Integrand size = 32, antiderivative size = 146 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=-\frac {c \sqrt {a+b x^2}}{6 a x^6}+\frac {(5 b c-6 a d) \sqrt {a+b x^2}}{24 a^2 x^4}-\frac {\left (5 b^2 c-6 a b d+8 a^2 e\right ) \sqrt {a+b x^2}}{16 a^3 x^2}+\frac {\left (5 b^3 c-6 a b^2 d+8 a^2 b e-16 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{7/2}} \]
1/16*(-16*a^3*f+8*a^2*b*e-6*a*b^2*d+5*b^3*c)*arctanh((b*x^2+a)^(1/2)/a^(1/ 2))/a^(7/2)-1/6*c*(b*x^2+a)^(1/2)/a/x^6+1/24*(-6*a*d+5*b*c)*(b*x^2+a)^(1/2 )/a^2/x^4-1/16*(8*a^2*e-6*a*b*d+5*b^2*c)*(b*x^2+a)^(1/2)/a^3/x^2
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a^2 c+10 a b c x^2-12 a^2 d x^2-15 b^2 c x^4+18 a b d x^4-24 a^2 e x^4\right )}{48 a^3 x^6}+\frac {\left (5 b^3 c-6 a b^2 d+8 a^2 b e-16 a^3 f\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 a^{7/2}} \]
(Sqrt[a + b*x^2]*(-8*a^2*c + 10*a*b*c*x^2 - 12*a^2*d*x^2 - 15*b^2*c*x^4 + 18*a*b*d*x^4 - 24*a^2*e*x^4))/(48*a^3*x^6) + ((5*b^3*c - 6*a*b^2*d + 8*a^2 *b*e - 16*a^3*f)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*a^(7/2))
Time = 0.46 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.25, 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, 25, 1471, 27, 298, 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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{2} \int \frac {f x^6+e x^4+d x^2+c}{x^8 \sqrt {b x^2+a}}dx^2\) |
| \(\Big \downarrow \) 2124 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-6 a f x^4-6 a e x^2+5 b c-6 a d}{2 x^6 \sqrt {b x^2+a}}dx^2}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {-6 a f x^4-6 a e x^2+5 b c-6 a d}{x^6 \sqrt {b x^2+a}}dx^2}{6 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int -\frac {-6 a f x^8-6 a (b e-2 a f) x^4+5 b^3 c-6 a b^2 d+6 a^2 b e-6 a^3 f}{\left (a-x^4\right )^3}d\sqrt {b x^2+a}}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {-6 a f x^8-6 a (b e-2 a f) x^4+5 b^3 c-6 a b^2 d+6 a^2 b e-6 a^3 f}{\left (a-x^4\right )^3}d\sqrt {b x^2+a}}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\int -\frac {3 \left (8 a^2 f x^4+5 b^3 c-6 a b^2 d+8 a^2 b e-8 a^3 f\right )}{\left (a-x^4\right )^2}d\sqrt {b x^2+a}}{4 a}-\frac {b^2 \sqrt {a+b x^2} (5 b c-6 a d)}{4 a \left (a-x^4\right )^2}}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 \int \frac {8 a^2 f x^4+5 b^3 c-6 a b^2 d+8 a^2 b e-8 a^3 f}{\left (a-x^4\right )^2}d\sqrt {b x^2+a}}{4 a}-\frac {b^2 \sqrt {a+b x^2} (5 b c-6 a d)}{4 a \left (a-x^4\right )^2}}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 \left (\frac {\left (-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right ) \int \frac {1}{a-x^4}d\sqrt {b x^2+a}}{2 a}+\frac {b \sqrt {a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{2 a \left (a-x^4\right )}\right )}{4 a}-\frac {b^2 \sqrt {a+b x^2} (5 b c-6 a d)}{4 a \left (a-x^4\right )^2}}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {3 \left (\frac {b \sqrt {a+b x^2} \left (8 a^2 e-6 a b d+5 b^2 c\right )}{2 a \left (a-x^4\right )}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (-16 a^3 f+8 a^2 b e-6 a b^2 d+5 b^3 c\right )}{2 a^{3/2}}\right )}{4 a}-\frac {b^2 \sqrt {a+b x^2} (5 b c-6 a d)}{4 a \left (a-x^4\right )^2}}{3 a}-\frac {c \sqrt {a+b x^2}}{3 a x^6}\right )\) |
