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

Optimal. Leaf size=148 \[ -\frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3} \]

[Out]

-363/64*a^4*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)+4*a^4*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1
/2)*c^(1/2)-1/4*(-a*c*x+c)^(1/2)/x^4+17/24*a*(-a*c*x+c)^(1/2)/x^3-107/96*a^2*(-a*c*x+c)^(1/2)/x^2+149/64*a^3*(
-a*c*x+c)^(1/2)/x

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Rubi [A]  time = 0.21, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6130, 21, 98, 151, 156, 63, 208, 206} \[ -\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {\sqrt {c-a c x}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[c - a*c*x]/(4*x^4) + (17*a*Sqrt[c - a*c*x])/(24*x^3) - (107*a^2*Sqrt[c - a*c*x])/(96*x^2) + (149*a^3*Sqr
t[c - a*c*x])/(64*x) - (363*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64 + 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[S
qrt[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 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 151

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] && LtQ[m, -1] && IntegerQ[m]

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 206

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

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\\ &=\frac {\int \frac {(c-a c x)^{3/2}}{x^5 (1+a x)} \, dx}{c}\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}-\frac {\int \frac {\frac {17 a c^2}{2}-\frac {15}{2} a^2 c^2 x}{x^4 (1+a x) \sqrt {c-a c x}} \, dx}{4 c}\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {\int \frac {\frac {107 a^2 c^3}{4}-\frac {85}{4} a^3 c^3 x}{x^3 (1+a x) \sqrt {c-a c x}} \, dx}{12 c^2}\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {\int \frac {\frac {447 a^3 c^4}{8}-\frac {321}{8} a^4 c^4 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{24 c^3}\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {\int \frac {\frac {1089 a^4 c^5}{16}-\frac {447}{16} a^5 c^5 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{24 c^4}\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {1}{128} \left (363 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx-\left (4 a^5 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {1}{64} \left (363 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 )+\left (8 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )\\ &=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^4 \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 = 0.11, size = 109, normalized size = 0.74 \[ -\frac {363}{64} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^4 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )+\frac {\left (447 a^3 x^3-214 a^2 x^2+136 a x-48\right ) \sqrt {c-a c x}}{192 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c - a*c*x]*(-48 + 136*a*x - 214*a^2*x^2 + 447*a^3*x^3))/(192*x^4) - (363*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*
c*x]/Sqrt[c]])/64 + 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

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fricas [A]  time = 0.56, size = 236, normalized size = 1.59 \[ \left [\frac {768 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 1089 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{384 \, x^{4}}, -\frac {768 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 1089 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{192 \, x^{4}}\right ] \]

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^5,x, algorithm="fricas")

[Out]

[1/384*(768*sqrt(2)*a^4*sqrt(c)*x^4*log((a*c*x - 2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(c) - 3*c)/(a*x + 1)) + 1089*a
^4*sqrt(c)*x^4*log((a*c*x + 2*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/x) + 2*(447*a^3*x^3 - 214*a^2*x^2 + 136*a*x - 48
)*sqrt(-a*c*x + c))/x^4, -1/192*(768*sqrt(2)*a^4*sqrt(-c)*x^4*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)*sqrt(-c)/c)
- 1089*a^4*sqrt(-c)*x^4*arctan(sqrt(-a*c*x + c)*sqrt(-c)/c) - (447*a^3*x^3 - 214*a^2*x^2 + 136*a*x - 48)*sqrt(
-a*c*x + c))/x^4]

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giac [A]  time = 0.18, size = 160, normalized size = 1.08 \[ -\frac {4 \, \sqrt {2} a^{4} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \frac {363 \, a^{4} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{64 \, \sqrt {-c}} + \frac {447 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{4} c + 1127 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{2} - 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{3} + 321 \, \sqrt {-a c x + c} a^{4} c^{4}}{192 \, a^{4} c^{4} x^{4}} \]

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^5,x, algorithm="giac")

[Out]

-4*sqrt(2)*a^4*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 363/64*a^4*c*arctan(sqrt(-a*c*x + c)
/sqrt(-c))/sqrt(-c) + 1/192*(447*(a*c*x - c)^3*sqrt(-a*c*x + c)*a^4*c + 1127*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^
4*c^2 - 1049*(-a*c*x + c)^(3/2)*a^4*c^3 + 321*sqrt(-a*c*x + c)*a^4*c^4)/(a^4*c^4*x^4)

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maple [A]  time = 0.04, size = 123, normalized size = 0.83 \[ -2 a^{4} c^{4} \left (-\frac {2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}-\frac {\frac {-\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}+\frac {1127 c \left (-a c x +c \right )^{\frac {5}{2}}}{384}-\frac {1049 \left (-a c x +c \right )^{\frac {3}{2}} c^{2}}{384}+\frac {107 \sqrt {-a c x +c}\, c^{3}}{128}}{x^{4} a^{4} c^{4}}-\frac {363 \arctanh \left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right ) \]

