3.1125 \(\int e^{2 \tanh ^{-1}(a x)} x^m (c-a^2 c x^2)^3 \, dx\)
Optimal. Leaf size=120 \[ -\frac{a^2 c^3 x^{m+3}}{m+3}-\frac{4 a^3 c^3 x^{m+4}}{m+4}-\frac{a^4 c^3 x^{m+5}}{m+5}+\frac{2 a^5 c^3 x^{m+6}}{m+6}+\frac{a^6 c^3 x^{m+7}}{m+7}+\frac{2 a c^3 x^{m+2}}{m+2}+\frac{c^3 x^{m+1}}{m+1} \]
[Out]
(c^3*x^(1 + m))/(1 + m) + (2*a*c^3*x^(2 + m))/(2 + m) - (a^2*c^3*x^(3 + m))/(3 + m) - (4*a^3*c^3*x^(4 + m))/(4
+ m) - (a^4*c^3*x^(5 + m))/(5 + m) + (2*a^5*c^3*x^(6 + m))/(6 + m) + (a^6*c^3*x^(7 + m))/(7 + m)
________________________________________________________________________________________
Rubi [A] time = 0.1127, antiderivative size = 120, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used =
{6150, 88} \[ -\frac{a^2 c^3 x^{m+3}}{m+3}-\frac{4 a^3 c^3 x^{m+4}}{m+4}-\frac{a^4 c^3 x^{m+5}}{m+5}+\frac{2 a^5 c^3 x^{m+6}}{m+6}+\frac{a^6 c^3 x^{m+7}}{m+7}+\frac{2 a c^3 x^{m+2}}{m+2}+\frac{c^3 x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
[In]
Int[E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^3,x]
[Out]
(c^3*x^(1 + m))/(1 + m) + (2*a*c^3*x^(2 + m))/(2 + m) - (a^2*c^3*x^(3 + m))/(3 + m) - (4*a^3*c^3*x^(4 + m))/(4
+ m) - (a^4*c^3*x^(5 + m))/(5 + m) + (2*a^5*c^3*x^(6 + m))/(6 + m) + (a^6*c^3*x^(7 + m))/(7 + m)
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^{2 \tanh ^{-1}(a x)} x^m \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^m (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^m+2 a x^{1+m}-a^2 x^{2+m}-4 a^3 x^{3+m}-a^4 x^{4+m}+2 a^5 x^{5+m}+a^6 x^{6+m}\right ) \, dx\\ &=\frac{c^3 x^{1+m}}{1+m}+\frac{2 a c^3 x^{2+m}}{2+m}-\frac{a^2 c^3 x^{3+m}}{3+m}-\frac{4 a^3 c^3 x^{4+m}}{4+m}-\frac{a^4 c^3 x^{5+m}}{5+m}+\frac{2 a^5 c^3 x^{6+m}}{6+m}+\frac{a^6 c^3 x^{7+m}}{7+m}\\ \end{align*}
Mathematica [A] time = 0.0638301, size = 88, normalized size = 0.73 \[ c^3 x^{m+1} \left (\frac{a^6 x^6}{m+7}+\frac{2 a^5 x^5}{m+6}-\frac{a^4 x^4}{m+5}-\frac{4 a^3 x^3}{m+4}-\frac{a^2 x^2}{m+3}+\frac{2 a x}{m+2}+\frac{1}{m+1}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[E^(2*ArcTanh[a*x])*x^m*(c - a^2*c*x^2)^3,x]
[Out]
c^3*x^(1 + m)*((1 + m)^(-1) + (2*a*x)/(2 + m) - (a^2*x^2)/(3 + m) - (4*a^3*x^3)/(4 + m) - (a^4*x^4)/(5 + m) +
(2*a^5*x^5)/(6 + m) + (a^6*x^6)/(7 + m))
________________________________________________________________________________________
Maple [B] time = 0.031, size = 476, normalized size = 4. \begin{align*}{\frac{{c}^{3}{x}^{1+m} \left ({a}^{6}{m}^{6}{x}^{6}+21\,{a}^{6}{m}^{5}{x}^{6}+175\,{a}^{6}{m}^{4}{x}^{6}+2\,{a}^{5}{m}^{6}{x}^{5}+735\,{a}^{6}{m}^{3}{x}^{6}+44\,{a}^{5}{m}^{5}{x}^{5}+1624\,{a}^{6}{m}^{2}{x}^{6}+380\,{a}^{5}{m}^{4}{x}^{5}-{a}^{4}{m}^{6}{x}^{4}+1764\,{a}^{6}m{x}^{6}+1640\,{a}^{5}{m}^{3}{x}^{5}-23\,{a}^{4}{m}^{5}{x}^{4}+720\,{x}^{6}{a}^{6}+3698\,{a}^{5}{m}^{2}{x}^{5}-207\,{a}^{4}{m}^{4}{x}^{4}-4\,{a}^{3}{m}^{6}{x}^{3}+4076\,{a}^{5}m{x}^{5}-925\,{a}^{4}{m}^{3}{x}^{4}-96\,{a}^{3}{m}^{5}{x}^{3}+1680\,{x}^{5}{a}^{5}-2144\,{a}^{4}{m}^{2}{x}^{4}-904\,{a}^{3}{m}^{4}{x}^{3}-{a}^{2}{m}^{6}{x}^{2}-2412\,{a}^{4}m{x}^{4}-4224\,{a}^{3}{m}^{3}{x}^{3}-25\,{a}^{2}{m}^{5}{x}^{2}-1008\,{x}^{4}{a}^{4}-10180\,{a}^{3}{m}^{2}{x}^{3}-247\,{a}^{2}{m}^{4}{x}^{2}+2\,a{m}^{6}x-11808\,{a}^{3}m{x}^{3}-1219\,{a}^{2}{m}^{3}{x}^{2}+52\,a{m}^{5}x-5040\,{x}^{3}{a}^{3}-3112\,{a}^{2}{m}^{2}{x}^{2}+540\,a{m}^{4}x+{m}^{6}-3796\,{a}^{2}m{x}^{2}+2840\,a{m}^{3}x+27\,{m}^{5}-1680\,{a}^{2}{x}^{2}+7858\,a{m}^{2}x+295\,{m}^{4}+10548\,amx+1665\,{m}^{3}+5040\,ax+5104\,{m}^{2}+8028\,m+5040 \right ) }{ \left ( 7+m \right ) \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x)
[Out]
c^3*x^(1+m)*(a^6*m^6*x^6+21*a^6*m^5*x^6+175*a^6*m^4*x^6+2*a^5*m^6*x^5+735*a^6*m^3*x^6+44*a^5*m^5*x^5+1624*a^6*
m^2*x^6+380*a^5*m^4*x^5-a^4*m^6*x^4+1764*a^6*m*x^6+1640*a^5*m^3*x^5-23*a^4*m^5*x^4+720*a^6*x^6+3698*a^5*m^2*x^
5-207*a^4*m^4*x^4-4*a^3*m^6*x^3+4076*a^5*m*x^5-925*a^4*m^3*x^4-96*a^3*m^5*x^3+1680*a^5*x^5-2144*a^4*m^2*x^4-90
4*a^3*m^4*x^3-a^2*m^6*x^2-2412*a^4*m*x^4-4224*a^3*m^3*x^3-25*a^2*m^5*x^2-1008*a^4*x^4-10180*a^3*m^2*x^3-247*a^
2*m^4*x^2+2*a*m^6*x-11808*a^3*m*x^3-1219*a^2*m^3*x^2+52*a*m^5*x-5040*a^3*x^3-3112*a^2*m^2*x^2+540*a*m^4*x+m^6-
3796*a^2*m*x^2+2840*a*m^3*x+27*m^5-1680*a^2*x^2+7858*a*m^2*x+295*m^4+10548*a*m*x+1665*m^3+5040*a*x+5104*m^2+80
28*m+5040)/(7+m)/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)
________________________________________________________________________________________
Maxima [B] time = 1.14391, size = 410, normalized size = 3.42 \begin{align*} \frac{{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} a^{6} c^{3} x^{7} + 2 \,{\left (m^{6} + 22 \, m^{5} + 190 \, m^{4} + 820 \, m^{3} + 1849 \, m^{2} + 2038 \, m + 840\right )} a^{5} c^{3} x^{6} -{\left (m^{6} + 23 \, m^{5} + 207 \, m^{4} + 925 \, m^{3} + 2144 \, m^{2} + 2412 \, m + 1008\right )} a^{4} c^{3} x^{5} - 4 \,{\left (m^{6} + 24 \, m^{5} + 226 \, m^{4} + 1056 \, m^{3} + 2545 \, m^{2} + 2952 \, m + 1260\right )} a^{3} c^{3} x^{4} -{\left (m^{6} + 25 \, m^{5} + 247 \, m^{4} + 1219 \, m^{3} + 3112 \, m^{2} + 3796 \, m + 1680\right )} a^{2} c^{3} x^{3} + 2 \,{\left (m^{6} + 26 \, m^{5} + 270 \, m^{4} + 1420 \, m^{3} + 3929 \, m^{2} + 5274 \, m + 2520\right )} a c^{3} x^{2} +{\left (m^{6} + 27 \, m^{5} + 295 \, m^{4} + 1665 \, m^{3} + 5104 \, m^{2} + 8028 \, m + 5040\right )} c^{3} x\right )} x^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm="maxima")
[Out]
((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*a^6*c^3*x^7 + 2*(m^6 + 22*m^5 + 190*m^4 + 820*m^
3 + 1849*m^2 + 2038*m + 840)*a^5*c^3*x^6 - (m^6 + 23*m^5 + 207*m^4 + 925*m^3 + 2144*m^2 + 2412*m + 1008)*a^4*c
^3*x^5 - 4*(m^6 + 24*m^5 + 226*m^4 + 1056*m^3 + 2545*m^2 + 2952*m + 1260)*a^3*c^3*x^4 - (m^6 + 25*m^5 + 247*m^