(-1/3*(c*Sqrt[a + b*x^2])/(a*x^6) - (-1/4*(b^2*(5*b*c - 6*a*d)*Sqrt[a + b* x^2])/(a*(a - x^4)^2) - (3*((b*(5*b^2*c - 6*a*b*d + 8*a^2*e)*Sqrt[a + b*x^ 2])/(2*a*(a - x^4)) + ((5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*ArcTan h[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(3/2))))/(4*a))/(3*a))/2
3.2.49.3.1 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)^(-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 = 116, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(-\frac {x^{6} \left (f \,a^{3}-\frac {1}{2} a^{2} b e +\frac {3}{8} a \,b^{2} d -\frac {5}{16} b^{3} c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )+\frac {5 \left (\frac {4 \left (2 e \,x^{4}+d \,x^{2}+\frac {2}{3} c \right ) a^{\frac {5}{2}}}{5}+b \left (2 \left (-\frac {3 d \,x^{2}}{5}-\frac {c}{3}\right ) a^{\frac {3}{2}}+b c \,x^{2} \sqrt {a}\right ) x^{2}\right ) \sqrt {b \,x^{2}+a}}{16}}{a^{\frac {7}{2}} x^{6}}\) | \(116\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (24 a^{2} e \,x^{4}-18 a b d \,x^{4}+15 b^{2} c \,x^{4}+12 a^{2} d \,x^{2}-10 a b c \,x^{2}+8 a^{2} c \right )}{48 a^{3} x^{6}}-\frac {\left (16 f \,a^{3}-8 a^{2} b e +6 a \,b^{2} d -5 b^{3} c \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{16 a^{\frac {7}{2}}}\) | \(124\) |
| default | \(-\frac {f \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{\sqrt {a}}+c \left (-\frac {\sqrt {b \,x^{2}+a}}{6 a \,x^{6}}-\frac {5 b \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )}{6 a}\right )+e \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )+d \left (-\frac {\sqrt {b \,x^{2}+a}}{4 a \,x^{4}}-\frac {3 b \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )\) | \(250\) |
-(x^6*(f*a^3-1/2*a^2*b*e+3/8*a*b^2*d-5/16*b^3*c)*arctanh((b*x^2+a)^(1/2)/a ^(1/2))+5/16*(4/5*(2*e*x^4+d*x^2+2/3*c)*a^(5/2)+b*(2*(-3/5*d*x^2-1/3*c)*a^ (3/2)+b*c*x^2*a^(1/2))*x^2)*(b*x^2+a)^(1/2))/a^(7/2)/x^6
Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.79 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c - 6 \, a b^{2} d + 8 \, a^{2} b e - 16 \, a^{3} f\right )} \sqrt {a} x^{6} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (3 \, {\left (5 \, a b^{2} c - 6 \, a^{2} b d + 8 \, a^{3} e\right )} x^{4} + 8 \, a^{3} c - 2 \, {\left (5 \, a^{2} b c - 6 \, a^{3} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{96 \, a^{4} x^{6}}, -\frac {3 \, {\left (5 \, b^{3} c - 6 \, a b^{2} d + 8 \, a^{2} b e - 16 \, a^{3} f\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (3 \, {\left (5 \, a b^{2} c - 6 \, a^{2} b d + 8 \, a^{3} e\right )} x^{4} + 8 \, a^{3} c - 2 \, {\left (5 \, a^{2} b c - 6 \, a^{3} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{48 \, a^{4} x^{6}}\right ] \]
[-1/96*(3*(5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*sqrt(a)*x^6*log(-(b *x^2 - 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(3*(5*a*b^2*c - 6*a^2*b*d + 8*a^3*e)*x^4 + 8*a^3*c - 2*(5*a^2*b*c - 6*a^3*d)*x^2)*sqrt(b*x^2 + a))/ (a^4*x^6), -1/48*(3*(5*b^3*c - 6*a*b^2*d + 8*a^2*b*e - 16*a^3*f)*sqrt(-a)* x^6*arctan(sqrt(-a)/sqrt(b*x^2 + a)) + (3*(5*a*b^2*c - 6*a^2*b*d + 8*a^3*e )*x^4 + 8*a^3*c - 2*(5*a^2*b*c - 6*a^3*d)*x^2)*sqrt(b*x^2 + a))/(a^4*x^6)]
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (141) = 282\).
