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

Optimal. Leaf size=250 \[ \frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}-\frac {a^4 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}+\frac {a^2 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}} \]

[Out]

1/24*a^3*(1-1/a/x)^(5/2)*(1+1/a/x)^(3/2)*x^2/(a-1/x)^2/(-a*c*x+c)^(5/2)-1/6*a^4*(1-1/a/x)^(5/2)*(1+1/a/x)^(5/2
)*x^2/(a-1/x)^3/(-a*c*x+c)^(5/2)+1/32*a^(3/2)*(1-1/a/x)^(5/2)*arctanh(2^(1/2)*(1/x)^(1/2)/a^(1/2)/(1+1/a/x)^(1
/2))/(1/x)^(5/2)/(-a*c*x+c)^(5/2)*2^(1/2)+1/16*a^2*(1-1/a/x)^(5/2)*x^2*(1+1/a/x)^(1/2)/(a-1/x)/(-a*c*x+c)^(5/2
)

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Rubi [A]  time = 0.22, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6176, 6181, 94, 93, 206} \[ -\frac {a^4 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{3/2}}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}+\frac {a^2 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^2)/(16*(a - x^(-1))*(c - a*c*x)^(5/2)) + (a^3*(1 - 1/(a*x))^(5/2)
*(1 + 1/(a*x))^(3/2)*x^2)/(24*(a - x^(-1))^2*(c - a*c*x)^(5/2)) - (a^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)
*x^2)/(6*(a - x^(-1))^3*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt
[a]*Sqrt[1 + 1/(a*x)])])/(16*Sqrt[2]*(x^(-1))^(5/2)*(c - a*c*x)^(5/2))

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 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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 6176

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

Rule 6181

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

Rubi steps

\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{5/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}} \, dx}{(c-a c x)^{5/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{5/2} \operatorname {Subst}\left (\int \frac {\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^4} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{\sqrt {x} \left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{12 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{16 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{32 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {\left (a \left (1-\frac {1}{a x}\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{16 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ &=\frac {a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}{16 \left (a-\frac {1}{x}\right ) (c-a c x)^{5/2}}+\frac {a^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{3/2} x^2}{24 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{5/2}}-\frac {a^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^2}{6 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}+\frac {a^{3/2} \left (1-\frac {1}{a x}\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{16 \sqrt {2} \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 142, normalized size = 0.57 \[ -\frac {\sqrt {1-\frac {1}{a x}} \left (2 \sqrt {a} \sqrt {\frac {1}{x}} \sqrt {\frac {1}{a x}+1} \left (3 a^2 x^2+22 a x+7\right )-\frac {3 \sqrt {2} (a x-1)^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{x}\right )}{96 \sqrt {a} c^2 \left (\frac {1}{x}\right )^{3/2} (a x-1)^3 \sqrt {c-a c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/96*(Sqrt[1 - 1/(a*x)]*(2*Sqrt[a]*Sqrt[1 + 1/(a*x)]*Sqrt[x^(-1)]*(7 + 22*a*x + 3*a^2*x^2) - (3*Sqrt[2]*(-1 +
a*x)^3*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/x))/(Sqrt[a]*c^2*(x^(-1))^(3/2)*(-1 + a*x
)^3*Sqrt[c - a*c*x])

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fricas [A]  time = 0.61, size = 393, normalized size = 1.57 \[ \left [-\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 4 \, {\left (3 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 29 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{192 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - 2 \, {\left (3 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 29 \, a x + 7\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{96 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/192*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2
)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) - 4*(3*a^3*x^3 +
25*a^2*x^2 + 29*a*x + 7)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(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), -1/96*(3*sqrt(2)*(a^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*sqrt(c)*arctan(sqr
t(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 2*(3*a^3*x^3 + 25*a^2*x^2 + 29*a*x + 7)
*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(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)]

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giac [A]  time = 0.22, size = 116, normalized size = 0.46 \[ -\frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left (3 \, {\left (a c x + c\right )}^{2} \sqrt {-a c x - c} - 16 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c - 12 \, \sqrt {-a c x - c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c}}{96 \, a c \mathrm {sgn}\left (-a c x - c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/c^(3/2) - 2*(3*(a*c*x + c)^2*sqrt(-a*c*x - c) -
16*(-a*c*x - c)^(3/2)*c - 12*sqrt(-a*c*x - c)*c^2)/((a*c*x - c)^3*c))/(a*c*sgn(-a*c*x - c))

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maple [A]  time = 0.07, size = 226, normalized size = 0.90 \[ \frac {\sqrt {-c \left (a x -1\right )}\, \left (-3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x^{3} a^{3} c +9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x^{2} a^{2} c +6 x^{2} a^{2} \sqrt {-c \left (a x +1\right )}\, \sqrt {c}-9 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c +44 x a \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +14 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{96 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right )^{2} \left (a x +1\right ) c^{\frac {7}{2}} \sqrt {-c \left (a x +1\right )}\, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(-c*(a*x-1))^(1/2)*(-3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x^3*a^3*c+9*2^(1/2)*arctan(
1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x^2*a^2*c+6*x^2*a^2*(-c*(a*x+1))^(1/2)*c^(1/2)-9*2^(1/2)*arctan(1/2*(-
c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a*c+44*x*a*(-c*(a*x+1))^(1/2)*c^(1/2)+3*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1
/2)*2^(1/2)/c^(1/2))*c+14*(-c*(a*x+1))^(1/2)*c^(1/2))/((a*x-1)/(a*x+1))^(3/2)/(a*x-1)^2/(a*x+1)/c^(7/2)/(-c*(a
*x+1))^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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