3.534 \(\int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^{5/2}} \, dx\)

Optimal. Leaf size=293 \[ \frac {11 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {249 (1-a x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{16 \sqrt {2} a^{7/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {103 \sqrt {a x+1} (1-a x)^{5/2}}{32 a^3 x^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {43 (a x+1)^{3/2} (1-a x)^{3/2}}{32 a^3 x^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {13 (a x+1)^{3/2} \sqrt {1-a x}}{24 a^2 x \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {(a x+1)^{3/2}}{3 a \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{5/2}} \]

[Out]

43/32*(-a*x+1)^(3/2)*(a*x+1)^(3/2)/a^3/(c-c/a/x)^(5/2)/x^2+11*(-a*x+1)^(5/2)*arcsinh(a^(1/2)*x^(1/2))/a^(7/2)/
(c-c/a/x)^(5/2)/x^(5/2)-249/32*(-a*x+1)^(5/2)*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))/a^(7/2)/(c-c/a/x)
^(5/2)/x^(5/2)*2^(1/2)+1/3*(a*x+1)^(3/2)/a/(c-c/a/x)^(5/2)/(-a*x+1)^(1/2)-13/24*(a*x+1)^(3/2)*(-a*x+1)^(1/2)/a
^2/(c-c/a/x)^(5/2)/x+103/32*(-a*x+1)^(5/2)*(a*x+1)^(1/2)/a^3/(c-c/a/x)^(5/2)/x^2

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6134, 6129, 97, 149, 154, 157, 54, 215, 93, 206} \[ \frac {103 \sqrt {a x+1} (1-a x)^{5/2}}{32 a^3 x^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {43 (a x+1)^{3/2} (1-a x)^{3/2}}{32 a^3 x^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {249 (1-a x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{16 \sqrt {2} a^{7/2} x^{5/2} \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {13 (a x+1)^{3/2} \sqrt {1-a x}}{24 a^2 x \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {(a x+1)^{3/2}}{3 a \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcTanh[a*x])/(c - c/(a*x))^(5/2),x]

[Out]

(103*(1 - a*x)^(5/2)*Sqrt[1 + a*x])/(32*a^3*(c - c/(a*x))^(5/2)*x^2) + (1 + a*x)^(3/2)/(3*a*(c - c/(a*x))^(5/2
)*Sqrt[1 - a*x]) - (13*Sqrt[1 - a*x]*(1 + a*x)^(3/2))/(24*a^2*(c - c/(a*x))^(5/2)*x) + (43*(1 - a*x)^(3/2)*(1
+ a*x)^(3/2))/(32*a^3*(c - c/(a*x))^(5/2)*x^2) + (11*(1 - a*x)^(5/2)*ArcSinh[Sqrt[a]*Sqrt[x]])/(a^(7/2)*(c - c
/(a*x))^(5/2)*x^(5/2)) - (249*(1 - a*x)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(16*Sqrt[2]*a^
(7/2)*(c - c/(a*x))^(5/2)*x^(5/2))

Rule 54

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

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 206

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[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 6134

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

Rubi steps

\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx &=\frac {(1-a x)^{5/2} \int \frac {e^{3 \tanh ^{-1}(a x)} x^{5/2}}{(1-a x)^{5/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {(1-a x)^{5/2} \int \frac {x^{5/2} (1+a x)^{3/2}}{(1-a x)^4} \, dx}{\left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {(1-a x)^{5/2} \int \frac {x^{3/2} \sqrt {1+a x} \left (\frac {5}{2}+4 a x\right )}{(1-a x)^3} \, dx}{3 a \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {13 \sqrt {1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}-\frac {(1-a x)^{5/2} \int \frac {\sqrt {x} \sqrt {1+a x} \left (-\frac {39 a}{4}-\frac {45 a^2 x}{2}\right )}{(1-a x)^2} \, dx}{12 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {13 \sqrt {1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}+\frac {43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}-\frac {(1-a x)^{5/2} \int \frac {\sqrt {1+a x} \left (\frac {129 a^2}{8}+\frac {309 a^3 x}{4}\right )}{\sqrt {x} (1-a x)} \, dx}{24 a^5 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {103 (1-a x)^{5/2} \sqrt {1+a x}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {13 \sqrt {1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}+\frac {43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1-a x)^{5/2} \int \frac {-\frac {219 a^3}{4}-132 a^4 x}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{24 a^6 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {103 (1-a x)^{5/2} \sqrt {1+a x}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {13 \sqrt {1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}+\frac {43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {\left (11 (1-a x)^{5/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}-\frac {\left (249 (1-a x)^{5/2}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {103 (1-a x)^{5/2} \sqrt {1+a x}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {13 \sqrt {1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}+\frac {43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {\left (11 (1-a x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}-\frac {\left (249 (1-a x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{16 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ &=\frac {103 (1-a x)^{5/2} \sqrt {1+a x}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {(1+a x)^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{5/2} \sqrt {1-a x}}-\frac {13 \sqrt {1-a x} (1+a x)^{3/2}}{24 a^2 \left (c-\frac {c}{a x}\right )^{5/2} x}+\frac {43 (1-a x)^{3/2} (1+a x)^{3/2}}{32 a^3 \left (c-\frac {c}{a x}\right )^{5/2} x^2}+\frac {11 (1-a x)^{5/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}-\frac {249 (1-a x)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{16 \sqrt {2} a^{7/2} \left (c-\frac {c}{a x}\right )^{5/2} x^{5/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 147, normalized size = 0.50 \[ \frac {2 \sqrt {a} \sqrt {x} \sqrt {a x+1} \left (-48 a^3 x^3+415 a^2 x^2-554 a x+219\right )-1056 (a x-1)^3 \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )+747 \sqrt {2} (a x-1)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{96 a^{3/2} c^2 \sqrt {x} (1-a x)^{5/2} \sqrt {c-\frac {c}{a x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(3*ArcTanh[a*x])/(c - c/(a*x))^(5/2),x]

