Integrand size = 32, antiderivative size = 189 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=-\frac {c \sqrt {a+b x^2}}{9 a x^9}+\frac {(8 b c-9 a d) \sqrt {a+b x^2}}{63 a^2 x^7}-\frac {\left (16 b^2 c-18 a b d+21 a^2 e\right ) \sqrt {a+b x^2}}{105 a^3 x^5}+\frac {\left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^4 x^3}-\frac {2 b \left (64 b^3 c-72 a b^2 d+84 a^2 b e-105 a^3 f\right ) \sqrt {a+b x^2}}{315 a^5 x} \]
-1/9*c*(b*x^2+a)^(1/2)/a/x^9+1/63*(-9*a*d+8*b*c)*(b*x^2+a)^(1/2)/a^2/x^7-1 /105*(21*a^2*e-18*a*b*d+16*b^2*c)*(b*x^2+a)^(1/2)/a^3/x^5+1/315*(-105*a^3* f+84*a^2*b*e-72*a*b^2*d+64*b^3*c)*(b*x^2+a)^(1/2)/a^4/x^3-2/315*b*(-105*a^ 3*f+84*a^2*b*e-72*a*b^2*d+64*b^3*c)*(b*x^2+a)^(1/2)/a^5/x
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.71 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {a+b x^2} \left (128 b^4 c x^8-16 a b^3 x^6 \left (4 c+9 d x^2\right )+24 a^2 b^2 x^4 \left (2 c+3 d x^2+7 e x^4\right )-2 a^3 b x^2 \left (20 c+27 d x^2+42 e x^4+105 f x^6\right )+a^4 \left (35 c+45 d x^2+63 e x^4+105 f x^6\right )\right )}{315 a^5 x^9} \]
-1/315*(Sqrt[a + b*x^2]*(128*b^4*c*x^8 - 16*a*b^3*x^6*(4*c + 9*d*x^2) + 24 *a^2*b^2*x^4*(2*c + 3*d*x^2 + 7*e*x^4) - 2*a^3*b*x^2*(20*c + 27*d*x^2 + 42 *e*x^4 + 105*f*x^6) + a^4*(35*c + 45*d*x^2 + 63*e*x^4 + 105*f*x^6)))/(a^5* x^9)
Time = 0.46 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2334, 2089, 1588, 27, 359, 245, 242}
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.
| \(\displaystyle \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2334 |
| \(\displaystyle -\frac {\int \frac {8 b c-9 a \left (f x^4+e x^2+d\right )}{x^8 \sqrt {b x^2+a}}dx}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
| \(\Big \downarrow \) 2089 |
| \(\displaystyle -\frac {\int \frac {-9 a f x^4-9 a e x^2+8 b c-9 a d}{x^8 \sqrt {b x^2+a}}dx}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 \left (21 f x^2 a^2+21 e a^2-18 b d a+16 b^2 c\right )}{x^6 \sqrt {b x^2+a}}dx}{7 a}-\frac {\sqrt {a+b x^2} (8 b c-9 a d)}{7 a x^7}}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {3 \int \frac {21 f x^2 a^2+21 e a^2-18 b d a+16 b^2 c}{x^6 \sqrt {b x^2+a}}dx}{7 a}-\frac {\sqrt {a+b x^2} (8 b c-9 a d)}{7 a x^7}}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right ) \int \frac {1}{x^4 \sqrt {b x^2+a}}dx}{5 a}-\frac {\sqrt {a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{5 a x^5}\right )}{7 a}-\frac {\sqrt {a+b x^2} (8 b c-9 a d)}{7 a x^7}}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right ) \left (-\frac {2 b \int \frac {1}{x^2 \sqrt {b x^2+a}}dx}{3 a}-\frac {\sqrt {a+b x^2}}{3 a x^3}\right )}{5 a}-\frac {\sqrt {a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{5 a x^5}\right )}{7 a}-\frac {\sqrt {a+b x^2} (8 b c-9 a d)}{7 a x^7}}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {-\frac {3 \left (-\frac {\sqrt {a+b x^2} \left (21 a^2 e-18 a b d+16 b^2 c\right )}{5 a x^5}-\frac {\left (\frac {2 b \sqrt {a+b x^2}}{3 a^2 x}-\frac {\sqrt {a+b x^2}}{3 a x^3}\right ) \left (-105 a^3 f+84 a^2 b e-72 a b^2 d+64 b^3 c\right )}{5 a}\right )}{7 a}-\frac {\sqrt {a+b x^2} (8 b c-9 a d)}{7 a x^7}}{9 a}-\frac {c \sqrt {a+b x^2}}{9 a x^9}\) |
-1/9*(c*Sqrt[a + b*x^2])/(a*x^9) - (-1/7*((8*b*c - 9*a*d)*Sqrt[a + b*x^2]) /(a*x^7) - (3*(-1/5*((16*b^2*c - 18*a*b*d + 21*a^2*e)*Sqrt[a + b*x^2])/(a* x^5) - ((64*b^3*c - 72*a*b^2*d + 84*a^2*b*e - 105*a^3*f)*(-1/3*Sqrt[a + b* x^2]/(a*x^3) + (2*b*Sqrt[a + b*x^2])/(3*a^2*x)))/(5*a)))/(7*a))/(9*a)
