} c^{14} a^{3}}\)	\(55\)
orering	\(\frac {x^{3} \left (a^{22} x^{22}-5 a^{21} x^{21}+40 a^{19} x^{19}-50 a^{18} x^{18}-126 a^{17} x^{17}+280 a^{16} x^{16}+160 a^{15} x^{15}-765 a^{14} x^{14}+105 a^{13} x^{13}+1248 a^{12} x^{12}-720 a^{11} x^{11}-1260 a^{10} x^{10}+1260 a^{9} x^{9}+720 a^{8} x^{8}-1248 a^{7} x^{7}-105 x^{6} a^{6}+765 a^{5} x^{5}-160 a^{4} x^{4}-280 a^{3} x^{3}+126 a^{2} x^{2}+50 a x -40\right ) \left (a x -1\right ) \left (a x +1\right )^{6}}{120 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (-a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\)	\(208\)

Input:

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

Output:

-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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (53) = 106\).

Time = 1.45 (sec) , antiderivative size = 496, normalized size of antiderivative = 8.27 \[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=-\frac {{\left (a^{22} x^{25} - 5 \, a^{21} x^{24} + 40 \, a^{19} x^{22} - 50 \, a^{18} x^{21} - 126 \, a^{17} x^{20} + 280 \, a^{16} x^{19} + 160 \, a^{15} x^{18} - 765 \, a^{14} x^{17} + 105 \, a^{13} x^{16} + 1248 \, a^{12} x^{15} - 720 \, a^{11} x^{14} - 1260 \, a^{10} x^{13} + 1260 \, a^{9} x^{12} + 720 \, a^{8} x^{11} - 1248 \, a^{7} x^{10} - 105 \, a^{6} x^{9} + 765 \, a^{5} x^{8} - 160 \, a^{4} x^{7} - 280 \, a^{3} x^{6} + 126 \, a^{2} x^{5} + 50 \, a x^{4} - 40 \, x^{3}\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} - 5 \, a^{26} c^{14} x^{26} - a^{25} c^{14} x^{25} + 45 \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} - 166 \, a^{22} c^{14} x^{22} + 330 \, a^{21} c^{14} x^{21} + 286 \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} - 55 \, a^{18} c^{14} x^{18} + 2013 \, a^{17} c^{14} x^{17} - 825 \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} + 1980 \, a^{14} c^{14} x^{14} + 1980 \, a^{13} c^{14} x^{13} - 2508 \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} + 2013 \, a^{10} c^{14} x^{10} - 55 \, a^{9} c^{14} x^{9} - 1045 \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} + 330 \, a^{6} c^{14} x^{6} - 166 \, a^{5} c^{14} x^{5} - 50 \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} - a^{2} c^{14} x^{2} - 5 \, a c^{14} x + c^{14}\right )}} \] Input:

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

Output:

-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^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^1 
4*x^20 - 1045*a^19*c^14*x^19 - 55*a^18*c^14*x^18 + 2013*a^17*c^14*x^17 - 8 
25*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)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{5} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{5} x^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 27.27 (sec) , antiderivative size = 583, normalized size of antiderivative = 9.72 \[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=-\frac {\sqrt {c-a^2\,c\,x^2}-5\,a\,x\,\sqrt {c-a^2\,c\,x^2}}{120\,a^3\,c^{14}\,\sqrt {1-a^2\,x^2}-600\,a^4\,c^{14}\,x\,\sqrt {1-a^2\,x^2}+4800\,a^6\,c^{14}\,x^3\,\sqrt {1-a^2\,x^2}-6000\,a^7\,c^{14}\,x^4\,\sqrt {1-a^2\,x^2}-15120\,a^8\,c^{14}\,x^5\,\sqrt {1-a^2\,x^2}+33600\,a^9\,c^{14}\,x^6\,\sqrt {1-a^2\,x^2}+19200\,a^{10}\,c^{14}\,x^7\,\sqrt {1-a^2\,x^2}-91800\,a^{11}\,c^{14}\,x^8\,\sqrt {1-a^2\,x^2}+12600\,a^{12}\,c^{14}\,x^9\,\sqrt {1-a^2\,x^2}+149760\,a^{13}\,c^{14}\,x^{10}\,\sqrt {1-a^2\,x^2}-86400\,a^{14}\,c^{14}\,x^{11}\,\sqrt {1-a^2\,x^2}-151200\,a^{15}\,c^{14}\,x^{12}\,\sqrt {1-a^2\,x^2}+151200\,a^{16}\,c^{14}\,x^{13}\,\sqrt {1-a^2\,x^2}+86400\,a^{17}\,c^{14}\,x^{14}\,\sqrt {1-a^2\,x^2}-149760\,a^{18}\,c^{14}\,x^{15}\,\sqrt {1-a^2\,x^2}-12600\,a^{19}\,c^{14}\,x^{16}\,\sqrt {1-a^2\,x^2}+91800\,a^{20}\,c^{14}\,x^{17}\,\sqrt {1-a^2\,x^2}-19200\,a^{21}\,c^{14}\,x^{18}\,\sqrt {1-a^2\,x^2}-33600\,a^{22}\,c^{14}\,x^{19}\,\sqrt {1-a^2\,x^2}+15120\,a^{23}\,c^{14}\,x^{20}\,\sqrt {1-a^2\,x^2}+6000\,a^{24}\,c^{14}\,x^{21}\,\sqrt {1-a^2\,x^2}-4800\,a^{25}\,c^{14}\,x^{22}\,\sqrt {1-a^2\,x^2}+600\,a^{27}\,c^{14}\,x^{24}\,\sqrt {1-a^2\,x^2}-120\,a^{28}\,c^{14}\,x^{25}\,\sqrt {1-a^2\,x^2}} \] Input:

