3.704 \(\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{9/2}} \, dx\)

Optimal. Leaf size=363 \[ \frac {(a x+1)^2}{9 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {2 (a x+1)^4 (1019 a x+704) (1-a x)^5}{315 a^{10} x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}-\frac {2 (a x+1)^{9/2} (1-a x)^{9/2} \sin ^{-1}(a x)}{a^{10} x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {1334 (a x+1)^3 (1-a x)^5}{315 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {2458 (a x+1)^2 (1-a x)^5}{315 a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {302 (a x+1)^2 (1-a x)^4}{21 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}-\frac {646 (a x+1)^2 (1-a x)^3}{315 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {214 (a x+1)^2 (1-a x)^2}{315 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}-\frac {2 (a x+1)^2 (1-a x)}{7 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}} \]

[Out]

1/9*(a*x+1)^2/a^2/(c-c/a^2/x^2)^(9/2)/x-2/7*(-a*x+1)*(a*x+1)^2/a^3/(c-c/a^2/x^2)^(9/2)/x^2+214/315*(-a*x+1)^2*
(a*x+1)^2/a^4/(c-c/a^2/x^2)^(9/2)/x^3-646/315*(-a*x+1)^3*(a*x+1)^2/a^5/(c-c/a^2/x^2)^(9/2)/x^4+302/21*(-a*x+1)
^4*(a*x+1)^2/a^6/(c-c/a^2/x^2)^(9/2)/x^5+2458/315*(-a*x+1)^5*(a*x+1)^2/a^7/(c-c/a^2/x^2)^(9/2)/x^6+1334/315*(-
a*x+1)^5*(a*x+1)^3/a^8/(c-c/a^2/x^2)^(9/2)/x^7+2/315*(-a*x+1)^5*(a*x+1)^4*(1019*a*x+704)/a^10/(c-c/a^2/x^2)^(9
/2)/x^9-2*(-a*x+1)^(9/2)*(a*x+1)^(9/2)*arcsin(a*x)/a^10/(c-c/a^2/x^2)^(9/2)/x^9

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6159, 6129, 98, 150, 143, 41, 216} \[ \frac {1334 (a x+1)^3 (1-a x)^5}{315 a^8 x^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {2458 (a x+1)^2 (1-a x)^5}{315 a^7 x^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {2 (a x+1)^4 (1019 a x+704) (1-a x)^5}{315 a^{10} x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {302 (a x+1)^2 (1-a x)^4}{21 a^6 x^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}-\frac {646 (a x+1)^2 (1-a x)^3}{315 a^5 x^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {214 (a x+1)^2 (1-a x)^2}{315 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}-\frac {2 (a x+1)^2 (1-a x)}{7 a^3 x^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}+\frac {(a x+1)^2}{9 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{9/2}}-\frac {2 (a x+1)^{9/2} (1-a x)^{9/2} \sin ^{-1}(a x)}{a^{10} x^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(9/2),x]

[Out]

(1 + a*x)^2/(9*a^2*(c - c/(a^2*x^2))^(9/2)*x) - (2*(1 - a*x)*(1 + a*x)^2)/(7*a^3*(c - c/(a^2*x^2))^(9/2)*x^2)
+ (214*(1 - a*x)^2*(1 + a*x)^2)/(315*a^4*(c - c/(a^2*x^2))^(9/2)*x^3) - (646*(1 - a*x)^3*(1 + a*x)^2)/(315*a^5
*(c - c/(a^2*x^2))^(9/2)*x^4) + (302*(1 - a*x)^4*(1 + a*x)^2)/(21*a^6*(c - c/(a^2*x^2))^(9/2)*x^5) + (2458*(1
- a*x)^5*(1 + a*x)^2)/(315*a^7*(c - c/(a^2*x^2))^(9/2)*x^6) + (1334*(1 - a*x)^5*(1 + a*x)^3)/(315*a^8*(c - c/(
a^2*x^2))^(9/2)*x^7) + (2*(1 - a*x)^5*(1 + a*x)^4*(704 + 1019*a*x))/(315*a^10*(c - c/(a^2*x^2))^(9/2)*x^9) - (
2*(1 - a*x)^(9/2)*(1 + a*x)^(9/2)*ArcSin[a*x])/(a^10*(c - c/(a^2*x^2))^(9/2)*x^9)

