Integrand size = 20, antiderivative size = 115 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=-\frac {5 B c^2 \sqrt {a+c x^2}}{16 x^2}-\frac {5 B c \left (a+c x^2\right )^{3/2}}{24 x^4}-\frac {B \left (a+c x^2\right )^{5/2}}{6 x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac {5 B c^3 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{16 \sqrt {a}} \]
-5/24*B*c*(c*x^2+a)^(3/2)/x^4-1/6*B*(c*x^2+a)^(5/2)/x^6-1/7*A*(c*x^2+a)^(7 /2)/a/x^7-5/16*B*c^3*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(1/2)-5/16*B*c^2*( c*x^2+a)^(1/2)/x^2
Time = 0.46 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=-\frac {\frac {\sqrt {a+c x^2} \left (48 A c^3 x^6+8 a^3 (6 A+7 B x)+3 a c^2 x^4 (48 A+77 B x)+2 a^2 c x^2 (72 A+91 B x)\right )}{x^7}+105 \sqrt {a} B c^3 \log (x)-105 \sqrt {a} B c^3 \log \left (-\sqrt {a}+\sqrt {a+c x^2}\right )}{336 a} \]
-1/336*((Sqrt[a + c*x^2]*(48*A*c^3*x^6 + 8*a^3*(6*A + 7*B*x) + 3*a*c^2*x^4 *(48*A + 77*B*x) + 2*a^2*c*x^2*(72*A + 91*B*x)))/x^7 + 105*Sqrt[a]*B*c^3*L og[x] - 105*Sqrt[a]*B*c^3*Log[-Sqrt[a] + Sqrt[a + c*x^2]])/a
Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {534, 243, 51, 51, 51, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2} (A+B x)}{x^8} \, dx\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle B \int \frac {\left (c x^2+a\right )^{5/2}}{x^7}dx-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{2} B \int \frac {\left (c x^2+a\right )^{5/2}}{x^8}dx^2-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} B \left (\frac {5}{6} c \int \frac {\left (c x^2+a\right )^{3/2}}{x^6}dx^2-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \int \frac {\sqrt {c x^2+a}}{x^4}dx^2-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {1}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{x^2 \sqrt {c x^2+a}}dx^2-\frac {\sqrt {a+c x^2}}{x^2}\right )-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\int \frac {1}{\frac {x^4}{c}-\frac {a}{c}}d\sqrt {c x^2+a}-\frac {\sqrt {a+c x^2}}{x^2}\right )-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{x^2}\right )-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {A \left (a+c x^2\right )^{7/2}}{7 a x^7}\) |
-1/7*(A*(a + c*x^2)^(7/2))/(a*x^7) + (B*(-1/3*(a + c*x^2)^(5/2)/x^6 + (5*c *(-1/2*(a + c*x^2)^(3/2)/x^4 + (3*c*(-(Sqrt[a + c*x^2]/x^2) - (c*ArcTanh[S qrt[a + c*x^2]/Sqrt[a]])/Sqrt[a]))/4))/6))/2
3.4.51.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (48 A \,c^{3} x^{6}+231 a B \,c^{2} x^{5}+144 a A \,c^{2} x^{4}+182 a^{2} B c \,x^{3}+144 a^{2} A c \,x^{2}+56 a^{3} B x +48 A \,a^{3}\right )}{336 x^{7} a}-\frac {5 B \,c^{3} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{16 \sqrt {a}}\) | \(114\) |
| default | \(B \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 c \left (\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )-\frac {A \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{7 a \,x^{7}}\) | \(160\) |
-1/336*(c*x^2+a)^(1/2)*(48*A*c^3*x^6+231*B*a*c^2*x^5+144*A*a*c^2*x^4+182*B *a^2*c*x^3+144*A*a^2*c*x^2+56*B*a^3*x+48*A*a^3)/x^7/a-5/16*B*c^3/a^(1/2)*l n((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)
Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=\left [\frac {105 \, B \sqrt {a} c^{3} x^{7} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{672 \, a x^{7}}, \frac {105 \, B \sqrt {-a} c^{3} x^{7} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (48 \, A c^{3} x^{6} + 231 \, B a c^{2} x^{5} + 144 \, A a c^{2} x^{4} + 182 \, B a^{2} c x^{3} + 144 \, A a^{2} c x^{2} + 56 \, B a^{3} x + 48 \, A a^{3}\right )} \sqrt {c x^{2} + a}}{336 \, a x^{7}}\right ] \]
[1/672*(105*B*sqrt(a)*c^3*x^7*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2* a)/x^2) - 2*(48*A*c^3*x^6 + 231*B*a*c^2*x^5 + 144*A*a*c^2*x^4 + 182*B*a^2* c*x^3 + 144*A*a^2*c*x^2 + 56*B*a^3*x + 48*A*a^3)*sqrt(c*x^2 + a))/(a*x^7), 1/336*(105*B*sqrt(-a)*c^3*x^7*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (48*A*c^ 3*x^6 + 231*B*a*c^2*x^5 + 144*A*a*c^2*x^4 + 182*B*a^2*c*x^3 + 144*A*a^2*c* x^2 + 56*B*a^3*x + 48*A*a^3)*sqrt(c*x^2 + a))/(a*x^7)]
Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (109) = 218\).
