3.407 \(\int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx\)

Optimal. Leaf size=124 \[ -\frac {\sqrt {a x+1} (c-a c x)^{3/2}}{c x (1-a x)^{3/2}}-\frac {5 a (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{c (1-a x)^{3/2}}+\frac {4 \sqrt {2} a (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{c (1-a x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 124, 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, 23, 98, 156, 63, 208} \[ -\frac {\sqrt {a x+1} (c-a c x)^{3/2}}{c x (1-a x)^{3/2}}-\frac {5 a (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{c (1-a x)^{3/2}}+\frac {4 \sqrt {2} a (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{c (1-a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 23

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

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

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Mathematica [A]  time = 0.04, size = 75, normalized size = 0.60 \[ -\frac {\sqrt {c-a c x} \left (\sqrt {a x+1}+5 a x \tanh ^{-1}\left (\sqrt {a x+1}\right )-4 \sqrt {2} a x \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )\right )}{x \sqrt {1-a x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[c - a*c*x]*(Sqrt[1 + a*x] + 5*a*x*ArcTanh[Sqrt[1 + a*x]] - 4*Sqrt[2]*a*x*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]
]))/(x*Sqrt[1 - a*x]))

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fricas [A]  time = 0.52, size = 358, normalized size = 2.89 \[ \left [\frac {4 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 5 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{2 \, {\left (a x^{2} - x\right )}}, \frac {4 \, \sqrt {2} {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - 5 \, {\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{a x^{2} - x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(4*sqrt(2)*(a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x +
c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 5*(a^2*x^2 - a*x)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x + 2*sqrt(-a^2*x^
2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x), (4*s
qrt(2)*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) -
5*(a^2*x^2 - a*x)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*x
^2 + 1)*sqrt(-a*c*x + c))/(a*x^2 - x)]

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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*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^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 = 0.05, size = 105, normalized size = 0.85 \[ \frac {\left (-4 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) x a c +5 \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) x a c +\sqrt {c \left (a x +1\right )}\, \sqrt {c}\right ) \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}{\left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, \sqrt {c}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*x*a*c+5*arctanh((c*(a*x+1))^(1/2)/c^(1/2))*x*a*c+(c
*(a*x+1))^(1/2)*c^(1/2))*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)/(a*x-1)/(c*(a*x+1))^(1/2)/c^(1/2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{x^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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