3.1.55 \(\int \frac {A+B x+C x^2}{x (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=138 \[ -\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1805, 823, 12, 266, 63, 208} \begin {gather*} \frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {-a C+A b+b B x}{7 a b \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b - a*C + b*B*x)/(7*a*b*(a + b*x^2)^(7/2)) + (7*A + 6*B*x)/(35*a^2*(a + b*x^2)^(5/2)) + (35*A + 24*B*x)/(10
5*a^3*(a + b*x^2)^(3/2)) + (35*A + 16*B*x)/(35*a^4*Sqrt[a + b*x^2]) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(
9/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 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 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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{x \left (a+b x^2\right )^{9/2}} \, dx &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {-7 A-6 B x}{x \left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {35 a A b+24 a b B x}{x \left (a+b x^2\right )^{5/2}} \, dx}{35 a^3 b}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-105 a^2 A b^2-48 a^2 b^2 B x}{x \left (a+b x^2\right )^{3/2}} \, dx}{105 a^5 b^2}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}+\frac {\int \frac {105 a^3 A b^3}{x \sqrt {a+b x^2}} \, dx}{105 a^7 b^3}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}+\frac {A \int \frac {1}{x \sqrt {a+b x^2}} \, dx}{a^4}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^4}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}+\frac {A \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a^4 b}\\ &=\frac {A b-a C+b B x}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {7 A+6 B x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 A+24 B x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 A+16 B x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 120, normalized size = 0.87 \begin {gather*} \frac {-15 a^4 C+a^3 b (176 A+105 B x)+14 a^2 b^2 x^2 (29 A+15 B x)+14 a b^3 x^4 (25 A+12 B x)+3 b^4 x^6 (35 A+16 B x)}{105 a^4 b \left (a+b x^2\right )^{7/2}}-\frac {A \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*a^4*C + 14*a*b^3*x^4*(25*A + 12*B*x) + 14*a^2*b^2*x^2*(29*A + 15*B*x) + 3*b^4*x^6*(35*A + 16*B*x) + a^3*b
*(176*A + 105*B*x))/(105*a^4*b*(a + b*x^2)^(7/2)) - (A*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(9/2)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.20, size = 146, normalized size = 1.06 \begin {gather*} \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {-15 a^4 C+176 a^3 A b+105 a^3 b B x+406 a^2 A b^2 x^2+210 a^2 b^2 B x^3+350 a A b^3 x^4+168 a b^3 B x^5+105 A b^4 x^6+48 b^4 B x^7}{105 a^4 b \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x]

[Out]

(176*a^3*A*b - 15*a^4*C + 105*a^3*b*B*x + 406*a^2*A*b^2*x^2 + 210*a^2*b^2*B*x^3 + 350*a*A*b^3*x^4 + 168*a*b^3*
B*x^5 + 105*A*b^4*x^6 + 48*b^4*B*x^7)/(105*a^4*b*(a + b*x^2)^(7/2)) + (2*A*ArcTanh[(Sqrt[b]*x)/Sqrt[a] - Sqrt[
a + b*x^2]/Sqrt[a]])/a^(9/2)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 465, normalized size = 3.37 \begin {gather*} \left [\frac {105 \, {\left (A b^{5} x^{8} + 4 \, A a b^{4} x^{6} + 6 \, A a^{2} b^{3} x^{4} + 4 \, A a^{3} b^{2} x^{2} + A a^{4} b\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (48 \, B a b^{4} x^{7} + 105 \, A a b^{4} x^{6} + 168 \, B a^{2} b^{3} x^{5} + 350 \, A a^{2} b^{3} x^{4} + 210 \, B a^{3} b^{2} x^{3} + 406 \, A a^{3} b^{2} x^{2} + 105 \, B a^{4} b x - 15 \, C a^{5} + 176 \, A a^{4} b\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{5} b^{5} x^{8} + 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} + 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}, \frac {105 \, {\left (A b^{5} x^{8} + 4 \, A a b^{4} x^{6} + 6 \, A a^{2} b^{3} x^{4} + 4 \, A a^{3} b^{2} x^{2} + A a^{4} b\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (48 \, B a b^{4} x^{7} + 105 \, A a b^{4} x^{6} + 168 \, B a^{2} b^{3} x^{5} + 350 \, A a^{2} b^{3} x^{4} + 210 \, B a^{3} b^{2} x^{3} + 406 \, A a^{3} b^{2} x^{2} + 105 \, B a^{4} b x - 15 \, C a^{5} + 176 \, A a^{4} b\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{5} x^{8} + 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} + 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(105*(A*b^5*x^8 + 4*A*a*b^4*x^6 + 6*A*a^2*b^3*x^4 + 4*A*a^3*b^2*x^2 + A*a^4*b)*sqrt(a)*log(-(b*x^2 - 2*
sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(48*B*a*b^4*x^7 + 105*A*a*b^4*x^6 + 168*B*a^2*b^3*x^5 + 350*A*a^2*b^3*
x^4 + 210*B*a^3*b^2*x^3 + 406*A*a^3*b^2*x^2 + 105*B*a^4*b*x - 15*C*a^5 + 176*A*a^4*b)*sqrt(b*x^2 + a))/(a^5*b^
5*x^8 + 4*a^6*b^4*x^6 + 6*a^7*b^3*x^4 + 4*a^8*b^2*x^2 + a^9*b), 1/105*(105*(A*b^5*x^8 + 4*A*a*b^4*x^6 + 6*A*a^
2*b^3*x^4 + 4*A*a^3*b^2*x^2 + A*a^4*b)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (48*B*a*b^4*x^7 + 105*A*a*b
^4*x^6 + 168*B*a^2*b^3*x^5 + 350*A*a^2*b^3*x^4 + 210*B*a^3*b^2*x^3 + 406*A*a^3*b^2*x^2 + 105*B*a^4*b*x - 15*C*
a^5 + 176*A*a^4*b)*sqrt(b*x^2 + a))/(a^5*b^5*x^8 + 4*a^6*b^4*x^6 + 6*a^7*b^3*x^4 + 4*a^8*b^2*x^2 + a^9*b)]

