3.1.56 \(\int \frac {A+B x+C x^2}{x^2 (a+b x^2)^{9/2}} \, dx\)
Optimal. Leaf size=188 \[ -\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{35 a^4 \sqrt {a+b x^2}}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.38, antiderivative size = 188, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used =
{1805, 807, 266, 63, 208} \begin {gather*} \frac {35 B-x \left (\frac {93 A b}{a}-16 C\right )}{35 a^4 \sqrt {a+b x^2}}+\frac {35 B-3 x \left (\frac {29 A b}{a}-8 C\right )}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {7 B-x \left (\frac {13 A b}{a}-6 C\right )}{35 a^2 \left (a+b x^2\right )^{5/2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {B-x \left (\frac {A b}{a}-C\right )}{7 a \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(A + B*x + C*x^2)/(x^2*(a + b*x^2)^(9/2)),x]
[Out]
(B - ((A*b)/a - C)*x)/(7*a*(a + b*x^2)^(7/2)) + (7*B - ((13*A*b)/a - 6*C)*x)/(35*a^2*(a + b*x^2)^(5/2)) + (35*
B - 3*((29*A*b)/a - 8*C)*x)/(105*a^3*(a + b*x^2)^(3/2)) + (35*B - ((93*A*b)/a - 16*C)*x)/(35*a^4*Sqrt[a + b*x^
2]) - (A*Sqrt[a + b*x^2])/(a^5*x) - (B*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/a^(9/2)
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 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 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^2 \left (a+b x^2\right )^{9/2}} \, dx &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {-7 A-7 B x+6 \left (\frac {A b}{a}-C\right ) x^2}{x^2 \left (a+b x^2\right )^{7/2}} \, dx}{7 a}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {\int \frac {35 A+35 B x-4 \left (\frac {13 A b}{a}-6 C\right ) x^2}{x^2 \left (a+b x^2\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {-105 A-105 B x+6 \left (\frac {29 A b}{a}-8 C\right ) x^2}{x^2 \left (a+b x^2\right )^{3/2}} \, dx}{105 a^3}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}+\frac {\int \frac {105 A+105 B x}{x^2 \sqrt {a+b x^2}} \, dx}{105 a^4}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {B \int \frac {1}{x \sqrt {a+b x^2}} \, dx}{a^4}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {B \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a^4}\\ &=\frac {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}+\frac {B \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 {B-\left (\frac {A b}{a}-C\right ) x}{7 a \left (a+b x^2\right )^{7/2}}+\frac {7 B-\left (\frac {13 A b}{a}-6 C\right ) x}{35 a^2 \left (a+b x^2\right )^{5/2}}+\frac {35 B-3 \left (\frac {29 A b}{a}-8 C\right ) x}{105 a^3 \left (a+b x^2\right )^{3/2}}+\frac {35 B-\left (\frac {93 A b}{a}-16 C\right ) x}{35 a^4 \sqrt {a+b x^2}}-\frac {A \sqrt {a+b x^2}}{a^5 x}-\frac {B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 158, normalized size = 0.84 \begin {gather*} \frac {a^4 (x (176 B+105 C x)-105 A)+14 a^3 b x^2 (x (29 B+15 C x)-60 A)+14 a^2 b^2 x^4 (x (25 B+12 C x)-120 A)+3 a b^3 x^6 (x (35 B+16 C x)-448 A)-105 \sqrt {a} B x \left (a+b x^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )-384 A b^4 x^8}{105 a^5 x \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x + C*x^2)/(x^2*(a + b*x^2)^(9/2)),x]
[Out]
(-384*A*b^4*x^8 + 14*a^2*b^2*x^4*(-120*A + x*(25*B + 12*C*x)) + 14*a^3*b*x^2*(-60*A + x*(29*B + 15*C*x)) + 3*a
*b^3*x^6*(-448*A + x*(35*B + 16*C*x)) + a^4*(-105*A + x*(176*B + 105*C*x)) - 105*Sqrt[a]*B*x*(a + b*x^2)^(7/2)
*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(105*a^5*x*(a + b*x^2)^(7/2))
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.15, size = 190, normalized size = 1.01 \begin {gather*} \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {-105 a^4 A+176 a^4 B x+105 a^4 C x^2-840 a^3 A b x^2+406 a^3 b B x^3+210 a^3 b C x^4-1680 a^2 A b^2 x^4+350 a^2 b^2 B x^5+168 a^2 b^2 C x^6-1344 a A b^3 x^6+105 a b^3 B x^7+48 a b^3 C x^8-384 A b^4 x^8}{105 a^5 x \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(A + B*x + C*x^2)/(x^2*(a + b*x^2)^(9/2)),x]
[Out]
(-105*a^4*A + 176*a^4*B*x - 840*a^3*A*b*x^2 + 105*a^4*C*x^2 + 406*a^3*b*B*x^3 - 1680*a^2*A*b^2*x^4 + 210*a^3*b