3.2.58.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((f_.)*(x_))^(m_)*((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, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -1]
Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_)^(q_.), x_Symbol] :> Int[(f*x)^m*Expa ndToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && Binomi alQ[z, x] && TrinomialQ[u, x] && !(BinomialMatchQ[z, x] && TrinomialMatchQ [u, x])
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coef f[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A* x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[ x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
Time = 3.68 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.65
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (3 f \,x^{6}+\frac {9}{5} e \,x^{4}+\frac {9}{7} d \,x^{2}+c \right ) a^{4}-\frac {8 \left (\frac {21}{4} f \,x^{6}+\frac {21}{10} e \,x^{4}+\frac {27}{20} d \,x^{2}+c \right ) b \,x^{2} a^{3}}{7}+\frac {48 b^{2} x^{4} \left (\frac {7}{2} e \,x^{4}+\frac {3}{2} d \,x^{2}+c \right ) a^{2}}{35}-\frac {64 b^{3} x^{6} \left (\frac {9 d \,x^{2}}{4}+c \right ) a}{35}+\frac {128 b^{4} c \,x^{8}}{35}\right ) \sqrt {b \,x^{2}+a}}{9 x^{9} a^{5}}\) | \(123\) |
| gosper | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 a^{4} c \right )}{315 x^{9} a^{5}}\) | \(157\) |
| trager | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 a^{4} c \right )}{315 x^{9} a^{5}}\) | \(157\) |
| risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-210 a^{3} b f \,x^{8}+168 a^{2} b^{2} e \,x^{8}-144 a \,b^{3} d \,x^{8}+128 b^{4} c \,x^{8}+105 a^{4} f \,x^{6}-84 a^{3} b e \,x^{6}+72 a^{2} b^{2} d \,x^{6}-64 a \,b^{3} c \,x^{6}+63 a^{4} e \,x^{4}-54 a^{3} b d \,x^{4}+48 a^{2} b^{2} c \,x^{4}+45 a^{4} d \,x^{2}-40 a^{3} b c \,x^{2}+35 a^{4} c \right )}{315 x^{9} a^{5}}\) | \(157\) |
| default | \(d \left (-\frac {\sqrt {b \,x^{2}+a}}{7 a \,x^{7}}-\frac {6 b \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )+f \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )+e \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )+c \left (-\frac {\sqrt {b \,x^{2}+a}}{9 a \,x^{9}}-\frac {8 b \left (-\frac {\sqrt {b \,x^{2}+a}}{7 a \,x^{7}}-\frac {6 b \left (-\frac {\sqrt {b \,x^{2}+a}}{5 a \,x^{5}}-\frac {4 b \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )\) | \(298\) |
-1/9*((3*f*x^6+9/5*e*x^4+9/7*d*x^2+c)*a^4-8/7*(21/4*f*x^6+21/10*e*x^4+27/2 0*d*x^2+c)*b*x^2*a^3+48/35*b^2*x^4*(7/2*e*x^4+3/2*d*x^2+c)*a^2-64/35*b^3*x ^6*(9/4*d*x^2+c)*a+128/35*b^4*c*x^8)*(b*x^2+a)^(1/2)/x^9/a^5
Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.75 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=-\frac {{\left (2 \, {\left (64 \, b^{4} c - 72 \, a b^{3} d + 84 \, a^{2} b^{2} e - 105 \, a^{3} b f\right )} x^{8} - {\left (64 \, a b^{3} c - 72 \, a^{2} b^{2} d + 84 \, a^{3} b e - 105 \, a^{4} f\right )} x^{6} + 35 \, a^{4} c + 3 \, {\left (16 \, a^{2} b^{2} c - 18 \, a^{3} b d + 21 \, a^{4} e\right )} x^{4} - 5 \, {\left (8 \, a^{3} b c - 9 \, a^{4} d\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, a^{5} x^{9}} \]
-1/315*(2*(64*b^4*c - 72*a*b^3*d + 84*a^2*b^2*e - 105*a^3*b*f)*x^8 - (64*a *b^3*c - 72*a^2*b^2*d + 84*a^3*b*e - 105*a^4*f)*x^6 + 35*a^4*c + 3*(16*a^2 *b^2*c - 18*a^3*b*d + 21*a^4*e)*x^4 - 5*(8*a^3*b*c - 9*a^4*d)*x^2)*sqrt(b* x^2 + a)/(a^5*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 1642 vs. \(2 (190) = 380\).
