3.352 \(\int \frac{(A+B x) (a+c x^2)^{5/2}}{x^9} \, dx\)

Optimal. Leaf size=149 \[ \frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]

[Out]

(5*A*c^3*Sqrt[a + c*x^2])/(128*a*x^2) + (5*A*c^2*(a + c*x^2)^(3/2))/(192*a*x^4) + (A*c*(a + c*x^2)^(5/2))/(48*
a*x^6) - (A*(a + c*x^2)^(7/2))/(8*a*x^8) - (B*(a + c*x^2)^(7/2))/(7*a*x^7) + (5*A*c^4*ArcTanh[Sqrt[a + c*x^2]/
Sqrt[a]])/(128*a^(3/2))

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Rubi [A]  time = 0.0989955, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {835, 807, 266, 47, 63, 208} \[ \frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}+\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^(5/2))/x^9,x]

[Out]

(5*A*c^3*Sqrt[a + c*x^2])/(128*a*x^2) + (5*A*c^2*(a + c*x^2)^(3/2))/(192*a*x^4) + (A*c*(a + c*x^2)^(5/2))/(48*
a*x^6) - (A*(a + c*x^2)^(7/2))/(8*a*x^8) - (B*(a + c*x^2)^(7/2))/(7*a*x^7) + (5*A*c^4*ArcTanh[Sqrt[a + c*x^2]/
Sqrt[a]])/(128*a^(3/2))

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 47

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] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 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]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{x^9} \, dx &=-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{\int \frac{(-8 a B+A c x) \left (a+c x^2\right )^{5/2}}{x^8} \, dx}{8 a}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{(A c) \int \frac{\left (a+c x^2\right )^{5/2}}{x^7} \, dx}{8 a}\\ &=-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{(A c) \operatorname{Subst}\left (\int \frac{(a+c x)^{5/2}}{x^4} \, dx,x,x^2\right )}{16 a}\\ &=\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^2\right ) \operatorname{Subst}\left (\int \frac{(a+c x)^{3/2}}{x^3} \, dx,x,x^2\right )}{96 a}\\ &=\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )}{128 a}\\ &=\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{256 a}\\ &=\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}-\frac{\left (5 A c^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{128 a}\\ &=\frac{5 A c^3 \sqrt{a+c x^2}}{128 a x^2}+\frac{5 A c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac{A c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac{A \left (a+c x^2\right )^{7/2}}{8 a x^8}-\frac{B \left (a+c x^2\right )^{7/2}}{7 a x^7}+\frac{5 A c^4 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{128 a^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0189238, size = 53, normalized size = 0.36 \[ -\frac{\left (a+c x^2\right )^{7/2} \left (a^4 B+A c^4 x^7 \, _2F_1\left (\frac{7}{2},5;\frac{9}{2};\frac{c x^2}{a}+1\right )\right )}{7 a^5 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^(5/2))/x^9,x]

[Out]

-((a + c*x^2)^(7/2)*(a^4*B + A*c^4*x^7*Hypergeometric2F1[7/2, 5, 9/2, 1 + (c*x^2)/a]))/(7*a^5*x^7)

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Maple [A]  time = 0.019, size = 185, normalized size = 1.2 \begin{align*} -{\frac{B}{7\,a{x}^{7}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A}{8\,a{x}^{8}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{Ac}{48\,{a}^{2}{x}^{6}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{2}}{192\,{a}^{3}{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{c}^{3}}{128\,{a}^{4}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{A{c}^{4}}{128\,{a}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,A{c}^{4}}{384\,{a}^{3}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{c}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{5\,A{c}^{4}}{128\,{a}^{2}}\sqrt{c{x}^{2}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^(5/2)/x^9,x)

[Out]

-1/7*B*(c*x^2+a)^(7/2)/a/x^7-1/8*A*(c*x^2+a)^(7/2)/a/x^8+1/48*A/a^2*c/x^6*(c*x^2+a)^(7/2)+1/192*A/a^3*c^2/x^4*
(c*x^2+a)^(7/2)+1/128*A/a^4*c^3/x^2*(c*x^2+a)^(7/2)-1/128*A/a^4*c^4*(c*x^2+a)^(5/2)-5/384*A/a^3*c^4*(c*x^2+a)^
(3/2)+5/128*A/a^(3/2)*c^4*ln((2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)-5/128*A/a^2*c^4*(c*x^2+a)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0236, size = 666, normalized size = 4.47 \begin{align*} \left [\frac{105 \, A \sqrt{a} c^{4} x^{8} \log \left (-\frac{c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) - 2 \,{\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt{c x^{2} + a}}{5376 \, a^{2} x^{8}}, -\frac{105 \, A \sqrt{-a} c^{4} x^{8} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (384 \, B a c^{3} x^{7} + 105 \, A a c^{3} x^{6} + 1152 \, B a^{2} c^{2} x^{5} + 826 \, A a^{2} c^{2} x^{4} + 1152 \, B a^{3} c x^{3} + 952 \, A a^{3} c x^{2} + 384 \, B a^{4} x + 336 \, A a^{4}\right )} \sqrt{c x^{2} + a}}{2688 \, a^{2} x^{8}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^9,x, algorithm="fricas")

[Out]

