3.562 \(\int \frac {e^{-3 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^{7/2}} \, dx\)

Optimal. Leaf size=251 \[ \frac {(1-a x)^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 (1-a x)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{4 \sqrt {2} a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {7 \sqrt {a x+1} (1-a x)^{7/2}}{4 a^4 x^3 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {(1-a x)^{7/2}}{4 a^3 x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {(1-a x)^{5/2}}{2 a^2 x \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{7/2}} \]

[Out]

(-a*x+1)^(7/2)*arcsinh(a^(1/2)*x^(1/2))/a^(9/2)/(c-c/a/x)^(7/2)/x^(7/2)-11/8*(-a*x+1)^(7/2)*arctanh(2^(1/2)*a^
(1/2)*x^(1/2)/(a*x+1)^(1/2))/a^(9/2)/(c-c/a/x)^(7/2)/x^(7/2)*2^(1/2)+1/2*(-a*x+1)^(5/2)/a^2/(c-c/a/x)^(7/2)/x/
(a*x+1)^(1/2)-1/4*(-a*x+1)^(7/2)/a^3/(c-c/a/x)^(7/2)/x^2/(a*x+1)^(1/2)+7/4*(-a*x+1)^(7/2)*(a*x+1)^(1/2)/a^4/(c
-c/a/x)^(7/2)/x^3

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Rubi [A]  time = 0.22, antiderivative size = 251, 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 = {6134, 6129, 98, 150, 154, 157, 54, 215, 93, 206} \[ \frac {7 \sqrt {a x+1} (1-a x)^{7/2}}{4 a^4 x^3 \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {(1-a x)^{7/2}}{4 a^3 x^2 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {(1-a x)^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}-\frac {11 (1-a x)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )}{4 \sqrt {2} a^{9/2} x^{7/2} \left (c-\frac {c}{a x}\right )^{7/2}}+\frac {(1-a x)^{5/2}}{2 a^2 x \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(7/2)),x]

[Out]

(1 - a*x)^(5/2)/(2*a^2*(c - c/(a*x))^(7/2)*x*Sqrt[1 + a*x]) - (1 - a*x)^(7/2)/(4*a^3*(c - c/(a*x))^(7/2)*x^2*S
qrt[1 + a*x]) + (7*(1 - a*x)^(7/2)*Sqrt[1 + a*x])/(4*a^4*(c - c/(a*x))^(7/2)*x^3) + ((1 - a*x)^(7/2)*ArcSinh[S
qrt[a]*Sqrt[x]])/(a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2)) - (11*(1 - a*x)^(7/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])
/Sqrt[1 + a*x]])/(4*Sqrt[2]*a^(9/2)*(c - c/(a*x))^(7/2)*x^(7/2))

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 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 150

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] && IntegersQ[2*m, 2*n, 2*p]

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 )^{7/2}} \, dx &=\frac {(1-a x)^{7/2} \int \frac {e^{-3 \tanh ^{-1}(a x)} x^{7/2}}{(1-a x)^{7/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{7/2} \int \frac {x^{7/2}}{(1-a x)^2 (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2} \int \frac {x^{3/2} \left (\frac {5}{2}+3 a x\right )}{(1-a x) (1+a x)^{3/2}} \, dx}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2}}{4 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}-\frac {(1-a x)^{7/2} \int \frac {\sqrt {x} \left (-\frac {3 a}{4}+\frac {7 a^2 x}{2}\right )}{(1-a x) \sqrt {1+a x}} \, dx}{2 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2}}{4 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}+\frac {7 (1-a x)^{7/2} \sqrt {1+a x}}{4 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}+\frac {(1-a x)^{7/2} \int \frac {-\frac {7 a^2}{4}-a^3 x}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{2 a^6 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2}}{4 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}+\frac {7 (1-a x)^{7/2} \sqrt {1+a x}}{4 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}+\frac {(1-a x)^{7/2} \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}-\frac {\left (11 (1-a x)^{7/2}\right ) \int \frac {1}{\sqrt {x} (1-a x) \sqrt {1+a x}} \, dx}{8 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2}}{4 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}+\frac {7 (1-a x)^{7/2} \sqrt {1+a x}}{4 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}+\frac {(1-a x)^{7/2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}-\frac {\left (11 (1-a x)^{7/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-2 a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+a x}}\right )}{4 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ &=\frac {(1-a x)^{5/2}}{2 a^2 \left (c-\frac {c}{a x}\right )^{7/2} x \sqrt {1+a x}}-\frac {(1-a x)^{7/2}}{4 a^3 \left (c-\frac {c}{a x}\right )^{7/2} x^2 \sqrt {1+a x}}+\frac {7 (1-a x)^{7/2} \sqrt {1+a x}}{4 a^4 \left (c-\frac {c}{a x}\right )^{7/2} x^3}+\frac {(1-a x)^{7/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}-\frac {11 (1-a x)^{7/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {1+a x}}\right )}{4 \sqrt {2} a^{9/2} \left (c-\frac {c}{a x}\right )^{7/2} x^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.65, size = 234, normalized size = 0.93 \[ \frac {-40 a^{7/2} x^{7/2} (a x-1) \sqrt {a x+1} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};-a x\right )-56 a^{5/2} x^{5/2} (a x-1) \sqrt {a x+1} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};-a x\right )+35 \left (\sqrt {a} \sqrt {x} \left (2 a^3 x^3+13 a^2 x^2+2 a x-25\right )+19 (a x-1) \sqrt {a x+1} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )-22 (a x-1) \sqrt {2 a x+2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {a x+1}}\right )\right )}{560 a^{3/2} c^3 \sqrt {x} \sqrt {1-a^2 x^2} \sqrt {c-\frac {c}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a*x))^(7/2)),x]