Time = 2.92 (sec) , antiderivative size = 1642, normalized size of antiderivative = 8.69 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=\text {Too large to display} \]
-35*a**8*b**(33/2)*c*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8 *b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b* *20*x**16) - 100*a**7*b**(35/2)*c*x**2*sqrt(a/(b*x**2) + 1)/(315*a**9*b**1 6*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x **14 + 315*a**5*b**20*x**16) - 98*a**6*b**(37/2)*c*x**4*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 5*a**6*b**(19/2)*d*sqrt(a/ (b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11*x* *10 + 35*a**4*b**12*x**12) - 28*a**5*b**(39/2)*c*x**6*sqrt(a/(b*x**2) + 1) /(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 12 60*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 9*a**5*b**(21/2)*d*x**2*sqrt (a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b**11 *x**10 + 35*a**4*b**12*x**12) - 35*a**4*b**(41/2)*c*x**8*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17*x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20*x**16) - 5*a**4*b**(23/2)*d*x**4*s qrt(a/(b*x**2) + 1)/(35*a**7*b**9*x**6 + 105*a**6*b**10*x**8 + 105*a**5*b* *11*x**10 + 35*a**4*b**12*x**12) - 3*a**4*b**(9/2)*e*sqrt(a/(b*x**2) + 1)/ (15*a**5*b**4*x**4 + 30*a**4*b**5*x**6 + 15*a**3*b**6*x**8) - 280*a**3*b** (43/2)*c*x**10*sqrt(a/(b*x**2) + 1)/(315*a**9*b**16*x**8 + 1260*a**8*b**17 *x**10 + 1890*a**7*b**18*x**12 + 1260*a**6*b**19*x**14 + 315*a**5*b**20...
Time = 0.19 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.46 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=-\frac {128 \, \sqrt {b x^{2} + a} b^{4} c}{315 \, a^{5} x} + \frac {16 \, \sqrt {b x^{2} + a} b^{3} d}{35 \, a^{4} x} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} e}{15 \, a^{3} x} + \frac {2 \, \sqrt {b x^{2} + a} b f}{3 \, a^{2} x} + \frac {64 \, \sqrt {b x^{2} + a} b^{3} c}{315 \, a^{4} x^{3}} - \frac {8 \, \sqrt {b x^{2} + a} b^{2} d}{35 \, a^{3} x^{3}} + \frac {4 \, \sqrt {b x^{2} + a} b e}{15 \, a^{2} x^{3}} - \frac {\sqrt {b x^{2} + a} f}{3 \, a x^{3}} - \frac {16 \, \sqrt {b x^{2} + a} b^{2} c}{105 \, a^{3} x^{5}} + \frac {6 \, \sqrt {b x^{2} + a} b d}{35 \, a^{2} x^{5}} - \frac {\sqrt {b x^{2} + a} e}{5 \, a x^{5}} + \frac {8 \, \sqrt {b x^{2} + a} b c}{63 \, a^{2} x^{7}} - \frac {\sqrt {b x^{2} + a} d}{7 \, a x^{7}} - \frac {\sqrt {b x^{2} + a} c}{9 \, a x^{9}} \]
-128/315*sqrt(b*x^2 + a)*b^4*c/(a^5*x) + 16/35*sqrt(b*x^2 + a)*b^3*d/(a^4* x) - 8/15*sqrt(b*x^2 + a)*b^2*e/(a^3*x) + 2/3*sqrt(b*x^2 + a)*b*f/(a^2*x) + 64/315*sqrt(b*x^2 + a)*b^3*c/(a^4*x^3) - 8/35*sqrt(b*x^2 + a)*b^2*d/(a^3 *x^3) + 4/15*sqrt(b*x^2 + a)*b*e/(a^2*x^3) - 1/3*sqrt(b*x^2 + a)*f/(a*x^3) - 16/105*sqrt(b*x^2 + a)*b^2*c/(a^3*x^5) + 6/35*sqrt(b*x^2 + a)*b*d/(a^2* x^5) - 1/5*sqrt(b*x^2 + a)*e/(a*x^5) + 8/63*sqrt(b*x^2 + a)*b*c/(a^2*x^7) - 1/7*sqrt(b*x^2 + a)*d/(a*x^7) - 1/9*sqrt(b*x^2 + a)*c/(a*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (169) = 338\).
