Integrand size = 32, antiderivative size = 195 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=-\frac {c \sqrt {a+b x^2}}{8 a x^8}+\frac {(7 b c-8 a d) \sqrt {a+b x^2}}{48 a^2 x^6}-\frac {\left (35 b^2 c-40 a b d+48 a^2 e\right ) \sqrt {a+b x^2}}{192 a^3 x^4}+\frac {\left (35 b^3 c-40 a b^2 d+48 a^2 b e-64 a^3 f\right ) \sqrt {a+b x^2}}{128 a^4 x^2}-\frac {b \left (35 b^3 c-40 a b^2 d+48 a^2 b e-64 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{128 a^{9/2}} \]
-1/128*b*(-64*a^3*f+48*a^2*b*e-40*a*b^2*d+35*b^3*c)*arctanh((b*x^2+a)^(1/2 )/a^(1/2))/a^(9/2)-1/8*c*(b*x^2+a)^(1/2)/a/x^8+1/48*(-8*a*d+7*b*c)*(b*x^2+ a)^(1/2)/a^2/x^6-1/192*(48*a^2*e-40*a*b*d+35*b^2*c)*(b*x^2+a)^(1/2)/a^3/x^ 4+1/128*(-64*a^3*f+48*a^2*b*e-40*a*b^2*d+35*b^3*c)*(b*x^2+a)^(1/2)/a^4/x^2
Time = 0.37 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.82 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+b x^2} \left (105 b^3 c x^6-10 a b^2 x^4 \left (7 c+12 d x^2\right )+8 a^2 b x^2 \left (7 c+10 d x^2+18 e x^4\right )-16 a^3 \left (3 c+4 d x^2+6 e x^4+12 f x^6\right )\right )}{x^8}-3 b \left (35 b^3 c-40 a b^2 d+48 a^2 b e-64 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{384 a^{9/2}} \]
((Sqrt[a]*Sqrt[a + b*x^2]*(105*b^3*c*x^6 - 10*a*b^2*x^4*(7*c + 12*d*x^2) + 8*a^2*b*x^2*(7*c + 10*d*x^2 + 18*e*x^4) - 16*a^3*(3*c + 4*d*x^2 + 6*e*x^4 + 12*f*x^6)))/x^8 - 3*b*(35*b^3*c - 40*a*b^2*d + 48*a^2*b*e - 64*a^3*f)*A rcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(384*a^(9/2))
Time = 0.50 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2331, 2124, 27, 1192, 1471, 25, 298, 215, 219}
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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{2} \int \frac {f x^6+e x^4+d x^2+c}{x^{10} \sqrt {b x^2+a}}dx^2\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-8 a f x^4-8 a e x^2+7 b c-8 a d}{2 x^8 \sqrt {b x^2+a}}dx^2}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-8 a f x^4-8 a e x^2+7 b c-8 a d}{x^8 \sqrt {b x^2+a}}dx^2}{8 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {-8 a f x^8-8 a (b e-2 a f) x^4+7 b^3 c-8 a b^2 d+8 a^2 b e-8 a^3 f}{\left (a-x^4\right )^4}d\sqrt {b x^2+a}}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {b^2 \sqrt {a+b x^2} (7 b c-8 a d)}{6 a \left (a-x^4\right )^3}-\frac {\int -\frac {48 a^2 f x^4+35 b^3 c-40 a b^2 d+48 a^2 b e-48 a^3 f}{\left (a-x^4\right )^3}d\sqrt {b x^2+a}}{6 a}\right )}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\int \frac {48 a^2 f x^4+35 b^3 c-40 a b^2 d+48 a^2 b e-48 a^3 f}{\left (a-x^4\right )^3}d\sqrt {b x^2+a}}{6 a}+\frac {b^2 \sqrt {a+b x^2} (7 b c-8 a d)}{6 a \left (a-x^4\right )^3}\right )}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3 \left (-64 a^3 f+48 a^2 b e-40 a b^2 d+35 b^3 c\right ) \int \frac {1}{\left (a-x^4\right )^2}d\sqrt {b x^2+a}}{4 a}+\frac {b \sqrt {a+b x^2} \left (48 a^2 e-40 a b d+35 b^2 c\right )}{4 a \left (a-x^4\right )^2}}{6 a}+\frac {b^2 \sqrt {a+b x^2} (7 b c-8 a d)}{6 a \left (a-x^4\right )^3}\right )}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3 \left (-64 a^3 f+48 a^2 b e-40 a b^2 d+35 b^3 c\right ) \left (\frac {\int \frac {1}{a-x^4}d\sqrt {b x^2+a}}{2 a}+\frac {\sqrt {a+b x^2}}{2 a \left (a-x^4\right )}\right )}{4 a}+\frac {b \sqrt {a+b x^2} \left (48 a^2 e-40 a b d+35 b^2 c\right )}{4 a \left (a-x^4\right )^2}}{6 a}+\frac {b^2 \sqrt {a+b x^2} (7 b c-8 a d)}{6 a \left (a-x^4\right )^3}\right )}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {b \sqrt {a+b x^2} \left (48 a^2 e-40 a b d+35 b^2 c\right )}{4 a \left (a-x^4\right )^2}+\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2}}+\frac {\sqrt {a+b x^2}}{2 a \left (a-x^4\right )}\right ) \left (-64 a^3 f+48 a^2 b e-40 a b^2 d+35 b^3 c\right )}{4 a}}{6 a}+\frac {b^2 \sqrt {a+b x^2} (7 b c-8 a d)}{6 a \left (a-x^4\right )^3}\right )}{4 a}-\frac {c \sqrt {a+b x^2}}{4 a x^8}\right )\) |
(-1/4*(c*Sqrt[a + b*x^2])/(a*x^8) - (b*((b^2*(7*b*c - 8*a*d)*Sqrt[a + b*x^ 2])/(6*a*(a - x^4)^3) + ((b*(35*b^2*c - 40*a*b*d + 48*a^2*e)*Sqrt[a + b*x^ 2])/(4*a*(a - x^4)^2) + (3*(35*b^3*c - 40*a*b^2*d + 48*a^2*b*e - 64*a^3*f) *(Sqrt[a + b*x^2]/(2*a*(a - x^4)) + ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]]/(2*a^ (3/2))))/(4*a))/(6*a)))/(4*a))/2
3.2.50.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : > With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px , a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - a*d))), x] + Simp[1/((m + 1)*(b*c - a*d)) Int[(a + b*x)^(m + 1)*(c + d*x )^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] || ! ILtQ[n, -1])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Time = 3.57 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.76
| method | result | size |
| pseudoelliptic | \(\frac {b \,x^{8} \left (f \,a^{3}-\frac {3}{4} a^{2} b e +\frac {5}{8} a \,b^{2} d -\frac {35}{64} b^{3} c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )+\frac {35 \left (\left (-\frac {64}{35} f \,x^{6}-\frac {32}{35} e \,x^{4}-\frac {64}{105} d \,x^{2}-\frac {16}{35} c \right ) a^{\frac {7}{2}}+b \,x^{2} \left (\left (\frac {48}{35} e \,x^{4}+\frac {16}{21} d \,x^{2}+\frac {8}{15} c \right ) a^{\frac {5}{2}}+\left (\left (-\frac {8 d \,x^{2}}{7}-\frac {2 c}{3}\right ) a^{\frac {3}{2}}+b c \,x^{2} \sqrt {a}\right ) b \,x^{2}\right )\right ) \sqrt {b \,x^{2}+a}}{64}}{2 a^{\frac {9}{2}} x^{8}}\) | \(148\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (192 a^{3} f \,x^{6}-144 a^{2} b e \,x^{6}+120 a \,b^{2} d \,x^{6}-105 b^{3} c \,x^{6}+96 a^{3} e \,x^{4}-80 a^{2} b d \,x^{4}+70 a \,b^{2} c \,x^{4}+64 a^{3} d \,x^{2}-56 a^{2} b c \,x^{2}+48 c \,a^{3}\right )}{384 a^{4} x^{8}}+\frac {\left (64 f \,a^{3}-48 a^{2} b e +40 a \,b^{2} d -35 b^{3} c \right ) b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{128 a^{\frac {9}{2}}}\) | \(168\) |