[Out]

(35*(Sqrt[a]*Sqrt[x]*(-25 + 2*a*x + 13*a^2*x^2 + 2*a^3*x^3) + 19*(-1 + a*x)*Sqrt[1 + a*x]*ArcSinh[Sqrt[a]*Sqrt
[x]] - 22*(-1 + a*x)*Sqrt[2 + 2*a*x]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[x])/Sqrt[1 + a*x]]) - 56*a^(5/2)*x^(5/2)*(-
1 + a*x)*Sqrt[1 + a*x]*Hypergeometric2F1[3/2, 5/2, 7/2, -(a*x)] - 40*a^(7/2)*x^(7/2)*(-1 + a*x)*Sqrt[1 + a*x]*
Hypergeometric2F1[3/2, 7/2, 9/2, -(a*x)])/(560*a^(3/2)*c^3*Sqrt[c - c/(a*x)]*Sqrt[x]*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.59, size = 596, normalized size = 2.37 \[ \left [-\frac {11 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \log \left (-\frac {17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt {2} {\left (3 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 8 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 8 \, {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{32 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}, \frac {11 \, \sqrt {2} {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) - 8 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 4 \, {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - 7 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{16 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm="fricas")

[Out]

[-1/32*(11*sqrt(2)*(a^3*x^3 - a^2*x^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)) + 8*(a^3*x^3 - a^2*x^2 - 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*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sq
rt((a*c*x - c)/(a*x)))/(a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4), 1/16*(11*sqrt(2)*(a^3*x^3 - a^2*x^2 -
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)) - 8*(a^3*x^3 - a^2*x^2 - 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*(4*a^3*x^3 + a^2*x^2 - 7*a*x)*sqrt(-a^2*x^2 + 1)*sqrt((a*c*x - c)/(a*x)))/(
a^4*c^4*x^3 - a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]

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giac [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(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.07, size = 315, normalized size = 1.25 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \sqrt {2}\, \left (8 a^{\frac {7}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x^{2}+2 a^{\frac {5}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, x -4 a^{3} \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) \sqrt {2}\, \sqrt {-\frac {1}{a}}\, x^{2}+11 a^{\frac {5}{2}} \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) x^{2}-14 \sqrt {-\left (a x +1\right ) x}\, a^{\frac {3}{2}} \sqrt {2}\, \sqrt {-\frac {1}{a}}+4 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) a \sqrt {2}\, \sqrt {-\frac {1}{a}}-11 \ln \left (\frac {2 \sqrt {2}\, \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, a -3 a x -1}{a x -1}\right ) \sqrt {a}\right ) \sqrt {-a^{2} x^{2}+1}}{16 a^{\frac {3}{2}} c^{4} \left (a x +1\right ) \sqrt {-\frac {1}{a}}\, \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x)

[Out]

-1/16*(c*(a*x-1)/a/x)^(1/2)*x*2^(1/2)*(8*a^(7/2)*2^(1/2)*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*x^2+2*a^(5/2)*2^(1/2)
*(-1/a)^(1/2)*(-(a*x+1)*x)^(1/2)*x-4*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+11*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^2-14*(-(a*x+1)*x)^(1/2)*a^
(3/2)*2^(1/2)*(-1/a)^(1/2)+4*arctan(1/2/a^(1/2)*(2*a*x+1)/(-(a*x+1)*x)^(1/2))*a*2^(1/2)*(-1/a)^(1/2)-11*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))*(-a^2*x^2+1)^(1/2)/a^(3/2)/c^4/(a*x+1)/(-
1/a)^(1/2)/(-(a*x+1)*x)^(1/2)/(a*x-1)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a/x)^(7/2),x, algorithm="maxima")

[Out]

integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a*x))^(7/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{7/2}\,{\left (a\,x+1\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^(7/2)*(a*x + 1)^3),x)

[Out]

int((1 - a^2*x^2)^(3/2)/((c - c/(a*x))^(7/2)*(a*x + 1)^3), x)

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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)**3*(-a**2*x**2+1)**(3/2)/(c-c/a/x)**(7/2),x)

[Out]

Timed out

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