3.401 \(\int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx\)

Optimal. Leaf size=110 \[ -\frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {5 a \sqrt {c-a c x}}{8 x^3} \]

[Out]

-75/64*a^4*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)-1/4*(-a*c*x+c)^(1/2)/x^4-5/8*a*(-a*c*x+c)^(1/2)/x^3-25/32
*a^2*(-a*c*x+c)^(1/2)/x^2-75/64*a^3*(-a*c*x+c)^(1/2)/x

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Rubi [A]  time = 0.14, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6130, 21, 78, 51, 63, 208} \[ -\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {\sqrt {c-a c x}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[c - a*c*x]/(4*x^4) - (5*a*Sqrt[c - a*c*x])/(8*x^3) - (25*a^2*Sqrt[c - a*c*x])/(32*x^2) - (75*a^3*Sqrt[c
- a*c*x])/(64*x) - (75*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64

Rule 21

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

Rule 51

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] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 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 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 6130

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, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !(IntegerQ[p] || GtQ[c, 0]
)

Rubi steps

\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx &=\int \frac {(1+a x) \sqrt {c-a c x}}{x^5 (1-a x)} \, dx\\ &=c \int \frac {1+a x}{x^5 \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {1}{8} (15 a c) \int \frac {1}{x^4 \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {5 a \sqrt {c-a c x}}{8 x^3}+\frac {1}{16} \left (25 a^2 c\right ) \int \frac {1}{x^3 \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}+\frac {1}{64} \left (75 a^3 c\right ) \int \frac {1}{x^2 \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {75 a^3 \sqrt {c-a c x}}{64 x}+\frac {1}{128} \left (75 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {1}{64} \left (75 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {5 a \sqrt {c-a c x}}{8 x^3}-\frac {25 a^2 \sqrt {c-a c x}}{32 x^2}-\frac {75 a^3 \sqrt {c-a c x}}{64 x}-\frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 71, normalized size = 0.65 \[ -\frac {75}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-\frac {\left (75 a^3 x^3+50 a^2 x^2+40 a x+16\right ) \sqrt {c-a c x}}{64 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/64*(Sqrt[c - a*c*x]*(16 + 40*a*x + 50*a^2*x^2 + 75*a^3*x^3))/x^4 - (75*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/
Sqrt[c]])/64

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fricas [A]  time = 0.75, size = 149, normalized size = 1.35 \[ \left [\frac {75 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{128 \, x^{4}}, \frac {75 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (75 \, a^{3} x^{3} + 50 \, a^{2} x^{2} + 40 \, a x + 16\right )} \sqrt {-a c x + c}}{64 \, x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/128*(75*a^4*sqrt(c)*x^4*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) - 2*(75*a^3*x^3 + 50*a^2*x^2 + 40
*a*x + 16)*sqrt(-a*c*x + c))/x^4, 1/64*(75*a^4*sqrt(-c)*x^4*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (75*a^3*x^3
+ 50*a^2*x^2 + 40*a*x + 16)*sqrt(-a*c*x + c))/x^4]

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giac [A]  time = 0.39, size = 131, normalized size = 1.19 \[ \frac {\frac {75 \, a^{5} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {75 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{5} c + 275 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{5} c^{2} - 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{5} c^{3} + 181 \, \sqrt {-a c x + c} a^{5} c^{4}}{a^{4} c^{4} x^{4}}}{64 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(75*a^5*c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - (75*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^5*c + 275*(a*
c*x - c)^2*sqrt(-a*c*x + c)*a^5*c^2 - 365*(-a*c*x + c)^(3/2)*a^5*c^3 + 181*sqrt(-a*c*x + c)*a^5*c^4)/(a^4*c^4*
x^4))/a

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maple [A]  time = 0.04, size = 93, normalized size = 0.85 \[ -2 a^{4} c^{4} \left (\frac {-\frac {75 \left (-a c x +c \right )^{\frac {7}{2}}}{128 c^{3}}+\frac {275 \left (-a c x +c \right )^{\frac {5}{2}}}{128 c^{2}}-\frac {365 \left (-a c x +c \right )^{\frac {3}{2}}}{128 c}+\frac {181 \sqrt {-a c x +c}}{128}}{x^{4} a^{4} c^{4}}+\frac {75 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 c^{\frac {7}{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^4*c^4*((-75/128/c^3*(-a*c*x+c)^(7/2)+275/128/c^2*(-a*c*x+c)^(5/2)-365/128/c*(-a*c*x+c)^(3/2)+181/128*(-a*
c*x+c)^(1/2))/x^4/a^4/c^4+75/128/c^(7/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2)))