Time = 9.72 (sec) , antiderivative size = 605, normalized size of antiderivative = 5.26 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=- \frac {15 A a^{7} c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {33 A a^{6} c^{\frac {11}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {17 A a^{5} c^{\frac {13}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {3 A a^{4} c^{\frac {15}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {12 A a^{3} c^{\frac {17}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {8 A a^{2} c^{\frac {19}{2}} x^{10} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {2 A a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {7 A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 x^{2}} - \frac {A c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a} - \frac {B a^{3}}{6 \sqrt {c} x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {17 B a^{2} \sqrt {c}}{24 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {35 B a c^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {B c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{2 x} - \frac {3 B c^{\frac {5}{2}}}{16 x \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 B c^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{16 \sqrt {a}} \]
-15*A*a**7*c**(9/2)*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c* *5*x**8 + 105*a**3*c**6*x**10) - 33*A*a**6*c**(11/2)*x**2*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 17* A*a**5*c**(13/2)*x**4*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4* c**5*x**8 + 105*a**3*c**6*x**10) - 3*A*a**4*c**(15/2)*x**6*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 12 *A*a**3*c**(17/2)*x**8*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4 *c**5*x**8 + 105*a**3*c**6*x**10) - 8*A*a**2*c**(19/2)*x**10*sqrt(a/(c*x** 2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) - 2*A*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 7*A*c**(5/2)*sqrt(a/(c*x**2 ) + 1)/(15*x**2) - A*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a) - B*a**3/(6*sqrt (c)*x**7*sqrt(a/(c*x**2) + 1)) - 17*B*a**2*sqrt(c)/(24*x**5*sqrt(a/(c*x**2 ) + 1)) - 35*B*a*c**(3/2)/(48*x**3*sqrt(a/(c*x**2) + 1)) - B*c**(5/2)*sqrt (a/(c*x**2) + 1)/(2*x) - 3*B*c**(5/2)/(16*x*sqrt(a/(c*x**2) + 1)) - 5*B*c* *3*asinh(sqrt(a)/(sqrt(c)*x))/(16*sqrt(a))
Time = 0.22 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=-\frac {5 \, B c^{3} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{16 \, \sqrt {a}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{3}}{16 \, a^{3}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{3}}{48 \, a^{2}} + \frac {5 \, \sqrt {c x^{2} + a} B c^{3}}{16 \, a} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{2}}{16 \, a^{3} x^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{24 \, a^{2} x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{6 \, a x^{6}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{7 \, a x^{7}} \]
-5/16*B*c^3*arcsinh(a/(sqrt(a*c)*abs(x)))/sqrt(a) + 1/16*(c*x^2 + a)^(5/2) *B*c^3/a^3 + 5/48*(c*x^2 + a)^(3/2)*B*c^3/a^2 + 5/16*sqrt(c*x^2 + a)*B*c^3 /a - 1/16*(c*x^2 + a)^(7/2)*B*c^2/(a^3*x^2) - 1/24*(c*x^2 + a)^(7/2)*B*c/( a^2*x^4) - 1/6*(c*x^2 + a)^(7/2)*B/(a*x^6) - 1/7*(c*x^2 + a)^(7/2)*A/(a*x^ 7)
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (91) = 182\).
Time = 0.29 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.75 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=\frac {5 \, B c^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a}} + \frac {231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} B c^{3} + 336 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{12} A c^{\frac {7}{2}} - 196 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} B a c^{3} + 595 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B a^{2} c^{3} + 1680 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A a^{2} c^{\frac {7}{2}} - 595 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a^{4} c^{3} + 1008 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{4} c^{\frac {7}{2}} + 196 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{5} c^{3} - 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{6} c^{3} + 48 \, A a^{6} c^{\frac {7}{2}}}{168 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{7}} \]
5/8*B*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) + 1/168 *(231*(sqrt(c)*x - sqrt(c*x^2 + a))^13*B*c^3 + 336*(sqrt(c)*x - sqrt(c*x^2 + a))^12*A*c^(7/2) - 196*(sqrt(c)*x - sqrt(c*x^2 + a))^11*B*a*c^3 + 595*( sqrt(c)*x - sqrt(c*x^2 + a))^9*B*a^2*c^3 + 1680*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*a^2*c^(7/2) - 595*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*a^4*c^3 + 1008 *(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^4*c^(7/2) + 196*(sqrt(c)*x - sqrt(c*x ^2 + a))^3*B*a^5*c^3 - 231*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^6*c^3 + 48*A* a^6*c^(7/2))/((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^7
Time = 13.10 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.30 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx=\frac {5\,B\,a\,{\left (c\,x^2+a\right )}^{3/2}}{6\,x^6}-\frac {11\,B\,{\left (c\,x^2+a\right )}^{5/2}}{16\,x^6}-\frac {A\,a^2\,\sqrt {c\,x^2+a}}{7\,x^7}-\frac {5\,B\,a^2\,\sqrt {c\,x^2+a}}{16\,x^6}-\frac {3\,A\,c^2\,\sqrt {c\,x^2+a}}{7\,x^3}-\frac {A\,c^3\,\sqrt {c\,x^2+a}}{7\,a\,x}-\frac {3\,A\,a\,c\,\sqrt {c\,x^2+a}}{7\,x^5}+\frac {B\,c^3\,\mathrm {atan}\left (\frac {\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,\sqrt {a}} \]
(B*c^3*atan(((a + c*x^2)^(1/2)*1i)/a^(1/2))*5i)/(16*a^(1/2)) - (11*B*(a + c*x^2)^(5/2))/(16*x^6) + (5*B*a*(a + c*x^2)^(3/2))/(6*x^6) - (A*a^2*(a + c *x^2)^(1/2))/(7*x^7) - (5*B*a^2*(a + c*x^2)^(1/2))/(16*x^6) - (3*A*c^2*(a + c*x^2)^(1/2))/(7*x^3) - (A*c^3*(a + c*x^2)^(1/2))/(7*a*x) - (3*A*a*c*(a + c*x^2)^(1/2))/(7*x^5)