4 + 1219*m^3 + 3112*m^2 + 3796*m + 1680)*a^2*c^3*x^3 + 2*(m^6 + 26*m^5 + 270*m^4 + 1420*m^3 + 3929*m^2 + 5274*
m + 2520)*a*c^3*x^2 + (m^6 + 27*m^5 + 295*m^4 + 1665*m^3 + 5104*m^2 + 8028*m + 5040)*c^3*x)*x^m/(m^7 + 28*m^6
+ 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)
________________________________________________________________________________________
Fricas [B] time = 3.26204, size = 1233, normalized size = 10.28 \begin{align*} \frac{{\left ({\left (a^{6} c^{3} m^{6} + 21 \, a^{6} c^{3} m^{5} + 175 \, a^{6} c^{3} m^{4} + 735 \, a^{6} c^{3} m^{3} + 1624 \, a^{6} c^{3} m^{2} + 1764 \, a^{6} c^{3} m + 720 \, a^{6} c^{3}\right )} x^{7} + 2 \,{\left (a^{5} c^{3} m^{6} + 22 \, a^{5} c^{3} m^{5} + 190 \, a^{5} c^{3} m^{4} + 820 \, a^{5} c^{3} m^{3} + 1849 \, a^{5} c^{3} m^{2} + 2038 \, a^{5} c^{3} m + 840 \, a^{5} c^{3}\right )} x^{6} -{\left (a^{4} c^{3} m^{6} + 23 \, a^{4} c^{3} m^{5} + 207 \, a^{4} c^{3} m^{4} + 925 \, a^{4} c^{3} m^{3} + 2144 \, a^{4} c^{3} m^{2} + 2412 \, a^{4} c^{3} m + 1008 \, a^{4} c^{3}\right )} x^{5} - 4 \,{\left (a^{3} c^{3} m^{6} + 24 \, a^{3} c^{3} m^{5} + 226 \, a^{3} c^{3} m^{4} + 1056 \, a^{3} c^{3} m^{3} + 2545 \, a^{3} c^{3} m^{2} + 2952 \, a^{3} c^{3} m + 1260 \, a^{3} c^{3}\right )} x^{4} -{\left (a^{2} c^{3} m^{6} + 25 \, a^{2} c^{3} m^{5} + 247 \, a^{2} c^{3} m^{4} + 1219 \, a^{2} c^{3} m^{3} + 3112 \, a^{2} c^{3} m^{2} + 3796 \, a^{2} c^{3} m + 1680 \, a^{2} c^{3}\right )} x^{3} + 2 \,{\left (a c^{3} m^{6} + 26 \, a c^{3} m^{5} + 270 \, a c^{3} m^{4} + 1420 \, a c^{3} m^{3} + 3929 \, a c^{3} m^{2} + 5274 \, a c^{3} m + 2520 \, a c^{3}\right )} x^{2} +{\left (c^{3} m^{6} + 27 \, c^{3} m^{5} + 295 \, c^{3} m^{4} + 1665 \, c^{3} m^{3} + 5104 \, c^{3} m^{2} + 8028 \, c^{3} m + 5040 \, c^{3}\right )} x\right )} x^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm="fricas")
[Out]
((a^6*c^3*m^6 + 21*a^6*c^3*m^5 + 175*a^6*c^3*m^4 + 735*a^6*c^3*m^3 + 1624*a^6*c^3*m^2 + 1764*a^6*c^3*m + 720*a
^6*c^3)*x^7 + 2*(a^5*c^3*m^6 + 22*a^5*c^3*m^5 + 190*a^5*c^3*m^4 + 820*a^5*c^3*m^3 + 1849*a^5*c^3*m^2 + 2038*a^
5*c^3*m + 840*a^5*c^3)*x^6 - (a^4*c^3*m^6 + 23*a^4*c^3*m^5 + 207*a^4*c^3*m^4 + 925*a^4*c^3*m^3 + 2144*a^4*c^3*
m^2 + 2412*a^4*c^3*m + 1008*a^4*c^3)*x^5 - 4*(a^3*c^3*m^6 + 24*a^3*c^3*m^5 + 226*a^3*c^3*m^4 + 1056*a^3*c^3*m^
3 + 2545*a^3*c^3*m^2 + 2952*a^3*c^3*m + 1260*a^3*c^3)*x^4 - (a^2*c^3*m^6 + 25*a^2*c^3*m^5 + 247*a^2*c^3*m^4 +
1219*a^2*c^3*m^3 + 3112*a^2*c^3*m^2 + 3796*a^2*c^3*m + 1680*a^2*c^3)*x^3 + 2*(a*c^3*m^6 + 26*a*c^3*m^5 + 270*a
*c^3*m^4 + 1420*a*c^3*m^3 + 3929*a*c^3*m^2 + 5274*a*c^3*m + 2520*a*c^3)*x^2 + (c^3*m^6 + 27*c^3*m^5 + 295*c^3*
m^4 + 1665*c^3*m^3 + 5104*c^3*m^2 + 8028*c^3*m + 5040*c^3)*x)*x^m/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^
3 + 13132*m^2 + 13068*m + 5040)
