3.583 \(\int \frac {A+B x^2}{x^8 (a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=148 \[ \frac {16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {A}{7 a x^7 \sqrt {a+b x^2}} \]

[Out]

-1/7*A/a/x^7/(b*x^2+a)^(1/2)+1/35*(8*A*b-7*B*a)/a^2/x^5/(b*x^2+a)^(1/2)-2/35*b*(8*A*b-7*B*a)/a^3/x^3/(b*x^2+a)
^(1/2)+8/35*b^2*(8*A*b-7*B*a)/a^4/x/(b*x^2+a)^(1/2)+16/35*b^3*(8*A*b-7*B*a)*x/a^5/(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 191} \[ \frac {16 b^3 x (8 A b-7 a B)}{35 a^5 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {A}{7 a x^7 \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x]

[Out]

-A/(7*a*x^7*Sqrt[a + b*x^2]) + (8*A*b - 7*a*B)/(35*a^2*x^5*Sqrt[a + b*x^2]) - (2*b*(8*A*b - 7*a*B))/(35*a^3*x^
3*Sqrt[a + b*x^2]) + (8*b^2*(8*A*b - 7*a*B))/(35*a^4*x*Sqrt[a + b*x^2]) + (16*b^3*(8*A*b - 7*a*B)*x)/(35*a^5*S
qrt[a + b*x^2])

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {A+B x^2}{x^8 \left (a+b x^2\right )^{3/2}} \, dx &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}-\frac {(8 A b-7 a B) \int \frac {1}{x^6 \left (a+b x^2\right )^{3/2}} \, dx}{7 a}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}+\frac {(6 b (8 A b-7 a B)) \int \frac {1}{x^4 \left (a+b x^2\right )^{3/2}} \, dx}{35 a^2}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}-\frac {\left (8 b^2 (8 A b-7 a B)\right ) \int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{35 a^3}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}+\frac {\left (16 b^3 (8 A b-7 a B)\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a^4}\\ &=-\frac {A}{7 a x^7 \sqrt {a+b x^2}}+\frac {8 A b-7 a B}{35 a^2 x^5 \sqrt {a+b x^2}}-\frac {2 b (8 A b-7 a B)}{35 a^3 x^3 \sqrt {a+b x^2}}+\frac {8 b^2 (8 A b-7 a B)}{35 a^4 x \sqrt {a+b x^2}}+\frac {16 b^3 (8 A b-7 a B) x}{35 a^5 \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 71, normalized size = 0.48 \[ \frac {x^2 \left (a^3-2 a^2 b x^2+8 a b^2 x^4+16 b^3 x^6\right ) (8 A b-7 a B)-5 a^4 A}{35 a^5 x^7 \sqrt {a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x]

[Out]

(-5*a^4*A + (8*A*b - 7*a*B)*x^2*(a^3 - 2*a^2*b*x^2 + 8*a*b^2*x^4 + 16*b^3*x^6))/(35*a^5*x^7*Sqrt[a + b*x^2])

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fricas [A]  time = 0.59, size = 117, normalized size = 0.79 \[ -\frac {{\left (16 \, {\left (7 \, B a b^{3} - 8 \, A b^{4}\right )} x^{8} + 8 \, {\left (7 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 5 \, A a^{4} - 2 \, {\left (7 \, B a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + {\left (7 \, B a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{35 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

-1/35*(16*(7*B*a*b^3 - 8*A*b^4)*x^8 + 8*(7*B*a^2*b^2 - 8*A*a*b^3)*x^6 + 5*A*a^4 - 2*(7*B*a^3*b - 8*A*a^2*b^2)*
x^4 + (7*B*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a)/(a^5*b*x^9 + a^6*x^7)

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giac [B]  time = 0.54, size = 407, normalized size = 2.75 \[ -\frac {{\left (B a b^{3} - A b^{4}\right )} x}{\sqrt {b x^{2} + a} a^{5}} + \frac {2 \, {\left (35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} B a b^{\frac {5}{2}} - 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} A b^{\frac {7}{2}} - 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} B a^{2} b^{\frac {5}{2}} + 280 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} A a b^{\frac {7}{2}} + 1015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} B a^{3} b^{\frac {5}{2}} - 1015 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} A a^{2} b^{\frac {7}{2}} - 1680 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} B a^{4} b^{\frac {5}{2}} + 2240 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} A a^{3} b^{\frac {7}{2}} + 1337 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a^{5} b^{\frac {5}{2}} - 1673 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a^{4} b^{\frac {7}{2}} - 504 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{6} b^{\frac {5}{2}} + 616 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{5} b^{\frac {7}{2}} + 77 \, B a^{7} b^{\frac {5}{2}} - 93 \, A a^{6} b^{\frac {7}{2}}\right )}}{35 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{7} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