[1/5376*(105*A*sqrt(a)*c^4*x^8*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(384*B*a*c^3*x^7 + 105*
A*a*c^3*x^6 + 1152*B*a^2*c^2*x^5 + 826*A*a^2*c^2*x^4 + 1152*B*a^3*c*x^3 + 952*A*a^3*c*x^2 + 384*B*a^4*x + 336*
A*a^4)*sqrt(c*x^2 + a))/(a^2*x^8), -1/2688*(105*A*sqrt(-a)*c^4*x^8*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + (384*B*a
*c^3*x^7 + 105*A*a*c^3*x^6 + 1152*B*a^2*c^2*x^5 + 826*A*a^2*c^2*x^4 + 1152*B*a^3*c*x^3 + 952*A*a^3*c*x^2 + 384
*B*a^4*x + 336*A*a^4)*sqrt(c*x^2 + a))/(a^2*x^8)]

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Sympy [B]  time = 27.5989, size = 609, normalized size = 4.09 \begin{align*} - \frac{A a^{3}}{8 \sqrt{c} x^{9} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{23 A a^{2} \sqrt{c}}{48 x^{7} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{127 A a c^{\frac{3}{2}}}{192 x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{133 A c^{\frac{5}{2}}}{384 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{5 A c^{\frac{7}{2}}}{128 a x \sqrt{\frac{a}{c x^{2}} + 1}} + \frac{5 A c^{4} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{128 a^{\frac{3}{2}}} - \frac{15 B 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 B 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 B 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 B 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 B 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 B 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 B a c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{7 B c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 x^{2}} - \frac{B c^{\frac{7}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{15 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**(5/2)/x**9,x)

[Out]

-A*a**3/(8*sqrt(c)*x**9*sqrt(a/(c*x**2) + 1)) - 23*A*a**2*sqrt(c)/(48*x**7*sqrt(a/(c*x**2) + 1)) - 127*A*a*c**
(3/2)/(192*x**5*sqrt(a/(c*x**2) + 1)) - 133*A*c**(5/2)/(384*x**3*sqrt(a/(c*x**2) + 1)) - 5*A*c**(7/2)/(128*a*x
*sqrt(a/(c*x**2) + 1)) + 5*A*c**4*asinh(sqrt(a)/(sqrt(c)*x))/(128*a**(3/2)) - 15*B*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*B*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*B*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*B*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*B*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*B*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*B*a*c**(3/2)*sqrt(a/(c*x*
*2) + 1)/(5*x**4) - 7*B*c**(5/2)*sqrt(a/(c*x**2) + 1)/(15*x**2) - B*c**(7/2)*sqrt(a/(c*x**2) + 1)/(15*a)

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Giac [B]  time = 1.25284, size = 663, normalized size = 4.45 \begin{align*} -\frac{5 \, A c^{4} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{64 \, \sqrt{-a} a} + \frac{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{15} A c^{4} + 2688 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{14} B a c^{\frac{7}{2}} + 2779 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{13} A a c^{4} - 2688 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{12} B a^{2} c^{\frac{7}{2}} + 6265 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{11} A a^{2} c^{4} + 13440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{10} B a^{3} c^{\frac{7}{2}} + 12355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} A a^{3} c^{4} - 13440 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} B a^{4} c^{\frac{7}{2}} + 12355 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} A a^{4} c^{4} + 8064 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{6} B a^{5} c^{\frac{7}{2}} + 6265 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{5} A a^{5} c^{4} - 8064 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} B a^{6} c^{\frac{7}{2}} + 2779 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} A a^{6} c^{4} + 384 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} B a^{7} c^{\frac{7}{2}} + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} A a^{7} c^{4} - 384 \, B a^{8} c^{\frac{7}{2}}}{1344 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{8} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^(5/2)/x^9,x, algorithm="giac")

[Out]

-5/64*A*c^4*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/1344*(105*(sqrt(c)*x - sqrt(c*x^2
+ a))^15*A*c^4 + 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^14*B*a*c^(7/2) + 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^13*A*
a*c^4 - 2688*(sqrt(c)*x - sqrt(c*x^2 + a))^12*B*a^2*c^(7/2) + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^11*A*a^2*c^4
+ 13440*(sqrt(c)*x - sqrt(c*x^2 + a))^10*B*a^3*c^(7/2) + 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^9*A*a^3*c^4 - 134
40*(sqrt(c)*x - sqrt(c*x^2 + a))^8*B*a^4*c^(7/2) + 12355*(sqrt(c)*x - sqrt(c*x^2 + a))^7*A*a^4*c^4 + 8064*(sqr
t(c)*x - sqrt(c*x^2 + a))^6*B*a^5*c^(7/2) + 6265*(sqrt(c)*x - sqrt(c*x^2 + a))^5*A*a^5*c^4 - 8064*(sqrt(c)*x -
sqrt(c*x^2 + a))^4*B*a^6*c^(7/2) + 2779*(sqrt(c)*x - sqrt(c*x^2 + a))^3*A*a^6*c^4 + 384*(sqrt(c)*x - sqrt(c*x
^2 + a))^2*B*a^7*c^(7/2) + 105*(sqrt(c)*x - sqrt(c*x^2 + a))*A*a^7*c^4 - 384*B*a^8*c^(7/2))/(((sqrt(c)*x - sqr
t(c*x^2 + a))^2 - a)^8*a)