________________________________________________________________________________________
Sympy [A] time = 4.08118, size = 3009, normalized size = 25.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)**2/(-a**2*x**2+1)*x**m*(-a**2*c*x**2+c)**3,x)
[Out]
Piecewise((a**6*c**3*log(x) - 2*a**5*c**3/x + a**4*c**3/(2*x**2) + 4*a**3*c**3/(3*x**3) + a**2*c**3/(4*x**4) -
2*a*c**3/(5*x**5) - c**3/(6*x**6), Eq(m, -7)), (a**6*c**3*x + 2*a**5*c**3*log(x) + a**4*c**3/x + 2*a**3*c**3/
x**2 + a**2*c**3/(3*x**3) - a*c**3/(2*x**4) - c**3/(5*x**5), Eq(m, -6)), (a**6*c**3*x**2/2 + 2*a**5*c**3*x - a
**4*c**3*log(x) + 4*a**3*c**3/x + a**2*c**3/(2*x**2) - 2*a*c**3/(3*x**3) - c**3/(4*x**4), Eq(m, -5)), (a**6*c*
*3*x**3/3 + a**5*c**3*x**2 - a**4*c**3*x - 4*a**3*c**3*log(x) + a**2*c**3/x - a*c**3/x**2 - c**3/(3*x**3), Eq(
m, -4)), (a**6*c**3*x**4/4 + 2*a**5*c**3*x**3/3 - a**4*c**3*x**2/2 - 4*a**3*c**3*x - a**2*c**3*log(x) - 2*a*c*
*3/x - c**3/(2*x**2), Eq(m, -3)), (a**6*c**3*x**5/5 + a**5*c**3*x**4/2 - a**4*c**3*x**3/3 - 2*a**3*c**3*x**2 -
a**2*c**3*x + 2*a*c**3*log(x) - c**3/x, Eq(m, -2)), (a**6*c**3*x**6/6 + 2*a**5*c**3*x**5/5 - a**4*c**3*x**4/4
- 4*a**3*c**3*x**3/3 - a**2*c**3*x**2/2 + 2*a*c**3*x + c**3*log(x), Eq(m, -1)), (a**6*c**3*m**6*x**7*x**m/(m*
*7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 21*a**6*c**3*m**5*x**7*x**m/(
m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 175*a**6*c**3*m**4*x**7*x**
m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 735*a**6*c**3*m**3*x**7*
x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1624*a**6*c**3*m**2*x
**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1764*a**6*c**3*m*
x**7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 720*a**6*c**3*x*
*7*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 2*a**5*c**3*m**6*x
**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 44*a**5*c**3*m**5
*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 380*a**5*c**3*m
**4*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 1640*a**5*c*
*3*m**3*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 3698*a**
5*c**3*m**2*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 4076
*a**5*c**3*m*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 168
0*a**5*c**3*x**6*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - a**4
*c**3*m**6*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 23*a*
*4*c**3*m**5*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) - 207
*a**4*c**3*m**4*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) -
925*a**4*c**3*m**3*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040)
- 2144*a**4*c**3*m**2*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5
040) - 2412*a**4*c**3*m*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m +
5040) - 1008*a**4*c**3*x**5*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5