-(B*a*b^3 - A*b^4)*x/(sqrt(b*x^2 + a)*a^5) + 2/35*(35*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a*b^(5/2) - 35*(sqrt(
b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) - 280*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2) + 280*(sqrt(b)*x - s
qrt(b*x^2 + a))^10*A*a*b^(7/2) + 1015*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^3*b^(5/2) - 1015*(sqrt(b)*x - sqrt(b
*x^2 + a))^8*A*a^2*b^(7/2) - 1680*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 2240*(sqrt(b)*x - sqrt(b*x^2
+ a))^6*A*a^3*b^(7/2) + 1337*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^5*b^(5/2) - 1673*(sqrt(b)*x - sqrt(b*x^2 + a
))^4*A*a^4*b^(7/2) - 504*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^6*b^(5/2) + 616*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A
*a^5*b^(7/2) + 77*B*a^7*b^(5/2) - 93*A*a^6*b^(7/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7*a^4)

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maple [A]  time = 0.01, size = 107, normalized size = 0.72 \[ -\frac {-128 A \,b^{4} x^{8}+112 B a \,b^{3} x^{8}-64 A a \,b^{3} x^{6}+56 B \,a^{2} b^{2} x^{6}+16 A \,a^{2} b^{2} x^{4}-14 B \,a^{3} b \,x^{4}-8 A \,a^{3} b \,x^{2}+7 B \,a^{4} x^{2}+5 A \,a^{4}}{35 \sqrt {b \,x^{2}+a}\, a^{5} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x)

[Out]

-1/35*(-128*A*b^4*x^8+112*B*a*b^3*x^8-64*A*a*b^3*x^6+56*B*a^2*b^2*x^6+16*A*a^2*b^2*x^4-14*B*a^3*b*x^4-8*A*a^3*
b*x^2+7*B*a^4*x^2+5*A*a^4)/(b*x^2+a)^(1/2)/x^7/a^5

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maxima [A]  time = 1.06, size = 176, normalized size = 1.19 \[ -\frac {16 \, B b^{3} x}{5 \, \sqrt {b x^{2} + a} a^{4}} + \frac {128 \, A b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {8 \, B b^{2}}{5 \, \sqrt {b x^{2} + a} a^{3} x} + \frac {64 \, A b^{3}}{35 \, \sqrt {b x^{2} + a} a^{4} x} + \frac {2 \, B b}{5 \, \sqrt {b x^{2} + a} a^{2} x^{3}} - \frac {16 \, A b^{2}}{35 \, \sqrt {b x^{2} + a} a^{3} x^{3}} - \frac {B}{5 \, \sqrt {b x^{2} + a} a x^{5}} + \frac {8 \, A b}{35 \, \sqrt {b x^{2} + a} a^{2} x^{5}} - \frac {A}{7 \, \sqrt {b x^{2} + a} a x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^8/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

-16/5*B*b^3*x/(sqrt(b*x^2 + a)*a^4) + 128/35*A*b^4*x/(sqrt(b*x^2 + a)*a^5) - 8/5*B*b^2/(sqrt(b*x^2 + a)*a^3*x)
+ 64/35*A*b^3/(sqrt(b*x^2 + a)*a^4*x) + 2/5*B*b/(sqrt(b*x^2 + a)*a^2*x^3) - 16/35*A*b^2/(sqrt(b*x^2 + a)*a^3*
x^3) - 1/5*B/(sqrt(b*x^2 + a)*a*x^5) + 8/35*A*b/(sqrt(b*x^2 + a)*a^2*x^5) - 1/7*A/(sqrt(b*x^2 + a)*a*x^7)

