3.553 \(\int \frac {e^{-2 \tanh ^{-1}(a x)}}{(c-\frac {c}{a x})^{9/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}+\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}} \]

[Out]

6/5/a/c^2/(c-c/a/x)^(5/2)+11/6/a/c^3/(c-c/a/x)^(3/2)-x/c^2/(c-c/a/x)^(5/2)-5*arctanh((c-c/a/x)^(1/2)/c^(1/2))/
a/c^(9/2)-1/8*arctanh(1/2*(c-c/a/x)^(1/2)*2^(1/2)/c^(1/2))/a/c^(9/2)*2^(1/2)+21/4/a/c^4/(c-c/a/x)^(1/2)

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Rubi [A]  time = 0.26, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6133, 25, 514, 375, 103, 152, 156, 63, 208} \[ -\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(9/2)),x]

[Out]

6/(5*a*c^2*(c - c/(a*x))^(5/2)) + 11/(6*a*c^3*(c - c/(a*x))^(3/2)) + 21/(4*a*c^4*Sqrt[c - c/(a*x)]) - x/(c^2*(
c - c/(a*x))^(5/2)) - (5*ArcTanh[Sqrt[c - c/(a*x)]/Sqrt[c]])/(a*c^(9/2)) - ArcTanh[Sqrt[c - c/(a*x)]/(Sqrt[2]*
Sqrt[c])]/(4*Sqrt[2]*a*c^(9/2))

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 152

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] && IntegersQ[2*m, 2*n, 2*p]

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 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 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 514

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 6133

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

Rubi steps

\begin {align*} \int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{9/2}} \, dx &=\int \frac {1-a x}{\left (c-\frac {c}{a x}\right )^{9/2} (1+a x)} \, dx\\ &=-\frac {a \int \frac {x}{\left (c-\frac {c}{a x}\right )^{7/2} (1+a x)} \, dx}{c}\\ &=-\frac {a \int \frac {1}{\left (a+\frac {1}{x}\right ) \left (c-\frac {c}{a x}\right )^{7/2}} \, dx}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x) \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {5 c}{2}-\frac {7 c x}{2 a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {25 c^2}{2}+\frac {15 c^2 x}{a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{5 c^4}\\ &=\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {75 c^3}{2}-\frac {165 c^3 x}{4 a}}{x (a+x) \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{15 c^6}\\ &=\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {75 c^4}{2}+\frac {315 c^4 x}{8 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{15 c^8}\\ &=\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a c^4}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^4}\\ &=\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{4 c^5}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^5}\\ &=\frac {6}{5 a c^2 \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {11}{6 a c^3 \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {21}{4 a c^4 \sqrt {c-\frac {c}{a x}}}-\frac {x}{c^2 \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{9/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{4 \sqrt {2} a c^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 82, normalized size = 0.47 \[ -\frac {a x^2 \left (-\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {a-\frac {1}{x}}{2 a}\right )-5 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {1}{a x}\right )+5 a x\right )}{5 c^4 (a x-1)^2 \sqrt {c-\frac {c}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(E^(2*ArcTanh[a*x])*(c - c/(a*x))^(9/2)),x]

[Out]

-1/5*(a*x^2*(5*a*x - Hypergeometric2F1[-5/2, 1, -3/2, (a - x^(-1))/(2*a)] - 5*Hypergeometric2F1[-5/2, 1, -3/2,
1 - 1/(a*x)]))/(c^4*Sqrt[c - c/(a*x)]*(-1 + a*x)^2)

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fricas [A]  time = 0.42, size = 428, normalized size = 2.47 \[ \left [\frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) + 600 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 4 \, {\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{240 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}, \frac {15 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) + 600 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - 2 \, {\left (60 \, a^{4} x^{4} - 497 \, a^{3} x^{3} + 740 \, a^{2} x^{2} - 315 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{120 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(9/2),x, algorithm="fricas")

[Out]

[1/240*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log((2*sqrt(2)*a*sqrt(c)*x*sqrt((a*c*x - c)/(a*x)
) - 3*a*c*x + c)/(a*x + 1)) + 600*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(c)*log(-2*a*c*x + 2*a*sqrt(c)*x*sqrt(
(a*c*x - c)/(a*x)) + c) - 4*(60*a^4*x^4 - 497*a^3*x^3 + 740*a^2*x^2 - 315*a*x)*sqrt((a*c*x - c)/(a*x)))/(a^4*c
^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5), 1/120*(15*sqrt(2)*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*ar
ctan(1/2*sqrt(2)*sqrt(-c)*sqrt((a*c*x - c)/(a*x))/c) + 600*(a^3*x^3 - 3*a^2*x^2 + 3*a*x - 1)*sqrt(-c)*arctan(s
qrt(-c)*sqrt((a*c*x - c)/(a*x))/c) - 2*(60*a^4*x^4 - 497*a^3*x^3 + 740*a^2*x^2 - 315*a*x)*sqrt((a*c*x - c)/(a*
x)))/(a^4*c^5*x^3 - 3*a^3*c^5*x^2 + 3*a^2*c^5*x - a*c^5)]

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giac [A]  time = 0.19, size = 207, normalized size = 1.20 \[ \frac {1}{120} \, a c {\left (\frac {2 \, {\left (12 \, c^{2} + \frac {50 \, {\left (a c x - c\right )} c}{a x} + \frac {255 \, {\left (a c x - c\right )}^{2}}{a^{2} x^{2}}\right )} x^{2}}{{\left (a c x - c\right )}^{2} c^{5} \sqrt {\frac {a c x - c}{a x}}} + \frac {15 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a c x - c}{a x}}}{2 \, \sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{5}} + \frac {600 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c^{5}} - \frac {120 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(9/2),x, algorithm="giac")

[Out]

