3.698 \(\int e^{2 \tanh ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{3/2} \, dx\)

Optimal. Leaf size=214 \[ -\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{1-a x}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x)}+\frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]

[Out]

-a*(c-c/a^2/x^2)^(3/2)*x^2/(-a*x+1)+5/2*a^2*(c-c/a^2/x^2)^(3/2)*x^3/(-a*x+1)/(a*x+1)-1/2*(c-c/a^2/x^2)^(3/2)*x
*(a*x+1)/(-a*x+1)-2*a^2*(c-c/a^2/x^2)^(3/2)*x^3*arcsin(a*x)/(-a*x+1)^(3/2)/(a*x+1)^(3/2)-1/2*a^2*(c-c/a^2/x^2)
^(3/2)*x^3*arctanh((-a*x+1)^(1/2)*(a*x+1)^(1/2))/(-a*x+1)^(3/2)/(a*x+1)^(3/2)

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Rubi [A]  time = 0.39, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ \frac {5 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x) (a x+1)}-\frac {a x^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{1-a x}-\frac {x (a x+1) \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}{2 (1-a x)}-\frac {2 a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \sin ^{-1}(a x)}{(1-a x)^{3/2} (a x+1)^{3/2}}-\frac {a^2 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{2 (1-a x)^{3/2} (a x+1)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2),x]

[Out]

-((a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a*x)) + (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) - (
(c - c/(a^2*x^2))^(3/2)*x*(1 + a*x))/(2*(1 - a*x)) - (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcSin[a*x])/((1 - a*x
)^(3/2)*(1 + a*x)^(3/2)) - (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*ArcTanh[Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(2*(1 - a*x)
^(3/2)*(1 + a*x)^(3/2))

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 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 97

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

Rule 149

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] && IntegerQ[m]

Rule 154

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 208

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

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 e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^{3/2} (1+a x)^{3/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {\sqrt {1-a x} (1+a x)^{5/2}}{x^3} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {(1+a x)^{3/2} \left (2 a-3 a^2 x\right )}{x^2 \sqrt {1-a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {\sqrt {1+a x} \left (a^2-5 a^3 x\right )}{x \sqrt {1-a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac {\left (\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {-a^3+4 a^4 x}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 a (1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}+\frac {\left (a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2} (1+a x)^{3/2}}-\frac {\left (2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac {\left (a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}-\frac {\left (2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{(1-a x)^{3/2} (1+a x)^{3/2}}\\ &=-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^2}{1-a x}+\frac {5 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}{2 (1-a x) (1+a x)}-\frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x (1+a x)}{2 (1-a x)}-\frac {2 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \sin ^{-1}(a x)}{(1-a x)^{3/2} (1+a x)^{3/2}}-\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3 \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{2 (1-a x)^{3/2} (1+a x)^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 115, normalized size = 0.54 \[ -\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \left (\sqrt {a^2 x^2-1} \left (2 a^2 x^2-4 a x-1\right )+4 a^2 x^2 \log \left (\sqrt {a^2 x^2-1}+a x\right )+a^2 x^2 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )\right )}{2 a^2 x \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcTanh[a*x])*(c - c/(a^2*x^2))^(3/2),x]

[Out]

-1/2*(c*Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(-1 - 4*a*x + 2*a^2*x^2) + a^2*x^2*ArcTan[1/Sqrt[-1 + a^2*x^
2]] + 4*a^2*x^2*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(a^2*x*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 1.39, size = 316, normalized size = 1.48 \[ \left [\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + a \sqrt {-c} c x \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, a^{2} x}, -\frac {a c^{\frac {3}{2}} x \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) - 2 \, a c^{\frac {3}{2}} x \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (2 \, a^{2} c x^{2} - 4 \, a c x - c\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \, a^{2} x}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[1/4*(8*a*sqrt(-c)*c*x*arctan(a^2*sqrt(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + a*sqrt(-c)*c
*x*log(-(a^2*c*x^2 + 2*a*sqrt(-c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 2*c)/x^2) - 2*(2*a^2*c*x^2 - 4*a*c*x - c
)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x), -1/2*(a*c^(3/2)*x*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2
))/(a^2*c*x^2 - c)) - 2*a*c^(3/2)*x*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) +
(2*a^2*c*x^2 - 4*a*c*x - c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^2*x)]