Time = 0.34 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.49 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=\frac {4 \, {\left (315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{14} b^{\frac {3}{2}} f + 840 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} b^{\frac {5}{2}} e - 1995 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{12} a b^{\frac {3}{2}} f + 2520 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} b^{\frac {7}{2}} d - 3780 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a b^{\frac {5}{2}} e + 5355 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{10} a^{2} b^{\frac {3}{2}} f + 8064 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} b^{\frac {9}{2}} c - 6552 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a b^{\frac {7}{2}} d + 6804 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{2} b^{\frac {5}{2}} e - 7875 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{8} a^{3} b^{\frac {3}{2}} f - 5376 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a b^{\frac {9}{2}} c + 6048 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} b^{\frac {7}{2}} d - 6216 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{3} b^{\frac {5}{2}} e + 6825 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{4} b^{\frac {3}{2}} f + 2304 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {9}{2}} c - 2592 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} b^{\frac {7}{2}} d + 3024 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{4} b^{\frac {5}{2}} e - 3465 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{5} b^{\frac {3}{2}} f - 576 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {9}{2}} c + 648 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} b^{\frac {7}{2}} d - 756 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{5} b^{\frac {5}{2}} e + 945 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{6} b^{\frac {3}{2}} f + 64 \, a^{4} b^{\frac {9}{2}} c - 72 \, a^{5} b^{\frac {7}{2}} d + 84 \, a^{6} b^{\frac {5}{2}} e - 105 \, a^{7} b^{\frac {3}{2}} f\right )}}{315 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{9}} \]
4/315*(315*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(3/2)*f + 840*(sqrt(b)*x - s qrt(b*x^2 + a))^12*b^(5/2)*e - 1995*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^( 3/2)*f + 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^10*b^(7/2)*d - 3780*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a*b^(5/2)*e + 5355*(sqrt(b)*x - sqrt(b*x^2 + a))^10 *a^2*b^(3/2)*f + 8064*(sqrt(b)*x - sqrt(b*x^2 + a))^8*b^(9/2)*c - 6552*(sq rt(b)*x - sqrt(b*x^2 + a))^8*a*b^(7/2)*d + 6804*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(5/2)*e - 7875*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^3*b^(3/2)*f - 5376*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a*b^(9/2)*c + 6048*(sqrt(b)*x - sqrt (b*x^2 + a))^6*a^2*b^(7/2)*d - 6216*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^3*b^ (5/2)*e + 6825*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(3/2)*f + 2304*(sqrt( b)*x - sqrt(b*x^2 + a))^4*a^2*b^(9/2)*c - 2592*(sqrt(b)*x - sqrt(b*x^2 + a ))^4*a^3*b^(7/2)*d + 3024*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^4*b^(5/2)*e - 3465*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^5*b^(3/2)*f - 576*(sqrt(b)*x - sqrt (b*x^2 + a))^2*a^3*b^(9/2)*c + 648*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^4*b^( 7/2)*d - 756*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^5*b^(5/2)*e + 945*(sqrt(b)* x - sqrt(b*x^2 + a))^2*a^6*b^(3/2)*f + 64*a^4*b^(9/2)*c - 72*a^5*b^(7/2)*d + 84*a^6*b^(5/2)*e - 105*a^7*b^(3/2)*f)/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9
Time = 6.14 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^{10} \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b\,x^2+a}\,\left (-105\,f\,a^3+84\,e\,a^2\,b-72\,d\,a\,b^2+64\,c\,b^3\right )}{315\,a^4\,x^3}-\frac {\sqrt {b\,x^2+a}\,\left (9\,a\,d-8\,b\,c\right )}{63\,a^2\,x^7}-\frac {\sqrt {b\,x^2+a}\,\left (21\,e\,a^2-18\,d\,a\,b+16\,c\,b^2\right )}{105\,a^3\,x^5}-\frac {\sqrt {b\,x^2+a}\,\left (-210\,f\,a^3\,b+168\,e\,a^2\,b^2-144\,d\,a\,b^3+128\,c\,b^4\right )}{315\,a^5\,x}-\frac {c\,\sqrt {b\,x^2+a}}{9\,a\,x^9} \]
((a + b*x^2)^(1/2)*(64*b^3*c - 105*a^3*f - 72*a*b^2*d + 84*a^2*b*e))/(315* a^4*x^3) - ((a + b*x^2)^(1/2)*(9*a*d - 8*b*c))/(63*a^2*x^7) - ((a + b*x^2) ^(1/2)*(16*b^2*c + 21*a^2*e - 18*a*b*d))/(105*a^3*x^5) - ((a + b*x^2)^(1/2 )*(128*b^4*c + 168*a^2*b^2*e - 144*a*b^3*d - 210*a^3*b*f))/(315*a^5*x) - ( c*(a + b*x^2)^(1/2))/(9*a*x^9)