1/120*a*c*(2*(12*c^2 + 50*(a*c*x - c)*c/(a*x) + 255*(a*c*x - c)^2/(a^2*x^2))*x^2/((a*c*x - c)^2*c^5*sqrt((a*c*
x - c)/(a*x))) + 15*sqrt(2)*arctan(1/2*sqrt(2)*sqrt((a*c*x - c)/(a*x))/sqrt(-c))/(a^2*sqrt(-c)*c^5) + 600*arct
an(sqrt((a*c*x - c)/(a*x))/sqrt(-c))/(a^2*sqrt(-c)*c^5) - 120*sqrt((a*c*x - c)/(a*x))/(a^2*(c - (a*c*x - c)/(a
*x))*c^5))

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maple [B]  time = 0.05, size = 626, normalized size = 3.62 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (1260 a^{\frac {11}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{4}+600 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{4} a^{5}-15 a^{\frac {9}{2}} \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 ) x^{4}-1020 a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{2}-5040 a^{\frac {9}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{3}-2400 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{3} a^{4}+60 a^{\frac {7}{2}} \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 ) x^{3}+1792 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x +7560 a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x^{2}+3600 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x^{2} a^{3}-90 a^{\frac {5}{2}} \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 ) x^{2}-820 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}-5040 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, x -2400 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, x \,a^{2}+60 a^{\frac {3}{2}} \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 ) x +1260 \sqrt {\left (a x -1\right ) x}\, a^{\frac {3}{2}} \sqrt {\frac {1}{a}}+600 \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}}-15 \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}\right )}{240 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, c^{5} \left (a x -1\right )^{4} \sqrt {\frac {1}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(9/2),x)

[Out]

-1/240*(c*(a*x-1)/a/x)^(1/2)*x/a^(3/2)*(1260*a^(11/2)*(1/a)^(1/2)*((a*x-1)*x)^(1/2)*x^4+600*ln(1/2*(2*((a*x-1)
*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*(1/a)^(1/2)*x^4*a^5-15*a^(9/2)*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))*x^4-1020*a^(9/2)*(1/a)^(1/2)*((a*x-1)*x)^(3/2)*x^2-5040*a^(9/2)*(1/a)^(1/2)*((a*x
-1)*x)^(1/2)*x^3-2400*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*(1/a)^(1/2)*x^3*a^4+60*a^(7/2)*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))*x^3+1792*a^(7/2)*(1/a)^(1/2)*((a*x-1)*x)^
(3/2)*x+7560*a^(7/2)*(1/a)^(1/2)*((a*x-1)*x)^(1/2)*x^2+3600*ln(1/2*(2*((a*x-1)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/
2))*(1/a)^(1/2)*x^2*a^3-90*a^(5/2)*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))*x^2
-820*a^(5/2)*(1/a)^(1/2)*((a*x-1)*x)^(3/2)-5040*a^(5/2)*(1/a)^(1/2)*((a*x-1)*x)^(1/2)*x-2400*ln(1/2*(2*((a*x-1
)*x)^(1/2)*a^(1/2)+2*a*x-1)/a^(1/2))*(1/a)^(1/2)*x*a^2+60*a^(3/2)*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))*x+1260*((a*x-1)*x)^(1/2)*a^(3/2)*(1/a)^(1/2)+600*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)-15*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))/((a*x-1)*x)^(1/2)/c^5/(a*x-1)^4/(1/a)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(c-c/a/x)^(9/2),x, algorithm="maxima")

[Out]

-integrate((a^2*x^2 - 1)/((a*x + 1)^2*(c - c/(a*x))^(9/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {a^2\,x^2-1}{{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,{\left (a\,x+1\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(a^2*x^2 - 1)/((c - c/(a*x))^(9/2)*(a*x + 1)^2),x)

[Out]

-int((a^2*x^2 - 1)/((c - c/(a*x))^(9/2)*(a*x + 1)^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {a x}{a c^{4} x \sqrt {c - \frac {c}{a x}} - 3 c^{4} \sqrt {c - \frac {c}{a x}} + \frac {2 c^{4} \sqrt {c - \frac {c}{a x}}}{a x} + \frac {2 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} - \frac {3 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}} + \frac {c^{4} \sqrt {c - \frac {c}{a x}}}{a^{4} x^{4}}}\, dx - \int \left (- \frac {1}{a c^{4} x \sqrt {c - \frac {c}{a x}} - 3 c^{4} \sqrt {c - \frac {c}{a x}} + \frac {2 c^{4} \sqrt {c - \frac {c}{a x}}}{a x} + \frac {2 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}} - \frac {3 c^{4} \sqrt {c - \frac {c}{a x}}}{a^{3} x^{3}} + \frac {c^{4} \sqrt {c - \frac {c}{a x}}}{a^{4} x^{4}}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(c-c/a/x)**(9/2),x)

[Out]

-Integral(a*x/(a*c**4*x*sqrt(c - c/(a*x)) - 3*c**4*sqrt(c - c/(a*x)) + 2*c**4*sqrt(c - c/(a*x))/(a*x) + 2*c**4
*sqrt(c - c/(a*x))/(a**2*x**2) - 3*c**4*sqrt(c - c/(a*x))/(a**3*x**3) + c**4*sqrt(c - c/(a*x))/(a**4*x**4)), x
) - Integral(-1/(a*c**4*x*sqrt(c - c/(a*x)) - 3*c**4*sqrt(c - c/(a*x)) + 2*c**4*sqrt(c - c/(a*x))/(a*x) + 2*c*
*4*sqrt(c - c/(a*x))/(a**2*x**2) - 3*c**4*sqrt(c - c/(a*x))/(a**3*x**3) + c**4*sqrt(c - c/(a*x))/(a**4*x**4)),
x)

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