________________________________________________________________________________________

giac [A]  time = 0.57, size = 152, normalized size = 1.10 \begin {gather*} \frac {{\left ({\left ({\left ({\left (3 \, {\left ({\left (\frac {16 \, B b^{3} x}{a^{4}} + \frac {35 \, A b^{3}}{a^{4}}\right )} x + \frac {56 \, B b^{2}}{a^{3}}\right )} x + \frac {350 \, A b^{2}}{a^{3}}\right )} x + \frac {210 \, B b}{a^{2}}\right )} x + \frac {406 \, A b}{a^{2}}\right )} x + \frac {105 \, B}{a}\right )} x - \frac {15 \, C a^{14} b^{2} - 176 \, A a^{13} b^{3}}{a^{14} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, A \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*(((((3*((16*B*b^3*x/a^4 + 35*A*b^3/a^4)*x + 56*B*b^2/a^3)*x + 350*A*b^2/a^3)*x + 210*B*b/a^2)*x + 406*A*
b/a^2)*x + 105*B/a)*x - (15*C*a^14*b^2 - 176*A*a^13*b^3)/(a^14*b^3))/(b*x^2 + a)^(7/2) + 2*A*arctan(-(sqrt(b)*
x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a^4)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 169, normalized size = 1.22 \begin {gather*} \frac {B x}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a}+\frac {A}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a}+\frac {6 B x}{35 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2}}-\frac {C}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b}+\frac {A}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2}}+\frac {8 B x}{35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}+\frac {A}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}+\frac {16 B x}{35 \sqrt {b \,x^{2}+a}\, a^{4}}-\frac {A \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {9}{2}}}+\frac {A}{\sqrt {b \,x^{2}+a}\, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/x/(b*x^2+a)^(9/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.45, size = 157, normalized size = 1.14 \begin {gather*} \frac {16 \, B x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {A \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {A}{\sqrt {b x^{2} + a} a^{4}} + \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {A}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {C}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.62, size = 159, normalized size = 1.15 \begin {gather*} \frac {\frac {A}{7\,a}+\frac {A\,{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {A\,{\left (b\,x^2+a\right )}^3}{a^4}+\frac {A\,\left (b\,x^2+a\right )}{5\,a^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {C}{7\,b\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {A\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {16\,B\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,B\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,B\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {B\,x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/(x*(a + b*x^2)^(9/2)),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 107.41, size = 6613, normalized size = 47.92

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/x/(b*x**2+a)**(9/2),x)

[Out]