| default | \(d \left (-\frac {\sqrt {b \,x^{2}+a}}{6 a \,x^{6}}-\frac {5 b \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )+c \left (-\frac {\sqrt {b \,x^{2}+a}}{8 a \,x^{8}}-\frac {7 b \left (-\frac {\sqrt {b \,x^{2}+a}}{6 a \,x^{6}}-\frac {5 b \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+f \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )+e \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\) | \(342\) |
1/2*(b*x^8*(f*a^3-3/4*a^2*b*e+5/8*a*b^2*d-35/64*b^3*c)*arctanh((b*x^2+a)^( 1/2)/a^(1/2))+35/64*((-64/35*f*x^6-32/35*e*x^4-64/105*d*x^2-16/35*c)*a^(7/ 2)+b*x^2*((48/35*e*x^4+16/21*d*x^2+8/15*c)*a^(5/2)+((-8/7*d*x^2-2/3*c)*a^( 3/2)+b*c*x^2*a^(1/2))*b*x^2))*(b*x^2+a)^(1/2))/a^(9/2)/x^8
Time = 0.35 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.75 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=\left [-\frac {3 \, {\left (35 \, b^{4} c - 40 \, a b^{3} d + 48 \, a^{2} b^{2} e - 64 \, a^{3} b f\right )} \sqrt {a} x^{8} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (3 \, {\left (35 \, a b^{3} c - 40 \, a^{2} b^{2} d + 48 \, a^{3} b e - 64 \, a^{4} f\right )} x^{6} - 48 \, a^{4} c - 2 \, {\left (35 \, a^{2} b^{2} c - 40 \, a^{3} b d + 48 \, a^{4} e\right )} x^{4} + 8 \, {\left (7 \, a^{3} b c - 8 \, a^{4} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{768 \, a^{5} x^{8}}, \frac {3 \, {\left (35 \, b^{4} c - 40 \, a b^{3} d + 48 \, a^{2} b^{2} e - 64 \, a^{3} b f\right )} \sqrt {-a} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (35 \, a b^{3} c - 40 \, a^{2} b^{2} d + 48 \, a^{3} b e - 64 \, a^{4} f\right )} x^{6} - 48 \, a^{4} c - 2 \, {\left (35 \, a^{2} b^{2} c - 40 \, a^{3} b d + 48 \, a^{4} e\right )} x^{4} + 8 \, {\left (7 \, a^{3} b c - 8 \, a^{4} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{384 \, a^{5} x^{8}}\right ] \]
[-1/768*(3*(35*b^4*c - 40*a*b^3*d + 48*a^2*b^2*e - 64*a^3*b*f)*sqrt(a)*x^8 *log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(3*(35*a*b^3*c - 40*a^2*b^2*d + 48*a^3*b*e - 64*a^4*f)*x^6 - 48*a^4*c - 2*(35*a^2*b^2*c - 4 0*a^3*b*d + 48*a^4*e)*x^4 + 8*(7*a^3*b*c - 8*a^4*d)*x^2)*sqrt(b*x^2 + a))/ (a^5*x^8), 1/384*(3*(35*b^4*c - 40*a*b^3*d + 48*a^2*b^2*e - 64*a^3*b*f)*sq rt(-a)*x^8*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (3*(35*a*b^3*c - 40*a^2*b^2* d + 48*a^3*b*e - 64*a^4*f)*x^6 - 48*a^4*c - 2*(35*a^2*b^2*c - 40*a^3*b*d + 48*a^4*e)*x^4 + 8*(7*a^3*b*c - 8*a^4*d)*x^2)*sqrt(b*x^2 + a))/(a^5*x^8)]
Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (196) = 392\).