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

Output:

-((c - a^2*c*x^2)^(1/2) - 5*a*x*(c - a^2*c*x^2)^(1/2))/(120*a^3*c^14*(1 - 
a^2*x^2)^(1/2) - 600*a^4*c^14*x*(1 - a^2*x^2)^(1/2) + 4800*a^6*c^14*x^3*(1 
 - a^2*x^2)^(1/2) - 6000*a^7*c^14*x^4*(1 - a^2*x^2)^(1/2) - 15120*a^8*c^14 
*x^5*(1 - a^2*x^2)^(1/2) + 33600*a^9*c^14*x^6*(1 - a^2*x^2)^(1/2) + 19200* 
a^10*c^14*x^7*(1 - a^2*x^2)^(1/2) - 91800*a^11*c^14*x^8*(1 - a^2*x^2)^(1/2 
) + 12600*a^12*c^14*x^9*(1 - a^2*x^2)^(1/2) + 149760*a^13*c^14*x^10*(1 - a 
^2*x^2)^(1/2) - 86400*a^14*c^14*x^11*(1 - a^2*x^2)^(1/2) - 151200*a^15*c^1 
4*x^12*(1 - a^2*x^2)^(1/2) + 151200*a^16*c^14*x^13*(1 - a^2*x^2)^(1/2) + 8 
6400*a^17*c^14*x^14*(1 - a^2*x^2)^(1/2) - 149760*a^18*c^14*x^15*(1 - a^2*x 
^2)^(1/2) - 12600*a^19*c^14*x^16*(1 - a^2*x^2)^(1/2) + 91800*a^20*c^14*x^1 
7*(1 - a^2*x^2)^(1/2) - 19200*a^21*c^14*x^18*(1 - a^2*x^2)^(1/2) - 33600*a 
^22*c^14*x^19*(1 - a^2*x^2)^(1/2) + 15120*a^23*c^14*x^20*(1 - a^2*x^2)^(1/ 
2) + 6000*a^24*c^14*x^21*(1 - a^2*x^2)^(1/2) - 4800*a^25*c^14*x^22*(1 - a^ 
2*x^2)^(1/2) + 600*a^27*c^14*x^24*(1 - a^2*x^2)^(1/2) - 120*a^28*c^14*x^25 
*(1 - a^2*x^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 3.32 \[ \int \frac {e^{5 \text {arctanh}(a x)} x^2}{\left (c-a^2 c x^2\right )^{27/2}} \, dx=\frac {\sqrt {c}\, \left (-5 a x +1\right )}{120 a^{3} c^{14} \left (a^{25} x^{25}-5 a^{24} x^{24}+40 a^{22} x^{22}-50 a^{21} x^{21}-126 a^{20} x^{20}+280 a^{19} x^{19}+160 a^{18} x^{18}-765 a^{17} x^{17}+105 a^{16} x^{16}+1248 a^{15} x^{15}-720 a^{14} x^{14}-1260 a^{13} x^{13}+1260 a^{12} x^{12}+720 a^{11} x^{11}-1248 a^{10} x^{10}-105 a^{9} x^{9}+765 a^{8} x^{8}-160 a^{7} x^{7}-280 a^{6} x^{6}+126 a^{5} x^{5}+50 a^{4} x^{4}-40 a^{3} x^{3}+5 a x -1\right )} \] Input:

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

Output:

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