Time = 40.70 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.08 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=- \frac {c}{6 \sqrt {b} x^{7} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {d}{4 \sqrt {b} x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} c}{24 a x^{5} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {\sqrt {b} d}{8 a x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {\sqrt {b} e \sqrt {\frac {a}{b x^{2}} + 1}}{2 a x} - \frac {5 b^{\frac {3}{2}} c}{48 a^{2} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b^{\frac {3}{2}} d}{8 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {5 b^{\frac {5}{2}} c}{16 a^{3} x \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {f \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{\sqrt {a}} + \frac {b e \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {3}{2}}} - \frac {3 b^{2} d \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{8 a^{\frac {5}{2}}} + \frac {5 b^{3} c \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{16 a^{\frac {7}{2}}} \]
-c/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - d/(4*sqrt(b)*x**5*sqrt(a/(b*x** 2) + 1)) + sqrt(b)*c/(24*a*x**5*sqrt(a/(b*x**2) + 1)) + sqrt(b)*d/(8*a*x** 3*sqrt(a/(b*x**2) + 1)) - sqrt(b)*e*sqrt(a/(b*x**2) + 1)/(2*a*x) - 5*b**(3 /2)*c/(48*a**2*x**3*sqrt(a/(b*x**2) + 1)) + 3*b**(3/2)*d/(8*a**2*x*sqrt(a/ (b*x**2) + 1)) - 5*b**(5/2)*c/(16*a**3*x*sqrt(a/(b*x**2) + 1)) - f*asinh(s qrt(a)/(sqrt(b)*x))/sqrt(a) + b*e*asinh(sqrt(a)/(sqrt(b)*x))/(2*a**(3/2)) - 3*b**2*d*asinh(sqrt(a)/(sqrt(b)*x))/(8*a**(5/2)) + 5*b**3*c*asinh(sqrt(a )/(sqrt(b)*x))/(16*a**(7/2))
Time = 0.20 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=\frac {5 \, b^{3} c \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{16 \, a^{\frac {7}{2}}} - \frac {3 \, b^{2} d \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {b e \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {f \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{\sqrt {a}} - \frac {5 \, \sqrt {b x^{2} + a} b^{2} c}{16 \, a^{3} x^{2}} + \frac {3 \, \sqrt {b x^{2} + a} b d}{8 \, a^{2} x^{2}} - \frac {\sqrt {b x^{2} + a} e}{2 \, a x^{2}} + \frac {5 \, \sqrt {b x^{2} + a} b c}{24 \, a^{2} x^{4}} - \frac {\sqrt {b x^{2} + a} d}{4 \, a x^{4}} - \frac {\sqrt {b x^{2} + a} c}{6 \, a x^{6}} \]
5/16*b^3*c*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(7/2) - 3/8*b^2*d*arcsinh(a/(sq rt(a*b)*abs(x)))/a^(5/2) + 1/2*b*e*arcsinh(a/(sqrt(a*b)*abs(x)))/a^(3/2) - f*arcsinh(a/(sqrt(a*b)*abs(x)))/sqrt(a) - 5/16*sqrt(b*x^2 + a)*b^2*c/(a^3 *x^2) + 3/8*sqrt(b*x^2 + a)*b*d/(a^2*x^2) - 1/2*sqrt(b*x^2 + a)*e/(a*x^2) + 5/24*sqrt(b*x^2 + a)*b*c/(a^2*x^4) - 1/4*sqrt(b*x^2 + a)*d/(a*x^4) - 1/6 *sqrt(b*x^2 + a)*c/(a*x^6)