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 143

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x
)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] &&  !(SumSimplerQ[n, 1] &&  !SumSimplerQ[m, 1])

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 6159

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/(
(1 - a*x)^p*(1 + a*x)^p), Int[(u*(1 - a*x)^p*(1 + a*x)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d
, n, p}, x] && EqQ[c + a^2*d, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !GtQ[c, 0]

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{9/2}} \, dx &=\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} x^9}{(1-a x)^{9/2} (1+a x)^{9/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^9}{(1-a x)^{11/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^7 (8+10 a x)}{(1-a x)^{9/2} (1+a x)^{7/2}} \, dx}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^6 \left (-126 a-88 a^2 x\right )}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{63 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^5 \left (1284 a^2+654 a^3 x\right )}{(1-a x)^{5/2} (1+a x)^{7/2}} \, dx}{315 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^4 \left (-9690 a^3-3900 a^4 x\right )}{(1-a x)^{3/2} (1+a x)^{7/2}} \, dx}{945 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}+\frac {302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^3 \left (54360 a^4+17490 a^5 x\right )}{\sqrt {1-a x} (1+a x)^{7/2}} \, dx}{945 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}+\frac {302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}+\frac {2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x^2 \left (110610 a^5+50580 a^6 x\right )}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{4725 a^{12} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}+\frac {302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}+\frac {2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}+\frac {1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}-\frac {\left ((1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {x \left (120060 a^6+91710 a^7 x\right )}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{14175 a^{14} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}+\frac {302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}+\frac {2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}+\frac {1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}+\frac {2 (1-a x)^5 (1+a x)^4 (704+1019 a x)}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}-\frac {\left (2 (1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}+\frac {302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}+\frac {2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}+\frac {1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}+\frac {2 (1-a x)^5 (1+a x)^4 (704+1019 a x)}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}-\frac {\left (2 (1-a x)^{9/2} (1+a x)^{9/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^9 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ &=\frac {(1+a x)^2}{9 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x}-\frac {2 (1-a x) (1+a x)^2}{7 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^2}+\frac {214 (1-a x)^2 (1+a x)^2}{315 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^3}-\frac {646 (1-a x)^3 (1+a x)^2}{315 a^5 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^4}+\frac {302 (1-a x)^4 (1+a x)^2}{21 a^6 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^5}+\frac {2458 (1-a x)^5 (1+a x)^2}{315 a^7 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^6}+\frac {1334 (1-a x)^5 (1+a x)^3}{315 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^7}+\frac {2 (1-a x)^5 (1+a x)^4 (704+1019 a x)}{315 a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}-\frac {2 (1-a x)^{9/2} (1+a x)^{9/2} \sin ^{-1}(a x)}{a^{10} \left (c-\frac {c}{a^2 x^2}\right )^{9/2} x^9}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 151, normalized size = 0.42 \[ \frac {-315 a^8 x^8+1756 a^7 x^7+268 a^6 x^6-5784 a^5 x^5+2060 a^4 x^4+6200 a^3 x^3-3372 a^2 x^2-630 (a x-1)^4 (a x+1)^2 \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )-2186 a x+1408}{315 a^2 c^4 x (a x-1)^4 (a x+1)^2 \sqrt {c-\frac {c}{a^2 x^2}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcTanh[a*x])/(c - c/(a^2*x^2))^(9/2),x]

[Out]

(1408 - 2186*a*x - 3372*a^2*x^2 + 6200*a^3*x^3 + 2060*a^4*x^4 - 5784*a^5*x^5 + 268*a^6*x^6 + 1756*a^7*x^7 - 31
5*a^8*x^8 - 630*(-1 + a*x)^4*(1 + a*x)^2*Sqrt[-1 + a^2*x^2]*Log[a*x + Sqrt[-1 + a^2*x^2]])/(315*a^2*c^4*Sqrt[c
- c/(a^2*x^2)]*x*(-1 + a*x)^4*(1 + a*x)^2)

