3.6.82 \(\int e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx\) [582]

Optimal. Leaf size=292 \[ -\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {363 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{7/2} \sqrt {1-a x}} \]

[Out]

-21/32*x*(a*x+1)^(3/2)*(c-c/a/x)^(1/2)/a^3/(-a*x+1)^(1/2)-11/24*x^2*(a*x+1)^(3/2)*(c-c/a/x)^(1/2)/a^2/(-a*x+1)
^(1/2)-1/4*x^3*(a*x+1)^(3/2)*(c-c/a/x)^(1/2)/a/(-a*x+1)^(1/2)-363/64*arcsinh(a^(1/2)*x^(1/2))*(c-c/a/x)^(1/2)*
x^(1/2)/a^(7/2)/(-a*x+1)^(1/2)+4*arctanh(2^(1/2)*a^(1/2)*x^(1/2)/(a*x+1)^(1/2))*2^(1/2)*(c-c/a/x)^(1/2)*x^(1/2
)/a^(7/2)/(-a*x+1)^(1/2)-107/64*x*(c-c/a/x)^(1/2)*(a*x+1)^(1/2)/a^3/(-a*x+1)^(1/2)

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Rubi [A]
time = 0.21, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6269, 6264, 103, 159, 163, 56, 221, 95, 212} \begin {gather*} -\frac {363 \sqrt {x} \sqrt {c-\frac {c}{a x}} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {x} \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{a^{7/2} \sqrt {1-a x}}-\frac {21 x (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{32 a^3 \sqrt {1-a x}}-\frac {107 x \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{64 a^3 \sqrt {1-a x}}-\frac {11 x^2 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{24 a^2 \sqrt {1-a x}}-\frac {x^3 (a x+1)^{3/2} \sqrt {c-\frac {c}{a x}}}{4 a \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-107*Sqrt[c - c/(a*x)]*x*Sqrt[1 + a*x])/(64*a^3*Sqrt[1 - a*x]) - (21*Sqrt[c - c/(a*x)]*x*(1 + a*x)^(3/2))/(32
*a^3*Sqrt[1 - a*x]) - (11*Sqrt[c - c/(a*x)]*x^2*(1 + a*x)^(3/2))/(24*a^2*Sqrt[1 - a*x]) - (Sqrt[c - c/(a*x)]*x
^3*(1 + a*x)^(3/2))/(4*a*Sqrt[1 - a*x]) - (363*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcSinh[Sqrt[a]*Sqrt[x]])/(64*a^(7/2)
*Sqrt[1 - a*x]) + (4*Sqrt[2]*Sqrt[c - c/(a*x)]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]])/(a^(7
/2)*Sqrt[1 - a*x])

Rule 56

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 95

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 103

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

Rule 159

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 163

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 212

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

Rule 221

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 6264

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 6269

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 e^{3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int e^{3 \tanh ^{-1}(a x)} x^{5/2} \sqrt {1-a x} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{5/2} (1+a x)^{3/2}}{1-a x} \, dx}{\sqrt {1-a x}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {x^{3/2} \sqrt {1+a x} \left (\frac {5}{2}+\frac {11 a x}{2}\right )}{1-a x} \, dx}{4 a \sqrt {1-a x}}\\ &=-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {x} \sqrt {1+a x} \left (-\frac {33 a}{4}-\frac {63 a^2 x}{4}\right )}{1-a x} \, dx}{12 a^3 \sqrt {1-a x}}\\ &=-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}+\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1+a x} \left (\frac {63 a^2}{8}+\frac {321 a^3 x}{8}\right )}{\sqrt {x} (1-a x)} \, dx}{24 a^5 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-\frac {447 a^3}{16}-\frac {1089 a^4 x}{16}}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{24 a^6 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{128 a^3 \sqrt {1-a x}}+\frac {\left (4 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{a^3 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {\left (363 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{64 a^3 \sqrt {1-a x}}+\frac {\left (8 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{a^3 \sqrt {1-a x}}\\ &=-\frac {107 \sqrt {c-\frac {c}{a x}} x \sqrt {1+a x}}{64 a^3 \sqrt {1-a x}}-\frac {21 \sqrt {c-\frac {c}{a x}} x (1+a x)^{3/2}}{32 a^3 \sqrt {1-a x}}-\frac {11 \sqrt {c-\frac {c}{a x}} x^2 (1+a x)^{3/2}}{24 a^2 \sqrt {1-a x}}-\frac {\sqrt {c-\frac {c}{a x}} x^3 (1+a x)^{3/2}}{4 a \sqrt {1-a x}}-\frac {363 \sqrt {c-\frac {c}{a x}} \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {1-a x}}+\frac {4 \sqrt {2} \sqrt {c-\frac {c}{a x}} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{a^{7/2} \sqrt {1-a x}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 130, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {c-\frac {c}{a x}} \left (\sqrt {a} x \sqrt {1+a x} \left (447+214 a x+136 a^2 x^2+48 a^3 x^3\right )+1089 \sqrt {x} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-768 \sqrt {2} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )\right )}{192 a^{7/2} \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/192*(Sqrt[c - c/(a*x)]*(Sqrt[a]*x*Sqrt[1 + a*x]*(447 + 214*a*x + 136*a^2*x^2 + 48*a^3*x^3) + 1089*Sqrt[x]*A
rcSinh[Sqrt[a]*Sqrt[x]] - 768*Sqrt[2]*Sqrt[x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]))/(a^(7/2)*Sqrt
[1 - a*x])

