3.507 \(\int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx\)

Optimal. Leaf size=313 \[ \frac {363 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}+\frac {149 x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {x^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{4 \sqrt {1-\frac {1}{a x}}}+\frac {17 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{24 a \sqrt {1-\frac {1}{a x}}} \]

[Out]

363/64*arctanh((1+1/a/x)^(1/2))*(c-c/a/x)^(1/2)/a^4/(1-1/a/x)^(1/2)-4*arctanh(1/2*(1+1/a/x)^(1/2)*2^(1/2))*2^(
1/2)*(c-c/a/x)^(1/2)/a^4/(1-1/a/x)^(1/2)+149/64*x*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/a^3/(1-1/a/x)^(1/2)+107/96*x
^2*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/a^2/(1-1/a/x)^(1/2)+17/24*x^3*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/a/(1-1/a/x)^(
1/2)+1/4*x^4*(1+1/a/x)^(1/2)*(c-c/a/x)^(1/2)/(1-1/a/x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.35, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6182, 6180, 98, 151, 156, 63, 208, 206} \[ \frac {107 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {149 x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {363 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{a x}+1}}{\sqrt {2}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}+\frac {x^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{4 \sqrt {1-\frac {1}{a x}}}+\frac {17 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{24 a \sqrt {1-\frac {1}{a x}}} \]

Antiderivative was successfully verified.

[In]

Int[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^3,x]

[Out]

(149*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x)/(64*a^3*Sqrt[1 - 1/(a*x)]) + (107*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*
x)]*x^2)/(96*a^2*Sqrt[1 - 1/(a*x)]) + (17*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^3)/(24*a*Sqrt[1 - 1/(a*x)]) +
(Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]*x^4)/(4*Sqrt[1 - 1/(a*x)]) + (363*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a
*x)]])/(64*a^4*Sqrt[1 - 1/(a*x)]) - (4*Sqrt[2]*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/(a^4*Sqrt
[1 - 1/(a*x)])

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 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 6180

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

Rule 6182

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

Rubi steps

\begin {align*} \int e^{3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^3 \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^5 \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {-\frac {17}{2 a}-\frac {15 x}{2 a^2}}{x^4 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{4 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\frac {107}{4 a^2}+\frac {85 x}{4 a^3}}{x^3 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{12 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {-\frac {447}{8 a^3}-\frac {321 x}{8 a^4}}{x^2 \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{24 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\frac {1089}{16 a^4}+\frac {447 x}{16 a^5}}{x \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{24 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (4 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^5 \sqrt {1-\frac {1}{a x}}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a^4 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (8 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{64 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=\frac {149 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {107 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {17 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}}}+\frac {363 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {1+\frac {1}{a x}}}{\sqrt {2}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.99, size = 252, normalized size = 0.81 \[ \frac {1089 \sqrt {c} \log \left (2 a^2 \sqrt {c} x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-768 \sqrt {2} \sqrt {c} \log \left (2 \sqrt {2} a^2 \sqrt {c} x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )+\frac {2 a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (48 a^3 x^3+136 a^2 x^2+214 a x+447\right ) \sqrt {c-\frac {c}{a x}}}{a x-1}-1089 \sqrt {c} \log (1-a x)+768 \sqrt {2} \sqrt {c} \log \left ((a x-1)^2\right )}{384 a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(3*ArcCoth[a*x])*Sqrt[c - c/(a*x)]*x^3,x]

[Out]

((2*a^2*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2*(447 + 214*a*x + 136*a^2*x^2 + 48*a^3*x^3))/(-1 + a*x) - 1
089*Sqrt[c]*Log[1 - a*x] + 768*Sqrt[2]*Sqrt[c]*Log[(-1 + a*x)^2] + 1089*Sqrt[c]*Log[2*a^2*Sqrt[c]*Sqrt[1 - 1/(
a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - a*x + 2*a^2*x^2)] - 768*Sqrt[2]*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sq
rt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)])/(384*a^4)

________________________________________________________________________________________

fricas [A]  time = 0.88, size = 568, normalized size = 1.81 \[ \left [\frac {768 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt {2} {\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 1089 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 350 \, a^{3} x^{3} + 661 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{768 \, {\left (a^{5} x - a^{4}\right )}}, \frac {768 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {2} {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 1089 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (48 \, a^{5} x^{5} + 184 \, a^{4} x^{4} + 350 \, a^{3} x^{3} + 661 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} x - a^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/768*(768*sqrt(2)*(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^3*x^3 + 4*a
^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)
) + 1089*(a*x - 1)*sqrt(c)*log(-(8*a^3*c*x^3 - 7*a*c*x + 4*(2*a^3*x^3 + 3*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1
)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a*x - 1)) + 4*(48*a^5*x^5 + 184*a^4*x^4 + 350*a^3*x^3 + 661*a^2*x^2
+ 447*a*x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^5*x - a^4), 1/384*(768*sqrt(2)*(a*x - 1)*sqr
t(-c)*arctan(2*sqrt(2)*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2
- 2*a*c*x - c)) - 1089*(a*x - 1)*sqrt(-c)*arctan(2*(a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a
*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) + 2*(48*a^5*x^5 + 184*a^4*x^4 + 350*a^3*x^3 + 661*a^2*x^2 + 447*a*
x)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)))/(a^5*x - a^4)]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.06, size = 224, normalized size = 0.72 \[ -\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-96 \sqrt {\left (a x +1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, x^{3}-272 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, x^{2}-428 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, x -894 \sqrt {\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+768 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x +1\right ) x}\, a +3 a x +1}{a x -1}\right ) \sqrt {a}-1089 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a \sqrt {\frac {1}{a}}\right )}{384 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) a^{\frac {9}{2}} \sqrt {\left (a x +1\right ) x}\, \sqrt {\frac {1}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)*x/a^(9/2)*(-96*((a*x+1)*x)^(1/2)*a^(9/2)*
(1/a)^(1/2)*x^3-272*a^(7/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*x^2-428*a^(5/2)*(1/a)^(1/2)*((a*x+1)*x)^(1/2)*x-894*
((a*x+1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)+768*2^(1/2)*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)-1089*ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2)+2*a*x+1)/a^(1/2))*a*(1/a)^(1/2))/((a*x+1)*x)^(1/2)/(1/a)^
(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________