Time = 103.42 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.28 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=- \frac {c}{8 \sqrt {b} x^{9} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {d}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {e}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} c}{48 a x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} d}{24 a x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} e}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {\sqrt {b} f \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} - \frac {7 b^{\frac {3}{2}} c}{192 a^{2} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 b^{\frac {3}{2}} d}{48 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {3}{2}} e}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{\frac {5}{2}} c}{384 a^{3} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 b^{\frac {5}{2}} d}{16 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {35 b^{\frac {7}{2}} c}{128 a^{4} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {b f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {3 b^{2} e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} + \frac {5 b^{3} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {7}{2}}} - \frac {35 b^{4} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{128 a^{\frac {9}{2}}} \]
-c/(8*sqrt(b)*x**9*sqrt(a/(b*x**2) + 1)) - d/(6*sqrt(b)*x**7*sqrt(a/(b*x** 2) + 1)) - e/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) + sqrt(b)*c/(48*a*x**7* sqrt(a/(b*x**2) + 1)) + sqrt(b)*d/(24*a*x**5*sqrt(a/(b*x**2) + 1)) + sqrt( b)*e/(8*a*x**3*sqrt(a/(b*x**2) + 1)) - sqrt(b)*f*sqrt(a/(b*x**2) + 1)/(2*a *x) - 7*b**(3/2)*c/(192*a**2*x**5*sqrt(a/(b*x**2) + 1)) - 5*b**(3/2)*d/(48 *a**2*x**3*sqrt(a/(b*x**2) + 1)) + 3*b**(3/2)*e/(8*a**2*x*sqrt(a/(b*x**2) + 1)) + 35*b**(5/2)*c/(384*a**3*x**3*sqrt(a/(b*x**2) + 1)) - 5*b**(5/2)*d/ (16*a**3*x*sqrt(a/(b*x**2) + 1)) + 35*b**(7/2)*c/(128*a**4*x*sqrt(a/(b*x** 2) + 1)) + b*f*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/2)) - 3*b**2*e*asinh(sq rt(a)/(sqrt(b)*x))/(8*a**(5/2)) + 5*b**3*d*asinh(sqrt(a)/(sqrt(b)*x))/(16* a**(7/2)) - 35*b**4*c*asinh(sqrt(a)/(sqrt(b)*x))/(128*a**(9/2))
Time = 0.19 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.41 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=-\frac {35 \, b^{4} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{128 \, a^{\frac {9}{2}}} + \frac {5 \, b^{3} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {3 \, b^{2} e \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {b f \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} + \frac {35 \, \sqrt {b x^{2} + a} b^{3} c}{128 \, a^{4} x^{2}} - \frac {5 \, \sqrt {b x^{2} + a} b^{2} d}{16 \, a^{3} x^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b e}{8 \, a^{2} x^{2}} - \frac {\sqrt {b x^{2} + a} f}{2 \, a x^{2}} - \frac {35 \, \sqrt {b x^{2} + a} b^{2} c}{192 \, a^{3} x^{4}} + \frac {5 \, \sqrt {b x^{2} + a} b d}{24 \, a^{2} x^{4}} - \frac {\sqrt {b x^{2} + a} e}{4 \, a x^{4}} + \frac {7 \, \sqrt {b x^{2} + a} b c}{48 \, a^{2} x^{6}} - \frac {\sqrt {b x^{2} + a} d}{6 \, a x^{6}} - \frac {\sqrt {b x^{2} + a} c}{8 \, a x^{8}} \]
-35/128*b^4*c*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(9/2) + 5/16*b^3*d*arcsinh(a /(sqrt(a*b)*abs(x)))/a^(7/2) - 3/8*b^2*e*arcsinh(a/(sqrt(a*b)*abs(x)))/a^( 5/2) + 1/2*b*f*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) + 35/128*sqrt(b*x^2 + a)*b^3*c/(a^4*x^2) - 5/16*sqrt(b*x^2 + a)*b^2*d/(a^3*x^2) + 3/8*sqrt(b*x^ 2 + a)*b*e/(a^2*x^2) - 1/2*sqrt(b*x^2 + a)*f/(a*x^2) - 35/192*sqrt(b*x^2 + a)*b^2*c/(a^3*x^4) + 5/24*sqrt(b*x^2 + a)*b*d/(a^2*x^4) - 1/4*sqrt(b*x^2 + a)*e/(a*x^4) + 7/48*sqrt(b*x^2 + a)*b*c/(a^2*x^6) - 1/6*sqrt(b*x^2 + a)* d/(a*x^6) - 1/8*sqrt(b*x^2 + a)*c/(a*x^8)
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (171) = 342\).