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^5,x)

[Out]

-2*a^4*c^4*(-2/c^(7/2)*2^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))-1/c^3*((-149/128*(-a*c*x+c)^(7/2)
+1127/384*c*(-a*c*x+c)^(5/2)-1049/384*(-a*c*x+c)^(3/2)*c^2+107/128*(-a*c*x+c)^(1/2)*c^3)/x^4/a^4/c^4-363/128/c
^(1/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2))))

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maxima [A]  time = 0.60, size = 212, normalized size = 1.43 \[ -\frac {1}{384} \, a^{4} c^{4} {\left (\frac {2 \, {\left (447 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 1127 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 321 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {768 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} - \frac {1089 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*a^4*c^4*(2*(447*(-a*c*x + c)^(7/2) - 1127*(-a*c*x + c)^(5/2)*c + 1049*(-a*c*x + c)^(3/2)*c^2 - 321*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) +
768*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt(-a*c*x + c)))/c^(7/2) - 1089*log
((sqrt(-a*c*x + c) - sqrt(c))/(sqrt(-a*c*x + c) + sqrt(c)))/c^(7/2))

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mupad [B]  time = 0.89, size = 122, normalized size = 0.82 \[ \frac {107\,\sqrt {c-a\,c\,x}}{64\,x^4}+\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,363{}\mathrm {i}}{64}-\frac {1049\,{\left (c-a\,c\,x\right )}^{3/2}}{192\,c\,x^4}+\frac {1127\,{\left (c-a\,c\,x\right )}^{5/2}}{192\,c^2\,x^4}-\frac {149\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}-\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(107*(c - a*c*x)^(1/2))/(64*x^4) + (a^4*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i)/c^(1/2))*363i)/64 - (1049*(c - a*c
*x)^(3/2))/(192*c*x^4) + (1127*(c - a*c*x)^(5/2))/(192*c^2*x^4) - (149*(c - a*c*x)^(7/2))/(64*c^3*x^4) - 2^(1/
2)*a^4*c^(1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2*c^(1/2)))*4i

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sympy [B]  time = 49.27, size = 991, normalized size = 6.70 \[ - \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 {198 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 {240 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 {90 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 {40 a^{4} 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 {15 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 {15 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 {24 a^{4} 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 {3 a^{4} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (- c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{2} - \frac {3 a^{4} c^{3} \sqrt {\frac {1}{c^{5}}} \log {\left (c^{3} \sqrt {\frac {1}{c^{5}}} + \sqrt {- a c x + c} \right )}}{2} - 2 a^{4} c^{2} \sqrt {\frac {1}{c^{3}}} \log {\left (- c^{2} \sqrt {\frac {1}{c^{3}}} + \sqrt {- a c x + c} \right )} + 2 a^{4} 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^{4} c \operatorname {atan}{\left (\frac {\sqrt {- a c x + c}}{\sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {4 \sqrt {2} a^{4} c \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + \frac {4 a^{3} \sqrt {- a c x + c}}{x} \]

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**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) -
198*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) + 240*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 - 90*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) - 40*a**4*c**4*sqrt(-
a*c*x + c)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) - 15*a**4*c**4*sqrt(c**(-7))*log(-c**4*sqrt(c**(-7)
) + sqrt(-a*c*x + c))/16 + 15*a**4*c**4*sqrt(c**(-7))*log(c**4*sqrt(c**(-7)) + sqrt(-a*c*x + c))/16 + 24*a**4*
c**3*(-a*c*x + c)**(3/2)/(16*a*c**4*x - 8*c**4 + 8*c**2*(-a*c*x + c)**2) + 3*a**4*c**3*sqrt(c**(-5))*log(-c**3
*sqrt(c**(-5)) + sqrt(-a*c*x + c))/2 - 3*a**4*c**3*sqrt(c**(-5))*log(c**3*sqrt(c**(-5)) + sqrt(-a*c*x + c))/2
- 2*a**4*c**2*sqrt(c**(-3))*log(-c**2*sqrt(c**(-3)) + sqrt(-a*c*x + c)) + 2*a**4*c**2*sqrt(c**(-3))*log(c**2*s
qrt(c**(-3)) + sqrt(-a*c*x + c)) + 8*a**4*c*atan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 4*sqrt(2)*a**4*c*atan(s
qrt(2)*sqrt(-a*c*x + c)/(2*sqrt(-c)))/sqrt(-c) + 4*a**3*sqrt(-a*c*x + c)/x

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