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Maple [A]
time = 0.96, size = 247, normalized size = 0.85

method	result	size
default	\(\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (96 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{3}+272 a^{\frac {7}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+428 a^{\frac {5}{2}} \sqrt {-\left (a x +1\right ) x}\, \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x +894 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}-1089 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}+1536 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {2}}{768 a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}\, \sqrt {-\frac {1}{a}}}\)	\(247\)
risch	\(-\frac {\left (48 a^{3} x^{3}+136 a^{2} x^{2}+214 a x +447\right ) x \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{192 a^{3} \sqrt {-a c x \left (a x +1\right )}\, \sqrt {-a^{2} x^{2}+1}}-\frac {\left (\frac {363 \arctan \left (\frac {\sqrt {a^{2} c}\, \left (x +\frac {1}{2 a}\right )}{\sqrt {-a^{2} c \,x^{2}-c x a}}\right )}{128 a^{3} \sqrt {a^{2} c}}-\frac {4 \ln \left (\frac {-4 c -3 \left (x -\frac {1}{a}\right ) a c +2 \sqrt {-2 c}\, \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-3 \left (x -\frac {1}{a}\right ) a c -2 c}}{x -\frac {1}{a}}\right )}{a^{4} \sqrt {-2 c}}\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {\frac {a c x \left (-a^{2} x^{2}+1\right )}{a x -1}}}{\sqrt {-a^{2} x^{2}+1}}\)	\(270\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(c*(a*x-1)/a/x)^(1/2)*x*(-a^2*x^2+1)^(1/2)*(96*a^(9/2)*(-(a*x+1)*x)^(1/2)*2^(1/2)*(-1/a)^(1/2)*x^3+272*a
^(7/2)*(-(a*x+1)*x)^(1/2)*2^(1/2)*(-1/a)^(1/2)*x^2+428*a^(5/2)*(-(a*x+1)*x)^(1/2)*2^(1/2)*(-1/a)^(1/2)*x+894*(
-(a*x+1)*x)^(1/2)*a^(3/2)*2^(1/2)*(-1/a)^(1/2)-1089*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)
*(-1/a)^(1/2)+1536*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^(9/2)/
(a*x-1)/(-(a*x+1)*x)^(1/2)/(-1/a)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.46, size = 500, normalized size = 1.71 \begin {gather*} \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^{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 ) + 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^{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 ) + 4 \, {\left (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {-a^{2} x^{2} + 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} \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 ) - 1089 \, {\left (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 ) - 2 \, {\left (48 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 447 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} x - a^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+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^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)) + 1089*(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)) + 4*(48*a^4*x^4 + 136*a^3*x^3 + 214*a^2*x^2 + 447*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/
(a*x)))/(a^5*x - a^4), -1/384*(768*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(2*sqrt(2)*sqrt(-a^2*x^2 + 1)*a*sqrt(c)*x*s
qrt((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*sqrt(-a^2*x^2 + 1)*a*sqr
t(c)*x*sqrt((a*c*x - c)/(a*x))/(2*a^2*c*x^2 - a*c*x - c)) - 2*(48*a^4*x^4 + 136*a^3*x^3 + 214*a^2*x^2 + 447*a*
x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(a^5*x - a^4)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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