*C*x^4 + 350*a^2*b^2*B*x^5 - 1344*a*A*b^3*x^6 + 168*a^2*b^2*C*x^6 + 105*a*b^3*B*x^7 - 384*A*b^4*x^8 + 48*a*b^3
*C*x^8)/(105*a^5*x*(a + b*x^2)^(7/2)) + (2*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a] - Sqrt[a + b*x^2]/Sqrt[a]])/a^(9/2)
________________________________________________________________________________________
fricas [A] time = 0.71, size = 525, normalized size = 2.79 \begin {gather*} \left [\frac {105 \, {\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt {a} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \, {\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \, {\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \, {\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}, \frac {105 \, {\left (B b^{4} x^{9} + 4 \, B a b^{3} x^{7} + 6 \, B a^{2} b^{2} x^{5} + 4 \, B a^{3} b x^{3} + B a^{4} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, B a b^{3} x^{7} + 350 \, B a^{2} b^{2} x^{5} + 48 \, {\left (C a b^{3} - 8 \, A b^{4}\right )} x^{8} + 406 \, B a^{3} b x^{3} + 168 \, {\left (C a^{2} b^{2} - 8 \, A a b^{3}\right )} x^{6} + 176 \, B a^{4} x - 105 \, A a^{4} + 210 \, {\left (C a^{3} b - 8 \, A a^{2} b^{2}\right )} x^{4} + 105 \, {\left (C a^{4} - 8 \, A a^{3} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((C*x^2+B*x+A)/x^2/(b*x^2+a)^(9/2),x, algorithm="fricas")
[Out]
[1/210*(105*(B*b^4*x^9 + 4*B*a*b^3*x^7 + 6*B*a^2*b^2*x^5 + 4*B*a^3*b*x^3 + B*a^4*x)*sqrt(a)*log(-(b*x^2 - 2*sq
rt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(105*B*a*b^3*x^7 + 350*B*a^2*b^2*x^5 + 48*(C*a*b^3 - 8*A*b^4)*x^8 + 406*
B*a^3*b*x^3 + 168*(C*a^2*b^2 - 8*A*a*b^3)*x^6 + 176*B*a^4*x - 105*A*a^4 + 210*(C*a^3*b - 8*A*a^2*b^2)*x^4 + 10
5*(C*a^4 - 8*A*a^3*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b^4*x^9 + 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 + 4*a^8*b*x^3 + a^9*x
), 1/105*(105*(B*b^4*x^9 + 4*B*a*b^3*x^7 + 6*B*a^2*b^2*x^5 + 4*B*a^3*b*x^3 + B*a^4*x)*sqrt(-a)*arctan(sqrt(-a)
/sqrt(b*x^2 + a)) + (105*B*a*b^3*x^7 + 350*B*a^2*b^2*x^5 + 48*(C*a*b^3 - 8*A*b^4)*x^8 + 406*B*a^3*b*x^3 + 168*
(C*a^2*b^2 - 8*A*a*b^3)*x^6 + 176*B*a^4*x - 105*A*a^4 + 210*(C*a^3*b - 8*A*a^2*b^2)*x^4 + 105*(C*a^4 - 8*A*a^3
*b)*x^2)*sqrt(b*x^2 + a))/(a^5*b^4*x^9 + 4*a^6*b^3*x^7 + 6*a^7*b^2*x^5 + 4*a^8*b*x^3 + a^9*x)]
________________________________________________________________________________________
giac [A] time = 0.51, size = 239, normalized size = 1.27 \begin {gather*} \frac {{\left ({\left ({\left ({\left (3 \, {\left (x {\left (\frac {35 \, B b^{3}}{a^{4}} + \frac {{\left (16 \, C a^{20} b^{6} - 93 \, A a^{19} b^{7}\right )} x}{a^{24} b^{3}}\right )} + \frac {28 \, {\left (2 \, C a^{21} b^{5} - 11 \, A a^{20} b^{6}\right )}}{a^{24} b^{3}}\right )} x + \frac {350 \, B b^{2}}{a^{3}}\right )} x + \frac {210 \, {\left (C a^{22} b^{4} - 5 \, A a^{21} b^{5}\right )}}{a^{24} b^{3}}\right )} x + \frac {406 \, B b}{a^{2}}\right )} x + \frac {105 \, {\left (C a^{23} b^{3} - 4 \, A a^{22} b^{4}\right )}}{a^{24} b^{3}}\right )} x + \frac {176 \, B}{a}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, B \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {2 \, A \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((C*x^2+B*x+A)/x^2/(b*x^2+a)^(9/2),x, algorithm="giac")
[Out]
1/105*(((((3*(x*(35*B*b^3/a^4 + (16*C*a^20*b^6 - 93*A*a^19*b^7)*x/(a^24*b^3)) + 28*(2*C*a^21*b^5 - 11*A*a^20*b
^6)/(a^24*b^3))*x + 350*B*b^2/a^3)*x + 210*(C*a^22*b^4 - 5*A*a^21*b^5)/(a^24*b^3))*x + 406*B*b/a^2)*x + 105*(C
*a^23*b^3 - 4*A*a^22*b^4)/(a^24*b^3))*x + 176*B/a)/(b*x^2 + a)^(7/2) + 2*B*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a
))/sqrt(-a))/(sqrt(-a)*a^4) + 2*A*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^4)