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giac [A]  time = 0.45, size = 265, normalized size = 1.24 \[ {\left (\frac {c^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x)}{a^{2}} + \frac {2 \, c^{\frac {3}{2}} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\relax (x)}{a {\left | a \right |}} - \frac {\sqrt {a^{2} c x^{2} - c} c \mathrm {sgn}\relax (x)}{a^{2}} - \frac {{\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} c^{2} {\left | a \right |} \mathrm {sgn}\relax (x) - 4 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a c^{\frac {5}{2}} \mathrm {sgn}\relax (x) - {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} c^{3} {\left | a \right |} \mathrm {sgn}\relax (x) - 4 \, a c^{\frac {7}{2}} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{2} a^{2} {\left | a \right |}}\right )} {\left | a \right |} \]

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)^(3/2),x, algorithm="giac")

[Out]

(c^(3/2)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x)/a^2 + 2*c^(3/2)*log(abs(-sqrt(a^2*c)*x
+ sqrt(a^2*c*x^2 - c)))*sgn(x)/(a*abs(a)) - sqrt(a^2*c*x^2 - c)*c*sgn(x)/a^2 - ((sqrt(a^2*c)*x - sqrt(a^2*c*x^
2 - c))^3*c^2*abs(a)*sgn(x) - 4*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2*a*c^(5/2)*sgn(x) - (sqrt(a^2*c)*x - sq
rt(a^2*c*x^2 - c))*c^3*abs(a)*sgn(x) - 4*a*c^(7/2)*sgn(x))/(((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^2*a^
2*abs(a)))*abs(a)

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maple [B]  time = 0.05, size = 455, normalized size = 2.13 \[ -\frac {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} x \left (12 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{5} c -12 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} x \,a^{5}+4 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{4} c -\sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{4} c +6 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{3} c^{2}-3 a^{4} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {5}{2}} \sqrt {-\frac {c}{a^{2}}}-18 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{3} a^{3} c^{2}+18 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{2} a -6 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {5}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{2} a +3 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{2} c^{2}+3 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{2} c^{3}\right )}{6 a^{2} \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} c} \]

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)^(3/2),x)

[Out]

-1/6*(c*(a^2*x^2-1)/a^2/x^2)^(3/2)*x/a^2*(12*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^3*a^5*c-12*(-c/a^2)^(1
/2)*(c*(a^2*x^2-1)/a^2)^(5/2)*x*a^5+4*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(3/2)*x^2*a^4*c-(-c/a^2)^(1/2)*(c
*(a^2*x^2-1)/a^2)^(3/2)*x^2*a^4*c+6*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^3*a^3*c^2-3*a^4*(c*(a^2*x^2
-1)/a^2)^(5/2)*(-c/a^2)^(1/2)-18*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^3*a^3*c^2+18*(-c/a^2)^(1/2)*c^(5/2
)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x^2*a-6*(-c/a^2)^(1/2)*c^(5/2)*ln((c^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(
1/2)+c*x)/c^(1/2))*x^2*a+3*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^2*a^2*c^2+3*ln(2*((-c/a^2)^(1/2)*(c*(a^2
*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*x^2*c^3)/(-c/a^2)^(1/2)/(c*(a^2*x^2-1)/a^2)^(3/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}{a^{2} x^{2} - 1}\,{d x} \]

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)^(3/2),x, algorithm="maxima")

[Out]

-integrate((a*x + 1)^2*(c - c/(a^2*x^2))^(3/2)/(a^2*x^2 - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int -\frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{a^2\,x^2-1} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((c - c/(a^2*x^2))^(3/2)*(a*x + 1)^2)/(a^2*x^2 - 1),x)

[Out]

int(-((c - c/(a^2*x^2))^(3/2)*(a*x + 1)^2)/(a^2*x^2 - 1), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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)**(3/2),x)

[Out]

Exception raised: TypeError

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