3.5.9 \(\int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) [409]

Optimal. Leaf size=216 \[ -\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac {45 a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt {1+a x}\right )}{8 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )}{c (1-a x)^{3/2}} \]

[Out]

-45/8*a^3*(-a*c*x+c)^(3/2)*arctanh((a*x+1)^(1/2))/c/(-a*x+1)^(3/2)+4*a^3*(-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)-1/3*(-a*c*x+c)^(3/2)*(a*x+1)^(1/2)/c/x^3/(-a*x+1)^(3/2)-13/12*a*(-a*c*
x+c)^(3/2)*(a*x+1)^(1/2)/c/x^2/(-a*x+1)^(3/2)-19/8*a^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 = 0.12, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6265, 23, 100, 156, 162, 65, 214} \begin {gather*} -\frac {45 a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{8 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )}{c (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {a x+1} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac {\sqrt {a x+1} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {a x+1} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^4,x]

[Out]

-1/3*(Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(c*x^3*(1 - a*x)^(3/2)) - (13*a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(12*c*
x^2*(1 - a*x)^(3/2)) - (19*a^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(8*c*x*(1 - a*x)^(3/2)) - (45*a^3*(c - a*c*x)^
(3/2)*ArcTanh[Sqrt[1 + a*x]])/(8*c*(1 - a*x)^(3/2)) + (4*Sqrt[2]*a^3*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/S
qrt[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 65

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 100

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[((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] && ILtQ[m, -1]

Rule 162

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 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 6265

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^4} \, dx &=\int \frac {(1+a x)^{3/2} \sqrt {c-a c x}}{x^4 (1-a x)^{3/2}} \, dx\\ &=\frac {(c-a c x)^{3/2} \int \frac {(1+a x)^{3/2}}{x^4 (c-a c x)} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {(c-a c x)^{3/2} \int \frac {-\frac {13 a c}{2}-\frac {11}{2} a^2 c x}{x^3 \sqrt {1+a x} (c-a c x)} \, dx}{3 c (1-a x)^{3/2}}\\ &=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}+\frac {(c-a c x)^{3/2} \int \frac {\frac {57 a^2 c^2}{4}+\frac {39}{4} a^3 c^2 x}{x^2 \sqrt {1+a x} (c-a c x)} \, dx}{6 c^2 (1-a x)^{3/2}}\\ &=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac {(c-a c x)^{3/2} \int \frac {-\frac {135}{8} a^3 c^3-\frac {57}{8} a^4 c^3 x}{x \sqrt {1+a x} (c-a c x)} \, dx}{6 c^3 (1-a x)^{3/2}}\\ &=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}+\frac {\left (4 a^4 (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 (45 a^3 (c-a c x)^{3/2}\right ) \int \frac {1}{x \sqrt {1+a x}} \, dx}{16 c (1-a x)^{3/2}}\\ &=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}+\frac {\left (8 a^3 (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{2 c-c x^2} \, dx,x,\sqrt {1+a x}\right )}{(1-a x)^{3/2}}+\frac {\left (45 a^2 (c-a c x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {1+a x}\right )}{8 c (1-a x)^{3/2}}\\ &=-\frac {\sqrt {1+a x} (c-a c x)^{3/2}}{3 c x^3 (1-a x)^{3/2}}-\frac {13 a \sqrt {1+a x} (c-a c x)^{3/2}}{12 c x^2 (1-a x)^{3/2}}-\frac {19 a^2 \sqrt {1+a x} (c-a c x)^{3/2}}{8 c x (1-a x)^{3/2}}-\frac {45 a^3 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt {1+a x}\right )}{8 c (1-a x)^{3/2}}+\frac {4 \sqrt {2} a^3 (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 = 0.05, size = 100, normalized size = 0.46 \begin {gather*} -\frac {\sqrt {c-a c x} \left (\sqrt {1+a x} \left (8+26 a x+57 a^2 x^2\right )+135 a^3 x^3 \tanh ^{-1}\left (\sqrt {1+a x}\right )-96 \sqrt {2} a^3 x^3 \tanh ^{-1}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{24 x^3 \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^4,x]

[Out]

-1/24*(Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(8 + 26*a*x + 57*a^2*x^2) + 135*a^3*x^3*ArcTanh[Sqrt[1 + a*x]] - 96*Sqrt
[2]*a^3*x^3*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/(x^3*Sqrt[1 - a*x])

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Maple [A]
time = 1.06, size = 151, normalized size = 0.70

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (96 \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{3} c \,x^{3}-135 c \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}}{\sqrt {c}}\right ) a^{3} x^{3}-57 a^{2} x^{2} \sqrt {c}\, \sqrt {\left (a x +1\right ) c}-26 \sqrt {c}\, \sqrt {\left (a x +1\right ) c}\, a x -8 \sqrt {\left (a x +1\right ) c}\, \sqrt {c}\right )}{24 \sqrt {c}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) c}\, x^{3}}\)	\(151\)
risch	\(\frac {\left (57 a^{3} x^{3}+83 a^{2} x^{2}+34 a x +8\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{24 x^{3} \sqrt {\left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {a^{3} \left (-\frac {90 \arctanh \left (\frac {\sqrt {c x a +c}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {64 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{16 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\)	\(187\)

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^4,x,method=_RETURNVERBOSE)

[Out]

-1/24*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(96*2^(1/2)*arctanh(1/2*((a*x+1)*c)^(1/2)*2^(1/2)/c^(1/2))*a^3*c*x
^3-135*c*arctanh(((a*x+1)*c)^(1/2)/c^(1/2))*a^3*x^3-57*a^2*x^2*c^(1/2)*((a*x+1)*c)^(1/2)-26*c^(1/2)*((a*x+1)*c
)^(1/2)*a*x-8*((a*x+1)*c)^(1/2)*c^(1/2))/c^(1/2)/(a*x-1)/((a*x+1)*c)^(1/2)/x^3

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 407, normalized size = 1.88 \begin {gather*} \left [\frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\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 ) + 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\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 \, {\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{48 \, {\left (a x^{4} - x^{3}\right )}}, \frac {96 \, \sqrt {2} {\left (a^{4} x^{4} - a^{3} x^{3}\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 ) - 135 \, {\left (a^{4} x^{4} - a^{3} x^{3}\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 ) + {\left (57 \, a^{2} x^{2} + 26 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{24 \, {\left (a x^{4} - x^{3}\right )}}\right ] \end {gather*}

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^4,x, algorithm="fricas")

[Out]

[1/48*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*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)) + 135*(a^4*x^4 - a^3*x^3)*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*(57*a^2*x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 +
1)*sqrt(-a*c*x + c))/(a*x^4 - x^3), 1/24*(96*sqrt(2)*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(2)*sqrt(-a^2*x^2
+ 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 135*(a^4*x^4 - a^3*x^3)*sqrt(-c)*arctan(sqrt(-a^2*x^2 + 1)*
sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (57*a^2*x^2 + 26*a*x + 8)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a
*x^4 - x^3)]

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

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**4,x)

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{x^4\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - a*c*x)^(1/2)*(a*x + 1)^3)/(x^4*(1 - a^2*x^2)^(3/2)),x)

[Out]

int(((c - a*c*x)^(1/2)*(a*x + 1)^3)/(x^4*(1 - a^2*x^2)^(3/2)), x)

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