040) - 4*a**3*c**3*m**6*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m +
5040) - 96*a**3*c**3*m**5*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m
+ 5040) - 904*a**3*c**3*m**4*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068
*m + 5040) - 4224*a**3*c**3*m**3*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 1
3068*m + 5040) - 10180*a**3*c**3*m**2*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**
2 + 13068*m + 5040) - 11808*a**3*c**3*m*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m
**2 + 13068*m + 5040) - 5040*a**3*c**3*x**4*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m*
*2 + 13068*m + 5040) - a**2*c**3*m**6*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**
2 + 13068*m + 5040) - 25*a**2*c**3*m**5*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m
**2 + 13068*m + 5040) - 247*a**2*c**3*m**4*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 1313
2*m**2 + 13068*m + 5040) - 1219*a**2*c**3*m**3*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +
13132*m**2 + 13068*m + 5040) - 3112*a**2*c**3*m**2*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**
3 + 13132*m**2 + 13068*m + 5040) - 3796*a**2*c**3*m*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m*
*3 + 13132*m**2 + 13068*m + 5040) - 1680*a**2*c**3*x**3*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**
3 + 13132*m**2 + 13068*m + 5040) + 2*a*c**3*m**6*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + 52*a*c**3*m**5*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +
13132*m**2 + 13068*m + 5040) + 540*a*c**3*m**4*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +
13132*m**2 + 13068*m + 5040) + 2840*a*c**3*m**3*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + 7858*a*c**3*m**2*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + 10548*a*c**3*m*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3
+ 13132*m**2 + 13068*m + 5040) + 5040*a*c**3*x**2*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13
132*m**2 + 13068*m + 5040) + c**3*m**6*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2
+ 13068*m + 5040) + 27*c**3*m**5*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 1306
8*m + 5040) + 295*c**3*m**4*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m +
5040) + 1665*c**3*m**3*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 504
0) + 5104*c**3*m**2*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) +
8028*c**3*m*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 5040*c
**3*x*x**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040), True))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} - c\right )}^{3}{\left (a x + 1\right )}^{2} x^{m}}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a*x+1)^2/(-a^2*x^2+1)*x^m*(-a^2*c*x^2+c)^3,x, algorithm="giac")
[Out]
integrate((a^2*c*x^2 - c)^3*(a*x + 1)^2*x^m/(a^2*x^2 - 1), x)