A*(352*a**32*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(
67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*
a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 10
5*a**32*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3
*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*
b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 210*a**32*lo
g(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*
b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59
/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 2924*a**
31*b*x**2*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/
2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**
(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 1050*
a**31*b*x**2*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)
*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(5
9/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 2100*a*
*31*b*x**2*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 252
00*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 +
25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**2
0) + 10852*a**30*b**2*x**4*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**
4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*
x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**
10*x**20) + 4725*a**30*b**2*x**4*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x*
*4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6
*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b*
*10*x**20) - 9450*a**30*b**2*x**4*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a*
*(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100
*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 2
10*a**(53/2)*b**10*x**20) + 23630*a**29*b**3*x**6*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 +
9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10
+ 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x*
*18 + 210*a**(53/2)*b**10*x**20) + 12600*a**29*b**3*x**6*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2
+ 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**1
0 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*
x**18 + 210*a**(53/2)*b**10*x**20) - 25200*a**29*b**3*x**6*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a
**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**
(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100
*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 33280*a**28*b**4*x**8*sqrt(1 + b*x**2/a)/(210*a**(73/2) +
2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52
920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16
+ 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 22050*a**28*b**4*x**8*log(b*x**2/a)/(210*a**(73/2)
+ 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 +
52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**
16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 44100*a**28*b**4*x**8*log(sqrt(1 + b*x**2/a) + 1
)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65
/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a*
*(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 31442*a**27*b**5*x**10*sqrt(1 +
b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 4410
0*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 +
9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 26460*a**27*b**5*x**10*l
og(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 4
4100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**1
4 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 52920*a**27*b**5*x**1
0*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67
/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a*
*(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 1992
4*a**26*b**6*x**12*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 2520
0*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 +
25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20
) + 22050*a**26*b**6*x**12*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 2
5200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12
+ 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x*
*20) - 44100*a**26*b**6*x**12*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69
/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**
(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a
**(53/2)*b**10*x**20) + 8162*a**25*b**7*x**14*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450
*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44
100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18
+ 210*a**(53/2)*b**10*x**20) + 12600*a**25*b**7*x**14*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9
450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 +
44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**
18 + 210*a**(53/2)*b**10*x**20) - 25200*a**25*b**7*x**14*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**
(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(6
3/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a
**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 1960*a**24*b**8*x**16*sqrt(1 + b*x**2/a)/(210*a**(73/2) + 2
100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 5292
0*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 +
2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 4725*a**24*b**8*x**16*log(b*x**2/a)/(210*a**(73/2) +
2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52
920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16
+ 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 9450*a**24*b**8*x**16*log(sqrt(1 + b*x**2/a) + 1)/
(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2
)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(
57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 210*a**23*b**9*x**18*sqrt(1 + b*x*
*2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*a*
*(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 945
0*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 1050*a**23*b**9*x**18*log(b*
x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b**3*x**6 + 44100*
a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2)*b**7*x**14 + 9
450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 2100*a**23*b**9*x**18*log(
sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*b*
*3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59/2
)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) + 105*a**22*
b**10*x**20*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 + 25200*a**(67/2)*
b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**12 + 25200*a**(59
/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x**20) - 210*a**2
2*b**10*x**20*log(sqrt(1 + b*x**2/a) + 1)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 9450*a**(69/2)*b**2*x**4 +
25200*a**(67/2)*b**3*x**6 + 44100*a**(65/2)*b**4*x**8 + 52920*a**(63/2)*b**5*x**10 + 44100*a**(61/2)*b**6*x**1
2 + 25200*a**(59/2)*b**7*x**14 + 9450*a**(57/2)*b**8*x**16 + 2100*a**(55/2)*b**9*x**18 + 210*a**(53/2)*b**10*x
**20)) + B*(35*a**14*x/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33
/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1
+ b*x**2/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 175
*a**13*b*x**3/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*
x**4*sqrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2
/a) + 210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 371*a**12*b*
*2*x**5/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*s
qrt(1 + b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) +
210*a**(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 429*a**11*b**3*x**
7/(35*a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1
+ b*x**2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a*
*(27/2)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 286*a**10*b**4*x**9/(35*
a**(37/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x*
*2/a) + 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2
)*b**5*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 104*a**9*b**5*x**11/(35*a**(37
/2)*sqrt(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a)
+ 700*a**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5
*x**10*sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 16*a**8*b**6*x**13/(35*a**(37/2)*sqr
t(1 + b*x**2/a) + 210*a**(35/2)*b*x**2*sqrt(1 + b*x**2/a) + 525*a**(33/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 700*a
**(31/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 525*a**(29/2)*b**4*x**8*sqrt(1 + b*x**2/a) + 210*a**(27/2)*b**5*x**10*
sqrt(1 + b*x**2/a) + 35*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a))) + C*Piecewise((-1/(7*a**3*b*sqrt(a + b*x**2)
+ 21*a**2*b**2*x**2*sqrt(a + b*x**2) + 21*a*b**3*x**4*sqrt(a + b*x**2) + 7*b**4*x**6*sqrt(a + b*x**2)), Ne(b,
0)), (x**2/(2*a**(9/2)), True))

________________________________________________________________________________________