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mupad [B]  time = 1.07, size = 148, normalized size = 1.00 \[ -\frac {x^2\,\left (\frac {58\,A\,b^4-42\,B\,a\,b^3}{35\,a^5}-\frac {2\,b^3\,\left (93\,A\,b-77\,B\,a\right )}{35\,a^5}\right )-\frac {b^2\,\left (93\,A\,b-77\,B\,a\right )}{35\,a^4}}{x\,\sqrt {b\,x^2+a}}-\frac {\left (7\,B\,a^2-13\,A\,a\,b\right )\,\sqrt {b\,x^2+a}}{35\,a^4\,x^5}-\frac {A\,\sqrt {b\,x^2+a}}{7\,a^2\,x^7}-\frac {b\,\sqrt {b\,x^2+a}\,\left (29\,A\,b-21\,B\,a\right )}{35\,a^4\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^8*(a + b*x^2)^(3/2)),x)

[Out]

- (x^2*((58*A*b^4 - 42*B*a*b^3)/(35*a^5) - (2*b^3*(93*A*b - 77*B*a))/(35*a^5)) - (b^2*(93*A*b - 77*B*a))/(35*a
^4))/(x*(a + b*x^2)^(1/2)) - ((7*B*a^2 - 13*A*a*b)*(a + b*x^2)^(1/2))/(35*a^4*x^5) - (A*(a + b*x^2)^(1/2))/(7*
a^2*x^7) - (b*(a + b*x^2)^(1/2)*(29*A*b - 21*B*a))/(35*a^4*x^3)

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sympy [B]  time = 19.08, size = 1030, normalized size = 6.96 \[ A \left (- \frac {5 a^{7} b^{\frac {33}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} - \frac {7 a^{6} b^{\frac {35}{2}} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} - \frac {7 a^{5} b^{\frac {37}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {35 a^{4} b^{\frac {39}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {280 a^{3} b^{\frac {41}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {560 a^{2} b^{\frac {43}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {448 a b^{\frac {45}{2}} x^{12} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}} + \frac {128 b^{\frac {47}{2}} x^{14} \sqrt {\frac {a}{b x^{2}} + 1}}{35 a^{9} b^{16} x^{6} + 140 a^{8} b^{17} x^{8} + 210 a^{7} b^{18} x^{10} + 140 a^{6} b^{19} x^{12} + 35 a^{5} b^{20} x^{14}}\right ) + B \left (- \frac {a^{5} b^{\frac {19}{2}} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {5 a^{3} b^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {30 a^{2} b^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {40 a b^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}} - \frac {16 b^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{b x^{2}} + 1}}{5 a^{7} b^{9} x^{4} + 15 a^{6} b^{10} x^{6} + 15 a^{5} b^{11} x^{8} + 5 a^{4} b^{12} x^{10}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)/x**8/(b*x**2+a)**(3/2),x)

[Out]

A*(-5*a**7*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 1
40*a**6*b**19*x**12 + 35*a**5*b**20*x**14) - 7*a**6*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 +
140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) - 7*a**5*b**(37/2)*x*
*4*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**1
2 + 35*a**5*b**20*x**14) + 35*a**4*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x*
*8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 280*a**3*b**(41/2)*x**8*sqrt(a/(b*x*
*2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**
20*x**14) + 560*a**2*b**(43/2)*x**10*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7
*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 448*a*b**(45/2)*x**12*sqrt(a/(b*x**2) + 1)/(35*a*
*9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a**6*b**19*x**12 + 35*a**5*b**20*x**14) + 128
*b**(47/2)*x**14*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16*x**6 + 140*a**8*b**17*x**8 + 210*a**7*b**18*x**10 + 140*a
**6*b**19*x**12 + 35*a**5*b**20*x**14)) + B*(-a**5*b**(19/2)*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*
b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 5*a**3*b**(23/2)*x**4*sqrt(a/(b*x**2) + 1)/(5*a**7*b**
9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 30*a**2*b**(25/2)*x**6*sqrt(a/(b*x**2
) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 40*a*b**(27/2)*x**8
*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12*x**10) - 16*b
**(29/2)*x**10*sqrt(a/(b*x**2) + 1)/(5*a**7*b**9*x**4 + 15*a**6*b**10*x**6 + 15*a**5*b**11*x**8 + 5*a**4*b**12
*x**10))

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