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maxima [A]  time = 0.46, size = 163, normalized size = 1.48 \[ \frac {1}{128} \, a^{4} c^{4} {\left (\frac {2 \, {\left (75 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 275 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 365 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 181 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {75 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*a^4*c^4*(2*(75*(-a*c*x + c)^(7/2) - 275*(-a*c*x + c)^(5/2)*c + 365*(-a*c*x + c)^(3/2)*c^2 - 181*sqrt(-a*
c*x + c)*c^3)/((a*c*x - c)^4*c^3 + 4*(a*c*x - c)^3*c^4 + 6*(a*c*x - c)^2*c^5 + 4*(a*c*x - c)*c^6 + c^7) + 75*l
og((sqrt(-a*c*x + c) - sqrt(c))/(sqrt(-a*c*x + c) + sqrt(c)))/c^(7/2))

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mupad [B]  time = 0.10, size = 91, normalized size = 0.83 \[ \frac {365\,{\left (c-a\,c\,x\right )}^{3/2}}{64\,c\,x^4}-\frac {181\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {275\,{\left (c-a\,c\,x\right )}^{5/2}}{64\,c^2\,x^4}+\frac {75\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}+\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,75{}\mathrm {i}}{64} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^4*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i)/c^(1/2))*75i)/64 - (181*(c - a*c*x)^(1/2))/(64*x^4) + (365*(c - a*c*x
)^(3/2))/(64*c*x^4) - (275*(c - a*c*x)^(5/2))/(64*c^2*x^4) + (75*(c - a*c*x)^(7/2))/(64*c^3*x^4)

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sympy [B]  time = 78.97, size = 639, normalized size = 5.81 \[ - \frac {558 a^{4} c^{8} \sqrt {- a c x + c}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} + \frac {1022 a^{4} c^{7} \left (- a c x + c\right )^{\frac {3}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} - \frac {770 a^{4} c^{6} \left (- a c x + c\right )^{\frac {5}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} + \frac {66 a^{4} c^{6} \sqrt {- a c x + c}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac {210 a^{4} c^{5} \left (- a c x + c\right )^{\frac {7}{2}}}{1536 a c^{8} x - 1152 c^{8} + 2304 c^{6} \left (- a c x + c\right )^{2} - 1536 c^{5} \left (- a c x + c\right )^{3} + 384 c^{4} \left (- a c x + c\right )^{4}} - \frac {80 a^{4} c^{5} \left (- a c x + c\right )^{\frac {3}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac {35 a^{4} c^{5} \sqrt {\frac {1}{c^{9}}} \log {\left (- c^{5} \sqrt {\frac {1}{c^{9}}} + \sqrt {- a c x + c} \right )}}{128} - \frac {35 a^{4} c^{5} \sqrt {\frac {1}{c^{9}}} \log {\left (c^{5} \sqrt {\frac {1}{c^{9}}} + \sqrt {- a c x + c} \right )}}{128} + \frac {30 a^{4} c^{4} \left (- a c x + c\right )^{\frac {5}{2}}}{- 144 a c^{6} x + 96 c^{6} - 144 c^{4} \left (- a c x + c\right )^{2} + 48 c^{3} \left (- a c x + c\right )^{3}} + \frac {5 a^{4} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (- c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} - \frac {5 a^{4} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-558*a**4*c**8*sqrt(-a*c*x + c)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c
)**3 + 384*c**4*(-a*c*x + c)**4) + 1022*a**4*c**7*(-a*c*x + c)**(3/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(
-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) - 770*a**4*c**6*(-a*c*x + c)**(5/2)/(15
36*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x + c)**3 + 384*c**4*(-a*c*x + c)**4) +
66*a**4*c**6*sqrt(-a*c*x + c)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) +
210*a**4*c**5*(-a*c*x + c)**(7/2)/(1536*a*c**8*x - 1152*c**8 + 2304*c**6*(-a*c*x + c)**2 - 1536*c**5*(-a*c*x
+ c)**3 + 384*c**4*(-a*c*x + c)**4) - 80*a**4*c**5*(-a*c*x + c)**(3/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a
*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 35*a**4*c**5*sqrt(c**(-9))*log(-c**5*sqrt(c**(-9)) + sqrt(-a*c*x + c
))/128 - 35*a**4*c**5*sqrt(c**(-9))*log(c**5*sqrt(c**(-9)) + sqrt(-a*c*x + c))/128 + 30*a**4*c**4*(-a*c*x + c)
**(5/2)/(-144*a*c**6*x + 96*c**6 - 144*c**4*(-a*c*x + c)**2 + 48*c**3*(-a*c*x + c)**3) + 5*a**4*c**4*sqrt(c**(
-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 - 5*a**4*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(
-a*c*x + c))/16

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