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

Optimal. Leaf size=254 \[ -\frac {2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}} \]

[Out]

-32/35*(a-1/x)^3*(-a*c*x+c)^(5/2)/a^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(1/2)-2944/35*(-a*c*x+c)^(5/2)/a^4/(1-1/a/x)^(
5/2)/x^3/(1+1/a/x)^(1/2)-256/7*(-a*c*x+c)^(5/2)/a^3/(1-1/a/x)^(5/2)/x^2/(1+1/a/x)^(1/2)+128/35*(-a*c*x+c)^(5/2
)/a^2/(1-1/a/x)^(5/2)/x/(1+1/a/x)^(1/2)+2/7*(a-1/x)^4*x*(-a*c*x+c)^(5/2)/a^4/(1-1/a/x)^(5/2)/(1+1/a/x)^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac {256 (c-a c x)^{5/2}}{7 a^3 x^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {2944 (c-a c x)^{5/2}}{35 a^4 x^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^4 (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}}+\frac {128 (c-a c x)^{5/2}}{35 a^2 x \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {\frac {1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*(a - x^(-1))^3*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]) - (2944*(c - a*c*x)^(5/2
))/(35*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x^3) - (256*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2)*Sqr
t[1 + 1/(a*x)]*x^2) + (128*(c - a*c*x)^(5/2))/(35*a^2*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)]*x) + (2*(a - x^(-1
))^4*x*(c - a*c*x)^(5/2))/(7*a^4*(1 - 1/(a*x))^(5/2)*Sqrt[1 + 1/(a*x)])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 78

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

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 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 e^{-3 \coth ^{-1}(a x)} (c-a c x)^{5/2} \, dx &=\frac {(c-a c x)^{5/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{5/2} x^{5/2} \, dx}{\left (1-\frac {1}{a x}\right )^{5/2} x^{5/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4}{x^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{5/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (16 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{7 a \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (192 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (128 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1472 \left (\frac {1}{x}\right )^{5/2} (c-a c x)^{5/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2}}\\ &=-\frac {32 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}-\frac {2944 (c-a c x)^{5/2}}{35 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {256 (c-a c x)^{5/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 (c-a c x)^{5/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 x (c-a c x)^{5/2}}{7 a^4 \left (1-\frac {1}{a x}\right )^{5/2} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 68, normalized size = 0.27 \[ \frac {2 c^2 \left (5 a^4 x^4-36 a^3 x^3+142 a^2 x^2-708 a x-1451\right ) \sqrt {c-a c x}}{35 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c^2*Sqrt[c - a*c*x]*(-1451 - 708*a*x + 142*a^2*x^2 - 36*a^3*x^3 + 5*a^4*x^4))/(35*a^2*Sqrt[1 - 1/(a^2*x^2)]
*x)

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fricas [A]  time = 0.69, size = 83, normalized size = 0.33 \[ \frac {2 \, {\left (5 \, a^{4} c^{2} x^{4} - 36 \, a^{3} c^{2} x^{3} + 142 \, a^{2} c^{2} x^{2} - 708 \, a c^{2} x - 1451 \, c^{2}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{2} x - a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^4*c^2*x^4 - 36*a^3*c^2*x^3 + 142*a^2*c^2*x^2 - 708*a*c^2*x - 1451*c^2)*sqrt(-a*c*x + c)*sqrt((a*x -
1)/(a*x + 1))/(a^2*x - a)

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(a*x+1)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.04, size = 72, normalized size = 0.28 \[ \frac {2 \left (a x +1\right ) \left (5 x^{4} a^{4}-36 x^{3} a^{3}+142 a^{2} x^{2}-708 a x -1451\right ) \left (-a c x +c \right )^{\frac {5}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{35 a \left (a x -1\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(a*x+1)*(5*a^4*x^4-36*a^3*x^3+142*a^2*x^2-708*a*x-1451)*(-a*c*x+c)^(5/2)*((a*x-1)/(a*x+1))^(3/2)/a/(a*x-1
)^4

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maxima [A]  time = 0.35, size = 120, normalized size = 0.47 \[ \frac {2 \, {\left (5 \, a^{5} \sqrt {-c} c^{2} x^{5} - 31 \, a^{4} \sqrt {-c} c^{2} x^{4} + 106 \, a^{3} \sqrt {-c} c^{2} x^{3} - 566 \, a^{2} \sqrt {-c} c^{2} x^{2} - 2159 \, a \sqrt {-c} c^{2} x - 1451 \, \sqrt {-c} c^{2}\right )} {\left (a x - 1\right )}^{2}}{35 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*a^5*sqrt(-c)*c^2*x^5 - 31*a^4*sqrt(-c)*c^2*x^4 + 106*a^3*sqrt(-c)*c^2*x^3 - 566*a^2*sqrt(-c)*c^2*x^2 -
2159*a*sqrt(-c)*c^2*x - 1451*sqrt(-c)*c^2)*(a*x - 1)^2/((a^3*x^2 - 2*a^2*x + a)*(a*x + 1)^(3/2))

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mupad [B]  time = 1.38, size = 94, normalized size = 0.37 \[ \frac {2\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (5\,a^3\,x^3-31\,a^2\,x^2+111\,a\,x-597\right )}{35\,a}-\frac {4096\,c^2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{35\,a\,\left (a\,x-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2)*(111*a*x - 31*a^2*x^2 + 5*a^3*x^3 - 597))/(35*a) - (4096*
c^2*(c - a*c*x)^(1/2)*((a*x - 1)/(a*x + 1))^(1/2))/(35*a*(a*x - 1))

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

[Out]

Timed out

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