Time = 0.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.83 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=\frac {\frac {3 \, {\left (35 \, b^{5} c - 40 \, a b^{4} d + 48 \, a^{2} b^{3} e - 64 \, a^{3} b^{2} f\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5} c - 385 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b^{5} c + 511 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b^{5} c - 279 \, \sqrt {b x^{2} + a} a^{3} b^{5} c - 120 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a b^{4} d + 440 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b^{4} d - 584 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{4} d + 264 \, \sqrt {b x^{2} + a} a^{4} b^{4} d + 144 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} b^{3} e - 528 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{3} b^{3} e + 624 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{4} b^{3} e - 240 \, \sqrt {b x^{2} + a} a^{5} b^{3} e - 192 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} b^{2} f + 576 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4} b^{2} f - 576 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5} b^{2} f + 192 \, \sqrt {b x^{2} + a} a^{6} b^{2} f}{a^{4} b^{4} x^{8}}}{384 \, b} \]
1/384*(3*(35*b^5*c - 40*a*b^4*d + 48*a^2*b^3*e - 64*a^3*b^2*f)*arctan(sqrt (b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^4) + (105*(b*x^2 + a)^(7/2)*b^5*c - 385* (b*x^2 + a)^(5/2)*a*b^5*c + 511*(b*x^2 + a)^(3/2)*a^2*b^5*c - 279*sqrt(b*x ^2 + a)*a^3*b^5*c - 120*(b*x^2 + a)^(7/2)*a*b^4*d + 440*(b*x^2 + a)^(5/2)* a^2*b^4*d - 584*(b*x^2 + a)^(3/2)*a^3*b^4*d + 264*sqrt(b*x^2 + a)*a^4*b^4* d + 144*(b*x^2 + a)^(7/2)*a^2*b^3*e - 528*(b*x^2 + a)^(5/2)*a^3*b^3*e + 62 4*(b*x^2 + a)^(3/2)*a^4*b^3*e - 240*sqrt(b*x^2 + a)*a^5*b^3*e - 192*(b*x^2 + a)^(7/2)*a^3*b^2*f + 576*(b*x^2 + a)^(5/2)*a^4*b^2*f - 576*(b*x^2 + a)^ (3/2)*a^5*b^2*f + 192*sqrt(b*x^2 + a)*a^6*b^2*f)/(a^4*b^4*x^8))/b
Time = 7.85 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.42 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^9 \sqrt {a+b x^2}} \, dx=\frac {511\,c\,{\left (b\,x^2+a\right )}^{3/2}}{384\,a^2\,x^8}-\frac {93\,c\,\sqrt {b\,x^2+a}}{128\,a\,x^8}-\frac {385\,c\,{\left (b\,x^2+a\right )}^{5/2}}{384\,a^3\,x^8}+\frac {35\,c\,{\left (b\,x^2+a\right )}^{7/2}}{128\,a^4\,x^8}-\frac {11\,d\,\sqrt {b\,x^2+a}}{16\,a\,x^6}+\frac {5\,d\,{\left (b\,x^2+a\right )}^{3/2}}{6\,a^2\,x^6}-\frac {5\,d\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a^3\,x^6}-\frac {5\,e\,\sqrt {b\,x^2+a}}{8\,a\,x^4}+\frac {3\,e\,{\left (b\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {f\,\sqrt {b\,x^2+a}}{2\,a\,x^2}+\frac {b\,f\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {3\,b^2\,e\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}+\frac {b^4\,c\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,35{}\mathrm {i}}{128\,a^{9/2}}-\frac {b^3\,d\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,a^{7/2}} \]
(511*c*(a + b*x^2)^(3/2))/(384*a^2*x^8) - (93*c*(a + b*x^2)^(1/2))/(128*a* x^8) - (385*c*(a + b*x^2)^(5/2))/(384*a^3*x^8) + (35*c*(a + b*x^2)^(7/2))/ (128*a^4*x^8) - (11*d*(a + b*x^2)^(1/2))/(16*a*x^6) + (5*d*(a + b*x^2)^(3/ 2))/(6*a^2*x^6) - (5*d*(a + b*x^2)^(5/2))/(16*a^3*x^6) - (5*e*(a + b*x^2)^ (1/2))/(8*a*x^4) + (3*e*(a + b*x^2)^(3/2))/(8*a^2*x^4) - (f*(a + b*x^2)^(1 /2))/(2*a*x^2) + (b*f*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(2*a^(3/2)) + (b^4 *c*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*35i)/(128*a^(9/2)) - (b^3*d*atan(( (a + b*x^2)^(1/2)*1i)/a^(1/2))*5i)/(16*a^(7/2)) - (3*b^2*e*atanh((a + b*x^ 2)^(1/2)/a^(1/2)))/(8*a^(5/2))