________________________________________________________________________________________
maple [A] time = 0.01, size = 240, normalized size = 1.28 \begin {gather*} -\frac {8 A b x}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2}}+\frac {C x}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a}-\frac {48 A b x}{35 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{3}}+\frac {B}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a}+\frac {6 C x}{35 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2}}-\frac {A}{\left (b \,x^{2}+a \right )^{\frac {7}{2}} a x}-\frac {64 A b x}{35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{4}}+\frac {B}{5 \left (b \,x^{2}+a \right )^{\frac {5}{2}} a^{2}}+\frac {8 C x}{35 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}-\frac {128 A b x}{35 \sqrt {b \,x^{2}+a}\, a^{5}}+\frac {B}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{3}}+\frac {16 C x}{35 \sqrt {b \,x^{2}+a}\, a^{4}}-\frac {B \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{a^{\frac {9}{2}}}+\frac {B}{\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^2/(b*x^2+a)^(9/2),x)
[Out]
1/7*C*x/a/(b*x^2+a)^(7/2)+6/35*C/a^2*x/(b*x^2+a)^(5/2)+8/35*C/a^3*x/(b*x^2+a)^(3/2)+16/35*C/a^4*x/(b*x^2+a)^(1
/2)-A/a/x/(b*x^2+a)^(7/2)-8/7*A/a^2*b*x/(b*x^2+a)^(7/2)-48/35*A/a^3*b*x/(b*x^2+a)^(5/2)-64/35*A/a^4*b*x/(b*x^2
+a)^(3/2)-128/35*A/a^5*b*x/(b*x^2+a)^(1/2)+1/7*B/a/(b*x^2+a)^(7/2)+1/5*B/a^2/(b*x^2+a)^(5/2)+1/3*B/a^3/(b*x^2+
a)^(3/2)+B/a^4/(b*x^2+a)^(1/2)-B/a^(9/2)*ln((2*a+2*(b*x^2+a)^(1/2)*a^(1/2))/x)
________________________________________________________________________________________
maxima [A] time = 1.41, size = 228, normalized size = 1.21 \begin {gather*} \frac {16 \, C x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {C x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {128 \, A b x}{35 \, \sqrt {b x^{2} + a} a^{5}} - \frac {64 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4}} - \frac {48 \, A b x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3}} - \frac {8 \, A b x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2}} - \frac {B \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{a^{\frac {9}{2}}} + \frac {B}{\sqrt {b x^{2} + a} a^{4}} + \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {B}{5 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} - \frac {A}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((C*x^2+B*x+A)/x^2/(b*x^2+a)^(9/2),x, algorithm="maxima")
[Out]
16/35*C*x/(sqrt(b*x^2 + a)*a^4) + 8/35*C*x/((b*x^2 + a)^(3/2)*a^3) + 6/35*C*x/((b*x^2 + a)^(5/2)*a^2) + 1/7*C*
x/((b*x^2 + a)^(7/2)*a) - 128/35*A*b*x/(sqrt(b*x^2 + a)*a^5) - 64/35*A*b*x/((b*x^2 + a)^(3/2)*a^4) - 48/35*A*b
*x/((b*x^2 + a)^(5/2)*a^3) - 8/7*A*b*x/((b*x^2 + a)^(7/2)*a^2) - B*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(9/2) + B/(
sqrt(b*x^2 + a)*a^4) + 1/3*B/((b*x^2 + a)^(3/2)*a^3) + 1/5*B/((b*x^2 + a)^(5/2)*a^2) + 1/7*B/((b*x^2 + a)^(7/2
)*a) - A/((b*x^2 + a)^(7/2)*a*x)
________________________________________________________________________________________
mupad [B] time = 2.10, size = 225, normalized size = 1.20 \begin {gather*} \frac {\frac {B}{7\,a}+\frac {B\,{\left (b\,x^2+a\right )}^2}{3\,a^3}+\frac {B\,{\left (b\,x^2+a\right )}^3}{a^4}+\frac {B\,\left (b\,x^2+a\right )}{5\,a^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {A}{a^4}+\frac {128\,A\,b\,x^2}{35\,a^5}}{x\,\sqrt {b\,x^2+a}}-\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{a^{9/2}}+\frac {16\,C\,x}{35\,a^4\,\sqrt {b\,x^2+a}}+\frac {8\,C\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{3/2}}+\frac {6\,C\,x}{35\,a^2\,{\left (b\,x^2+a\right )}^{5/2}}+\frac {C\,x}{7\,a\,{\left (b\,x^2+a\right )}^{7/2}}-\frac {29\,A\,b\,x}{35\,a^4\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {13\,A\,b\,x}{35\,a^3\,{\left (b\,x^2+a\right )}^{5/2}}-\frac {A\,b\,x}{7\,a^2\,{\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^2*(a + b*x^2)^(9/2)),x)
[Out]
(B/(7*a) + (B*(a + b*x^2)^2)/(3*a^3) + (B*(a + b*x^2)^3)/a^4 + (B*(a + b*x^2))/(5*a^2))/(a + b*x^2)^(7/2) - (A
/a^4 + (128*A*b*x^2)/(35*a^5))/(x*(a + b*x^2)^(1/2)) - (B*atanh((a + b*x^2)^(1/2)/a^(1/2)))/a^(9/2) + (16*C*x)
/(35*a^4*(a + b*x^2)^(1/2)) + (8*C*x)/(35*a^3*(a + b*x^2)^(3/2)) + (6*C*x)/(35*a^2*(a + b*x^2)^(5/2)) + (C*x)/
(7*a*(a + b*x^2)^(7/2)) - (29*A*b*x)/(35*a^4*(a + b*x^2)^(3/2)) - (13*A*b*x)/(35*a^3*(a + b*x^2)^(5/2)) - (A*b
*x)/(7*a^2*(a + b*x^2)^(7/2))
