3.779 \(\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx\)

Optimal. Leaf size=181 \[ -\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {3 a^5 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{4 \sqrt {1-a x} \sqrt {a x+1}}-\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x} \]

[Out]

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

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Rubi [A]  time = 0.44, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6159, 6129, 98, 151, 12, 92, 208} \[ -\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {3 a^5 x \sqrt {c-\frac {c}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {a x+1}\right )}{4 \sqrt {1-a x} \sqrt {a x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^4*Sqrt[c - c/(a^2*x^2)])/5 - Sqrt[c - c/(a^2*x^2)]/(5*x^4) + (a*Sqrt[c - c/(a^2*x^2)])/(2*x^3) - (3*a^2*
Sqrt[c - c/(a^2*x^2)])/(5*x^2) + (3*a^3*Sqrt[c - c/(a^2*x^2)])/(4*x) + (3*a^5*Sqrt[c - c/(a^2*x^2)]*x*ArcTanh[
Sqrt[1 - a*x]*Sqrt[1 + a*x]])/(4*Sqrt[1 - a*x]*Sqrt[1 + a*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 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 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

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 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)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {1-a x} \sqrt {1+a x}}{x^6} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {(1-a x)^{3/2}}{x^6 \sqrt {1+a x}} \, dx}{\sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {10 a-9 a^2 x}{x^5 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{5 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {36 a^2-30 a^3 x}{x^4 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{20 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {90 a^3-72 a^4 x}{x^3 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{60 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {144 a^4-90 a^5 x}{x^2 \sqrt {1-a x} \sqrt {1+a x}} \, dx}{120 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {90 a^5}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{120 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}-\frac {\left (3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {1}{x \sqrt {1-a x} \sqrt {1+a x}} \, dx}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {\left (3 a^6 \sqrt {c-\frac {c}{a^2 x^2}} x\right ) \operatorname {Subst}\left (\int \frac {1}{a-a x^2} \, dx,x,\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ &=-\frac {6}{5} a^4 \sqrt {c-\frac {c}{a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 x^4}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}}}{2 x^3}-\frac {3 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{5 x^2}+\frac {3 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x}+\frac {3 a^5 \sqrt {c-\frac {c}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-a x} \sqrt {1+a x}\right )}{4 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 102, normalized size = 0.56 \[ -\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (15 a^5 x^5 \tan ^{-1}\left (\frac {1}{\sqrt {a^2 x^2-1}}\right )+\sqrt {a^2 x^2-1} \left (24 a^4 x^4-15 a^3 x^3+12 a^2 x^2-10 a x+4\right )\right )}{20 x^4 \sqrt {a^2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/20*(Sqrt[c - c/(a^2*x^2)]*(Sqrt[-1 + a^2*x^2]*(4 - 10*a*x + 12*a^2*x^2 - 15*a^3*x^3 + 24*a^4*x^4) + 15*a^5*
x^5*ArcTan[1/Sqrt[-1 + a^2*x^2]]))/(x^4*Sqrt[-1 + a^2*x^2])

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fricas [A]  time = 0.57, size = 232, normalized size = 1.28 \[ \left [\frac {15 \, a^{4} \sqrt {-c} x^{4} \log \left (-\frac {a^{2} c x^{2} + 2 \, a \sqrt {-c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{40 \, x^{4}}, -\frac {15 \, a^{4} \sqrt {c} x^{4} \arctan \left (\frac {a \sqrt {c} x \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (24 \, a^{4} x^{4} - 15 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 10 \, a x + 4\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{20 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/40*(15*a^4*sqrt(-c)*x^4*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
4*a^4*x^4 - 15*a^3*x^3 + 12*a^2*x^2 - 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4, -1/20*(15*a^4*sqrt(c)*
x^4*arctan(a*sqrt(c)*x*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)) + (24*a^4*x^4 - 15*a^3*x^3 + 12*a^2*x^
2 - 10*a*x + 4)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/x^4]

