3.5.26 \(\int \frac {e^{-2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) [426]

Optimal. Leaf size=127 \[ -\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}-\frac {19 a^2 \sqrt {c-a c x}}{8 x}+\frac {45}{8} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]

[Out]

45/8*a^3*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)-4*a^3*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)
*c^(1/2)-1/3*(-a*c*x+c)^(1/2)/x^3+13/12*a*(-a*c*x+c)^(1/2)/x^2-19/8*a^2*(-a*c*x+c)^(1/2)/x

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Rubi [A]
time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6265, 21, 100, 156, 162, 65, 214, 212} \begin {gather*} \frac {45}{8} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )-\frac {19 a^2 \sqrt {c-a c x}}{8 x}-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^4),x]

[Out]

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

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 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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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^{-2 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^4} \, dx &=\int \frac {(1-a x) \sqrt {c-a c x}}{x^4 (1+a x)} \, dx\\ &=\frac {\int \frac {(c-a c x)^{3/2}}{x^4 (1+a x)} \, dx}{c}\\ &=-\frac {\sqrt {c-a c x}}{3 x^3}-\frac {\int \frac {\frac {13 a c^2}{2}-\frac {11}{2} a^2 c^2 x}{x^3 (1+a x) \sqrt {c-a c x}} \, dx}{3 c}\\ &=-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}+\frac {\int \frac {\frac {57 a^2 c^3}{4}-\frac {39}{4} a^3 c^3 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{6 c^2}\\ &=-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}-\frac {19 a^2 \sqrt {c-a c x}}{8 x}-\frac {\int \frac {\frac {135 a^3 c^4}{8}-\frac {57}{8} a^4 c^4 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{6 c^3}\\ &=-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}-\frac {19 a^2 \sqrt {c-a c x}}{8 x}-\frac {1}{16} \left (45 a^3 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx+\left (4 a^4 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}-\frac {19 a^2 \sqrt {c-a c x}}{8 x}+\frac {1}{8} \left (45 a^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )-\left (8 a^3\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}-\frac {19 a^2 \sqrt {c-a c x}}{8 x}+\frac {45}{8} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 101, normalized size = 0.80 \begin {gather*} \frac {\sqrt {c-a c x} \left (-8+26 a x-57 a^2 x^2\right )}{24 x^3}+\frac {45}{8} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^3 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^4),x]

[Out]

(Sqrt[c - a*c*x]*(-8 + 26*a*x - 57*a^2*x^2))/(24*x^3) + (45*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8 -
4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

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Maple [A]
time = 0.78, size = 108, normalized size = 0.85

method	result	size
risch	\(\frac {\left (57 a^{3} x^{3}-83 a^{2} x^{2}+34 a x -8\right ) c}{24 x^{3} \sqrt {-c \left (a x -1\right )}}-\frac {a^{3} \left (-\frac {90 \arctanh \left (\frac {\sqrt {-c x a +c}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {64 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}\right ) c}{16}\)	\(92\)
derivativedivides	\(2 a^{3} c^{3} \left (\frac {\frac {-\frac {19 \left (-c x a +c \right )^{\frac {5}{2}}}{16}+\frac {11 c \left (-c x a +c \right )^{\frac {3}{2}}}{6}-\frac {13 c^{2} \sqrt {-c x a +c}}{16}}{c^{3} x^{3} a^{3}}+\frac {45 \arctanh \left (\frac {\sqrt {-c x a +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{c^{2}}-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )\)	\(108\)
default	\(2 a^{3} c^{3} \left (\frac {\frac {-\frac {19 \left (-c x a +c \right )^{\frac {5}{2}}}{16}+\frac {11 c \left (-c x a +c \right )^{\frac {3}{2}}}{6}-\frac {13 c^{2} \sqrt {-c x a +c}}{16}}{c^{3} x^{3} a^{3}}+\frac {45 \arctanh \left (\frac {\sqrt {-c x a +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{c^{2}}-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x,method=_RETURNVERBOSE)

[Out]

2*a^3*c^3*(1/c^2*((-19/16*(-a*c*x+c)^(5/2)+11/6*c*(-a*c*x+c)^(3/2)-13/16*c^2*(-a*c*x+c)^(1/2))/c^3/x^3/a^3+45/
16/c^(1/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2)))-2/c^(5/2)*2^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2)))

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Maxima [A]
time = 0.47, size = 183, normalized size = 1.44 \begin {gather*} -\frac {1}{48} \, a^{3} c^{3} {\left (\frac {2 \, {\left (57 \, {\left (-a c x + c\right )}^{\frac {5}{2}} - 88 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 39 \, \sqrt {-a c x + c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c^{2} + 3 \, {\left (a c x - c\right )}^{2} c^{3} + 3 \, {\left (a c x - c\right )} c^{4} + c^{5}} - \frac {96 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {5}{2}}} + \frac {135 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm="maxima")