[Out]

(2*Sqrt[a]*Sqrt[x]*Sqrt[1 + a*x]*(219 - 554*a*x + 415*a^2*x^2 - 48*a^3*x^3) - 1056*(-1 + a*x)^3*ArcSinh[Sqrt[a
]*Sqrt[x]] + 747*Sqrt[2]*(-1 + a*x)^3*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(96*a^(3/2)*c^2*Sqrt[c
- c/(a*x)]*Sqrt[x]*(1 - a*x)^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 668, normalized size = 2.28 \[ \left [-\frac {747 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 1056 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 8 \, {\left (48 \, a^{4} x^{4} - 415 \, a^{3} x^{3} + 554 \, a^{2} x^{2} - 219 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac {747 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 1056 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (48 \, a^{4} x^{4} - 415 \, a^{3} x^{3} + 554 \, a^{2} x^{2} - 219 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{192 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(747*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 -
13*a*c*x - 4*sqrt(2)*(3*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*
a^2*x^2 + 3*a*x - 1)) + 1056*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*log(-(8*a^3*c*x^3 - 7*a*c*
x + 4*(2*a^2*x^2 + a*x)*sqrt(-a^2*x^2 + 1)*sqrt(-c)*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) - 8*(48*a^4*x^4 -
415*a^3*x^3 + 554*a^2*x^2 - 219*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3
+ 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3), -1/192*(747*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt
(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) - 105
6*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*arctan(2*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*sqrt((a*c*x -
c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 4*(48*a^4*x^4 - 415*a^3*x^3 + 554*a^2*x^2 - 219*a*x)*sqrt(-a^2*x^2 + 1)
*sqrt((a*c*x - c)/(a*x)))/(a^5*c^3*x^4 - 4*a^4*c^3*x^3 + 6*a^3*c^3*x^2 - 4*a^2*c^3*x + a*c^3)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm="giac")

[Out]

integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^(5/2)), x)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 504, normalized size = 1.72 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (96 a^{\frac {9}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x^{3}-830 a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x^{2}+747 a^{\frac {7}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{3}-528 a^{4} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{3}+1108 a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x +1584 a^{3} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}-2241 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}-438 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-1584 a^{2} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +2241 a^{\frac {3}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x +528 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}-747 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{192 a^{\frac {3}{2}} c^{3} \left (a x -1\right )^{4} \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x)

[Out]

1/192*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(96*a^(9/2)*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*x^3-830*a
^(7/2)*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*x^2+747*a^(7/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a
-3*a*x-1)/(a*x-1))*x^3-528*a^4*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*2^(1/2)*(-1/a)^(1/2)*x^3+1108*
a^(5/2)*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*x+1584*a^3*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*2^
(1/2)*(-1/a)^(1/2)*x^2-2241*a^(5/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a-3*a*x-1)/(a*x-1))*x^2-438*
(-(a*x+1)*x)^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)-1584*a^2*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*2^(1
/2)*(-1/a)^(1/2)*x+2241*a^(3/2)*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*a-3*a*x-1)/(a*x-1))*x+528*arctan
(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)-747*ln((2*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^
(1/2)*a-3*a*x-1)/(a*x-1))*a^(1/2))*2^(1/2)/a^(3/2)/c^3/(a*x-1)^4/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a*x))^(5/2)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,x+1\right )}^3}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)^3/((c - c/(a*x))^(5/2)*(1 - a^2*x^2)^(3/2)),x)

[Out]

int((a*x + 1)^3/((c - c/(a*x))^(5/2)*(1 - a^2*x^2)^(3/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x + 1\right )^{3}}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {5}{2}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(5/2),x)

[Out]

Integral((a*x + 1)**3/((-c*(-1 + 1/(a*x)))**(5/2)*(-(a*x - 1)*(a*x + 1))**(3/2)), x)

________________________________________________________________________________________