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giac [B]  time = 7.93, size = 362, normalized size = 2.00 \[ \frac {1}{10} \, {\left (15 \, a^{3} \sqrt {c} \arctan \left (-\frac {\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}}{\sqrt {c}}\right ) \mathrm {sgn}\relax (x) - \frac {15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{9} a^{3} c \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{7} a^{3} c^{2} \mathrm {sgn}\relax (x) + 40 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{6} a^{2} c^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) + 200 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{4} a^{2} c^{\frac {7}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 70 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{3} a^{3} c^{4} \mathrm {sgn}\relax (x) + 120 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} a^{2} c^{\frac {9}{2}} {\left | a \right |} \mathrm {sgn}\relax (x) - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )} a^{3} c^{5} \mathrm {sgn}\relax (x) + 24 \, a^{2} c^{\frac {11}{2}} {\left | a \right |} \mathrm {sgn}\relax (x)}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{5}}\right )} {\left | a \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(15*a^3*sqrt(c)*arctan(-(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))/sqrt(c))*sgn(x) - (15*(sqrt(a^2*c)*x - sqrt
(a^2*c*x^2 - c))^9*a^3*c*sgn(x) + 70*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^7*a^3*c^2*sgn(x) + 40*(sqrt(a^2*c)*
x - sqrt(a^2*c*x^2 - c))^6*a^2*c^(5/2)*abs(a)*sgn(x) + 200*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^4*a^2*c^(7/2)
*abs(a)*sgn(x) - 70*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^3*a^3*c^4*sgn(x) + 120*(sqrt(a^2*c)*x - sqrt(a^2*c*x
^2 - c))^2*a^2*c^(9/2)*abs(a)*sgn(x) - 15*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))*a^3*c^5*sgn(x) + 24*a^2*c^(11/
2)*abs(a)*sgn(x))/((sqrt(a^2*c)*x - sqrt(a^2*c*x^2 - c))^2 + c)^5)*abs(a)

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maple [B]  time = 0.06, size = 447, normalized size = 2.47 \[ \frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, a^{2} \left (-40 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, x^{6} a^{4} c +40 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x^{4} a^{4}-15 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{5} a^{3} c +40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) x^{5} a^{2}-40 \sqrt {-\frac {c}{a^{2}}}\, c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+c x}{\sqrt {c}}\right ) x^{5} a^{2}+40 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{5} a^{3} c -25 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{3} a^{3}-15 \ln \left (\frac {2 \sqrt {-\frac {c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2}-2 c}{a^{2} x}\right ) x^{5} a \,c^{2}+16 \sqrt {-\frac {c}{a^{2}}}\, \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} x^{2} a^{2}-10 a \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\, x +4 \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )^{\frac {3}{2}} \sqrt {-\frac {c}{a^{2}}}\right )}{20 x^{4} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, \sqrt {-\frac {c}{a^{2}}}\, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/x^4*a^2*(-40*(c*(a^2*x^2-1)/a^2)^(1/2)*(-c/a^2)^(1/2)*x^6*a^4*c+40*(c*(a^2*
x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2)*x^4*a^4-15*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*x^5*a^3*c+40*(-c/a^2)^(1/
2)*c^(3/2)*ln(x*c^(1/2)+(c*(a^2*x^2-1)/a^2)^(1/2))*x^5*a^2-40*(-c/a^2)^(1/2)*c^(3/2)*ln((c^(1/2)*((a*x-1)*(a*x
+1)*c/a^2)^(1/2)+c*x)/c^(1/2))*x^5*a^2+40*(-c/a^2)^(1/2)*((a*x-1)*(a*x+1)*c/a^2)^(1/2)*x^5*a^3*c-25*(-c/a^2)^(
1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^3*a^3-15*ln(2*((-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(1/2)*a^2-c)/a^2/x)*x^5*a*c
^2+16*(-c/a^2)^(1/2)*(c*(a^2*x^2-1)/a^2)^(3/2)*x^2*a^2-10*a*(c*(a^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2)*x+4*(c*(a
^2*x^2-1)/a^2)^(3/2)*(-c/a^2)^(1/2))/(c*(a^2*x^2-1)/a^2)^(1/2)/(-c/a^2)^(1/2)/c

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{{\left (a x + 1\right )}^{2} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{x^5\,{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{6} + x^{5}}\right )\, dx - \int \frac {a x \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x^{6} + x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-sqrt(c - c/(a**2*x**2))/(a*x**6 + x**5), x) - Integral(a*x*sqrt(c - c/(a**2*x**2))/(a*x**6 + x**5),
x)

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