Time = 0.29 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.56 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=-\frac {\frac {3 \, {\left (5 \, b^{4} c - 6 \, a b^{3} d + 8 \, a^{2} b^{2} e - 16 \, a^{3} b f\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4} c - 40 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{4} c + 33 \, \sqrt {b x^{2} + a} a^{2} b^{4} c - 18 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b^{3} d + 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b^{3} d - 30 \, \sqrt {b x^{2} + a} a^{3} b^{3} d + 24 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} b^{2} e - 48 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} b^{2} e + 24 \, \sqrt {b x^{2} + a} a^{4} b^{2} e}{a^{3} b^{3} x^{6}}}{48 \, b} \]
-1/48*(3*(5*b^4*c - 6*a*b^3*d + 8*a^2*b^2*e - 16*a^3*b*f)*arctan(sqrt(b*x^ 2 + a)/sqrt(-a))/(sqrt(-a)*a^3) + (15*(b*x^2 + a)^(5/2)*b^4*c - 40*(b*x^2 + a)^(3/2)*a*b^4*c + 33*sqrt(b*x^2 + a)*a^2*b^4*c - 18*(b*x^2 + a)^(5/2)*a *b^3*d + 48*(b*x^2 + a)^(3/2)*a^2*b^3*d - 30*sqrt(b*x^2 + a)*a^3*b^3*d + 2 4*(b*x^2 + a)^(5/2)*a^2*b^2*e - 48*(b*x^2 + a)^(3/2)*a^3*b^2*e + 24*sqrt(b *x^2 + a)*a^4*b^2*e)/(a^3*b^3*x^6))/b
Time = 7.62 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.36 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^7 \sqrt {a+b x^2}} \, dx=\frac {5\,c\,{\left (b\,x^2+a\right )}^{3/2}}{6\,a^2\,x^6}-\frac {11\,c\,\sqrt {b\,x^2+a}}{16\,a\,x^6}-\frac {f\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {5\,c\,{\left (b\,x^2+a\right )}^{5/2}}{16\,a^3\,x^6}-\frac {5\,d\,\sqrt {b\,x^2+a}}{8\,a\,x^4}+\frac {3\,d\,{\left (b\,x^2+a\right )}^{3/2}}{8\,a^2\,x^4}-\frac {e\,\sqrt {b\,x^2+a}}{2\,a\,x^2}+\frac {b\,e\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{3/2}}-\frac {3\,b^2\,d\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{8\,a^{5/2}}-\frac {b^3\,c\,\mathrm {atan}\left (\frac {\sqrt {b\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{16\,a^{7/2}} \]
(5*c*(a + b*x^2)^(3/2))/(6*a^2*x^6) - (11*c*(a + b*x^2)^(1/2))/(16*a*x^6) - (f*atanh((a + b*x^2)^(1/2)/a^(1/2)))/a^(1/2) - (5*c*(a + b*x^2)^(5/2))/( 16*a^3*x^6) - (5*d*(a + b*x^2)^(1/2))/(8*a*x^4) + (3*d*(a + b*x^2)^(3/2))/ (8*a^2*x^4) - (e*(a + b*x^2)^(1/2))/(2*a*x^2) + (b*e*atanh((a + b*x^2)^(1/ 2)/a^(1/2)))/(2*a^(3/2)) - (b^3*c*atan(((a + b*x^2)^(1/2)*1i)/a^(1/2))*5i) /(16*a^(7/2)) - (3*b^2*d*atanh((a + b*x^2)^(1/2)/a^(1/2)))/(8*a^(5/2))