________________________________________________________________________________________
sympy [B] time = 165.70, size = 6922, normalized size = 36.82
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((C*x**2+B*x+A)/x**2/(b*x**2+a)**(9/2),x)
[Out]
A*(-35*a**4*b**(33/2)*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a*
*6*b**19*x**6 + 35*a**5*b**20*x**8) - 280*a**3*b**(35/2)*x**2*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b
**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 560*a**2*b**(37/2)*x**4*sqrt(a/(
b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*
x**8) - 448*a*b**(39/2)*x**6*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8*b**17*x**2 + 210*a**7*b**18*x**4 +
140*a**6*b**19*x**6 + 35*a**5*b**20*x**8) - 128*b**(41/2)*x**8*sqrt(a/(b*x**2) + 1)/(35*a**9*b**16 + 140*a**8
*b**17*x**2 + 210*a**7*b**18*x**4 + 140*a**6*b**19*x**6 + 35*a**5*b**20*x**8)) + B*(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**(6
5/2)*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**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*log(sqrt(1 + b*x**2/a) + 1)/(21
0*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)/(2
10*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) - 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 + 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) + 10852*a**30*b**2*x**4*sq
rt(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 + 210*a**(53/2)*b**10*x**20) + 2
3630*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 + 25
200*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**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) - 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 + 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) + 22050*a**28*b**4*x**8*log(b*x**2/a)/(210*a**(73/2) + 2100*a**(71/2)*b*x**2 + 94
50*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**1
8 + 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**(7
1/2)*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) + 21
00*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) + 26460*a**27*b**5*x**10*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) - 52920*a**27*b**5*x**10*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) + 19924*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 + 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) + 22050*a**26*b**6*x**12*lo
g(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 + 44
100*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 + 44100*a**(61/2)*b**6*x**12 + 25
200*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 + 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) - 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**(63/2)*b**5*x**10 + 44100*a**(6
1/2)*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) + 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 + 4410
0*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 + 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) - 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**(5
5/2)*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 + 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**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 + 9450*a**(57/2)*b**8*x**16 + 21
00*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)/(21
0*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**22*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 + 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)) + C*(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)*s
qrt(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**1
0*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*sqrt(1 + b*x**2/a) + 700*a**(3
1/2)*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*sqr
t(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) + 3
5*a**(25/2)*b**6*x**12*sqrt(1 + b*x**2/a)) + 16*a**8*b**6*x**13/(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**(2
5/2)*b**6*x**12*sqrt(1 + b*x**2/a)))
________________________________________________________________________________________