[Out]

-1/48*a^3*c^3*(2*(57*(-a*c*x + c)^(5/2) - 88*(-a*c*x + c)^(3/2)*c + 39*sqrt(-a*c*x + c)*c^2)/((a*c*x - c)^3*c^
2 + 3*(a*c*x - c)^2*c^3 + 3*(a*c*x - c)*c^4 + c^5) - 96*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqr
t(2)*sqrt(c) + sqrt(-a*c*x + c)))/c^(5/2) + 135*log((sqrt(-a*c*x + c) - sqrt(c))/(sqrt(-a*c*x + c) + sqrt(c)))
/c^(5/2))

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Fricas [A]
time = 0.37, size = 220, normalized size = 1.73 \begin {gather*} \left [\frac {96 \, \sqrt {2} a^{3} \sqrt {c} x^{3} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 135 \, a^{3} \sqrt {c} x^{3} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, {\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt {-a c x + c}}{48 \, x^{3}}, \frac {96 \, \sqrt {2} a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 135 \, a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt {-a c x + c}}{24 \, x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (119) = 238\).
time = 15.78, size = 614, normalized size = 4.83 \begin {gather*} \frac {66 a^{3} 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 {80 a^{3} 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 {30 a^{3} 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 {30 a^{3} c^{4} \sqrt {- a c x + c}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} + \frac {5 a^{3} 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^{3} c^{4} \sqrt {\frac {1}{c^{7}}} \log {\left (c^{4} \sqrt {\frac {1}{c^{7}}} + \sqrt {- a c x + c} \right )}}{16} - \frac {18 a^{3} c^{3} \left (- a c x + c\right )^{\frac {3}{2}}}{16 a c^{4} x - 8 c^{4} + 8 c^{2} \left (- a c x + c\right )^{2}} - \frac {9 a^{3} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} + \frac {9 a^{3} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{8} + 2 a^{3} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )} - 2 a^{3} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )} - \frac {8 a^{3} c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {4 \sqrt {2} a^{3} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {4 a^{2} \sqrt {- a c x + c}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

66*a**3*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) -
80*a**3*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) + 30*a**3*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) + 30*a**3*c**4*sqrt(-a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 5*a**3*c**4*sqrt(c**(
-7))*log(-c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 - 5*a**3*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(
-a*c*x + c))/16 - 18*a**3*c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) - 9*a**3*c*
*3*sqrt(c**(-5))*log(-c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/8 + 9*a**3*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-
5)) + sqrt(-a*c*x + c))/8 + 2*a**3*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) - 2*a**3*c**
2*sqrt(c**(-3))*log(c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) - 8*a**3*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c)
+ 4*sqrt(2)*a**3*c*atan(sqrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) - 4*a**2*sqrt(-a*c*x + c)/x

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Giac [A]
time = 0.40, size = 133, normalized size = 1.05 \begin {gather*} \frac {4 \, \sqrt {2} a^{3} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {45 \, a^{3} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{8 \, \sqrt {-c}} - \frac {57 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{3} c - 88 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{3} c^{2} + 39 \, \sqrt {-a c x + c} a^{3} c^{3}}{24 \, a^{3} c^{3} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x, algorithm="giac")

[Out]

4*sqrt(2)*a^3*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 45/8*a^3*c*arctan(sqrt(-a*c*x + c)/sq
rt(-c))/sqrt(-c) - 1/24*(57*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^3*c - 88*(-a*c*x + c)^(3/2)*a^3*c^2 + 39*sqrt(-a*
c*x + c)*a^3*c^3)/(a^3*c^3*x^3)

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Mupad [B]
time = 0.13, size = 105, normalized size = 0.83 \begin {gather*} \frac {11\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,c\,x^3}-\frac {a^3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,45{}\mathrm {i}}{8}-\frac {13\,\sqrt {c-a\,c\,x}}{8\,x^3}-\frac {19\,{\left (c-a\,c\,x\right )}^{5/2}}{8\,c^2\,x^3}+\sqrt {2}\,a^3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11*(c - a*c*x)^(3/2))/(3*c*x^3) - (a^3*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i)/c^(1/2))*45i)/8 - (13*(c - a*c*x)^
(1/2))/(8*x^3) - (19*(c - a*c*x)^(5/2))/(8*c^2*x^3) + 2^(1/2)*a^3*c^(1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/
(2*c^(1/2)))*4i

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