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fricas [A]  time = 0.99, size = 569, normalized size = 1.57 \[ \left [\frac {315 \, {\left (a^{8} x^{8} - 2 \, a^{7} x^{7} - 2 \, a^{6} x^{6} + 6 \, a^{5} x^{5} - 6 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) - {\left (315 \, a^{9} x^{9} - 1756 \, a^{8} x^{8} - 268 \, a^{7} x^{7} + 5784 \, a^{6} x^{6} - 2060 \, a^{5} x^{5} - 6200 \, a^{4} x^{4} + 3372 \, a^{3} x^{3} + 2186 \, a^{2} x^{2} - 1408 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{315 \, {\left (a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right )}}, \frac {630 \, {\left (a^{8} x^{8} - 2 \, a^{7} x^{7} - 2 \, a^{6} x^{6} + 6 \, a^{5} x^{5} - 6 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + 2 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - {\left (315 \, a^{9} x^{9} - 1756 \, a^{8} x^{8} - 268 \, a^{7} x^{7} + 5784 \, a^{6} x^{6} - 2060 \, a^{5} x^{5} - 6200 \, a^{4} x^{4} + 3372 \, a^{3} x^{3} + 2186 \, a^{2} x^{2} - 1408 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{315 \, {\left (a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(9/2),x, algorithm="fricas")

[Out]

[1/315*(315*(a^8*x^8 - 2*a^7*x^7 - 2*a^6*x^6 + 6*a^5*x^5 - 6*a^3*x^3 + 2*a^2*x^2 + 2*a*x - 1)*sqrt(c)*log(2*a^
2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c) - (315*a^9*x^9 - 1756*a^8*x^8 - 268*a^7*x^7 +
5784*a^6*x^6 - 2060*a^5*x^5 - 6200*a^4*x^4 + 3372*a^3*x^3 + 2186*a^2*x^2 - 1408*a*x)*sqrt((a^2*c*x^2 - c)/(a^
2*x^2)))/(a^9*c^5*x^8 - 2*a^8*c^5*x^7 - 2*a^7*c^5*x^6 + 6*a^6*c^5*x^5 - 6*a^4*c^5*x^3 + 2*a^3*c^5*x^2 + 2*a^2*
c^5*x - a*c^5), 1/315*(630*(a^8*x^8 - 2*a^7*x^7 - 2*a^6*x^6 + 6*a^5*x^5 - 6*a^3*x^3 + 2*a^2*x^2 + 2*a*x - 1)*s
qrt(-c)*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) - (315*a^9*x^9 - 1756*a^8*x^8
- 268*a^7*x^7 + 5784*a^6*x^6 - 2060*a^5*x^5 - 6200*a^4*x^4 + 3372*a^3*x^3 + 2186*a^2*x^2 - 1408*a*x)*sqrt((a^
2*c*x^2 - c)/(a^2*x^2)))/(a^9*c^5*x^8 - 2*a^8*c^5*x^7 - 2*a^7*c^5*x^6 + 6*a^6*c^5*x^5 - 6*a^4*c^5*x^3 + 2*a^3*
c^5*x^2 + 2*a^2*c^5*x - a*c^5)]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(9/2),x, algorithm="giac")

[Out]

integrate(-(a*x + 1)^2/((a^2*x^2 - 1)*(c - c/(a^2*x^2))^(9/2)), x)

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maple [B]  time = 0.11, size = 682, normalized size = 1.88 \[ \frac {\left (-315 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{9} a^{9}+256 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{8} a^{8}+1185 x^{8} c^{\frac {9}{2}} a^{8} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}}-256 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{7} a^{7}+2280 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{7} a^{7}-896 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{6} a^{6}-4620 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{6} a^{6}+896 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{5} a^{5}-4620 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{5} a^{5}-630 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x \,a^{8} c +1120 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{4} a^{4}+7140 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{4} a^{4}+630 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} a^{7} c -1120 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{3} a^{3}+3948 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{3} a^{3}-560 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x^{2} a^{2}-4998 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{2} a^{2}+560 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}} x a -1338 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x a +70 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {7}{2}} c^{\frac {9}{2}}+1338 c^{\frac {9}{2}} \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}}\right ) \left (a x +1\right )}{315 \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {7}{2}} x^{9} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {9}{2}} a^{10} c^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(9/2),x)

[Out]

1/315*(-315*c^(9/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x^9*a^9+256*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)*x^8*a^8+1185*x
^8*c^(9/2)*a^8*((a*x-1)*(a*x+1)*c/a^2)^(7/2)-256*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)*x^7*a^7+2280*c^(9/2)*((a*x-
1)*(a*x+1)*c/a^2)^(7/2)*x^7*a^7-896*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)*x^6*a^6-4620*c^(9/2)*((a*x-1)*(a*x+1)*c/
a^2)^(7/2)*x^6*a^6+896*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)*x^5*a^5-4620*c^(9/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x^
5*a^5-630*(c*(a^2*x^2-1)/a^2)^(7/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x*a^
8*c+1120*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)*x^4*a^4+7140*c^(9/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x^4*a^4+630*(c*(
a^2*x^2-1)/a^2)^(7/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*a^7*c-1120*(c*(a^2
*x^2-1)/a^2)^(7/2)*c^(9/2)*x^3*a^3+3948*c^(9/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x^3*a^3-560*(c*(a^2*x^2-1)/a^2)^
(7/2)*c^(9/2)*x^2*a^2-4998*c^(9/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x^2*a^2+560*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)
*x*a-1338*c^(9/2)*((a*x-1)*(a*x+1)*c/a^2)^(7/2)*x*a+70*(c*(a^2*x^2-1)/a^2)^(7/2)*c^(9/2)+1338*c^(9/2)*((a*x-1)
*(a*x+1)*c/a^2)^(7/2))*(a*x+1)/((a*x-1)*(a*x+1)*c/a^2)^(7/2)/x^9/(c*(a^2*x^2-1)/a^2/x^2)^(9/2)/a^10/c^(9/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^2/(-a^2*x^2+1)/(c-c/a^2/x^2)^(9/2),x, algorithm="maxima")

[Out]

-integrate((a*x + 1)^2/((a^2*x^2 - 1)*(c - c/(a^2*x^2))^(9/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{9/2}\,\left (a^2\,x^2-1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a*x + 1)^2/((c - c/(a^2*x^2))^(9/2)*(a^2*x^2 - 1)),x)

[Out]

int(-(a*x + 1)^2/((c - c/(a^2*x^2))^(9/2)*(a^2*x^2 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a x}{a c^{4} x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {6 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac {6 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{6} x^{6}} + \frac {c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{7} x^{7}} - \frac {c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{8} x^{8}}}\, dx - \int \frac {1}{a c^{4} x \sqrt {c - \frac {c}{a^{2} x^{2}}} - c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}} - \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x} + \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac {6 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{3} x^{3}} - \frac {6 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{5} x^{5}} + \frac {4 c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{6} x^{6}} + \frac {c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{7} x^{7}} - \frac {c^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a^{8} x^{8}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**2/(-a**2*x**2+1)/(c-c/a**2/x**2)**(9/2),x)

[Out]

-Integral(a*x/(a*c**4*x*sqrt(c - c/(a**2*x**2)) - c**4*sqrt(c - c/(a**2*x**2)) - 4*c**4*sqrt(c - c/(a**2*x**2)
)/(a*x) + 4*c**4*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 6*c**4*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 6*c**4*sqr
t(c - c/(a**2*x**2))/(a**4*x**4) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + 4*c**4*sqrt(c - c/(a**2*x**2))
/(a**6*x**6) + c**4*sqrt(c - c/(a**2*x**2))/(a**7*x**7) - c**4*sqrt(c - c/(a**2*x**2))/(a**8*x**8)), x) - Inte
gral(1/(a*c**4*x*sqrt(c - c/(a**2*x**2)) - c**4*sqrt(c - c/(a**2*x**2)) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a*x)
+ 4*c**4*sqrt(c - c/(a**2*x**2))/(a**2*x**2) + 6*c**4*sqrt(c - c/(a**2*x**2))/(a**3*x**3) - 6*c**4*sqrt(c - c
/(a**2*x**2))/(a**4*x**4) - 4*c**4*sqrt(c - c/(a**2*x**2))/(a**5*x**5) + 4*c**4*sqrt(c - c/(a**2*x**2))/(a**6*
x**6) + c**4*sqrt(c - c/(a**2*x**2))/(a**7*x**7) - c**4*sqrt(c - c/(a**2*x**2))/(a**8*x**8)), x)

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