Integrand size = 13, antiderivative size = 100 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=-\frac {a^8}{16 x^{16}}-\frac {4 a^7 b}{7 x^{14}}-\frac {7 a^6 b^2}{3 x^{12}}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {35 a^4 b^4}{4 x^8}-\frac {28 a^3 b^5}{3 x^6}-\frac {7 a^2 b^6}{x^4}-\frac {4 a b^7}{x^2}+b^8 \log (x) \] Output:
-1/16*a^8/x^16-4/7*a^7*b/x^14-7/3*a^6*b^2/x^12-28/5*a^5*b^3/x^10-35/4*a^4* b^4/x^8-28/3*a^3*b^5/x^6-7*a^2*b^6/x^4-4*a*b^7/x^2+b^8*ln(x)
Time = 0.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=-\frac {a^8}{16 x^{16}}-\frac {4 a^7 b}{7 x^{14}}-\frac {7 a^6 b^2}{3 x^{12}}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {35 a^4 b^4}{4 x^8}-\frac {28 a^3 b^5}{3 x^6}-\frac {7 a^2 b^6}{x^4}-\frac {4 a b^7}{x^2}+b^8 \log (x) \] Input:
Integrate[(a + b*x^2)^8/x^17,x]
Output:
-1/16*a^8/x^16 - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/ (5*x^10) - (35*a^4*b^4)/(4*x^8) - (28*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x^2 + b^8*Log[x]
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {243, 49, 2009}
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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^{18}}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^8}{x^{18}}+\frac {8 b a^7}{x^{16}}+\frac {28 b^2 a^6}{x^{14}}+\frac {56 b^3 a^5}{x^{12}}+\frac {70 b^4 a^4}{x^{10}}+\frac {56 b^5 a^3}{x^8}+\frac {28 b^6 a^2}{x^6}+\frac {8 b^7 a}{x^4}+\frac {b^8}{x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^8}{8 x^{16}}-\frac {8 a^7 b}{7 x^{14}}-\frac {14 a^6 b^2}{3 x^{12}}-\frac {56 a^5 b^3}{5 x^{10}}-\frac {35 a^4 b^4}{2 x^8}-\frac {56 a^3 b^5}{3 x^6}-\frac {14 a^2 b^6}{x^4}-\frac {8 a b^7}{x^2}+b^8 \log \left (x^2\right )\right )\) |
Input:
Int[(a + b*x^2)^8/x^17,x]
Output:
(-1/8*a^8/x^16 - (8*a^7*b)/(7*x^14) - (14*a^6*b^2)/(3*x^12) - (56*a^5*b^3) /(5*x^10) - (35*a^4*b^4)/(2*x^8) - (56*a^3*b^5)/(3*x^6) - (14*a^2*b^6)/x^4 - (8*a*b^7)/x^2 + b^8*Log[x^2])/2
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.25 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {a^{8}}{16 x^{16}}-\frac {4 a^{7} b}{7 x^{14}}-\frac {7 a^{6} b^{2}}{3 x^{12}}-\frac {28 a^{5} b^{3}}{5 x^{10}}-\frac {35 a^{4} b^{4}}{4 x^{8}}-\frac {28 a^{3} b^{5}}{3 x^{6}}-\frac {7 a^{2} b^{6}}{x^{4}}-\frac {4 a \,b^{7}}{x^{2}}+b^{8} \ln \left (x \right )\) | \(89\) |
norman | \(\frac {-\frac {1}{16} a^{8}-4 a \,b^{7} x^{14}-7 a^{2} b^{6} x^{12}-\frac {28}{3} a^{3} b^{5} x^{10}-\frac {35}{4} a^{4} b^{4} x^{8}-\frac {28}{5} a^{5} b^{3} x^{6}-\frac {7}{3} a^{6} b^{2} x^{4}-\frac {4}{7} a^{7} b \,x^{2}}{x^{16}}+b^{8} \ln \left (x \right )\) | \(91\) |
risch | \(\frac {-\frac {1}{16} a^{8}-4 a \,b^{7} x^{14}-7 a^{2} b^{6} x^{12}-\frac {28}{3} a^{3} b^{5} x^{10}-\frac {35}{4} a^{4} b^{4} x^{8}-\frac {28}{5} a^{5} b^{3} x^{6}-\frac {7}{3} a^{6} b^{2} x^{4}-\frac {4}{7} a^{7} b \,x^{2}}{x^{16}}+b^{8} \ln \left (x \right )\) | \(91\) |
parallelrisch | \(\frac {1680 b^{8} \ln \left (x \right ) x^{16}-6720 a \,b^{7} x^{14}-11760 a^{2} b^{6} x^{12}-15680 a^{3} b^{5} x^{10}-14700 a^{4} b^{4} x^{8}-9408 a^{5} b^{3} x^{6}-3920 a^{6} b^{2} x^{4}-960 a^{7} b \,x^{2}-105 a^{8}}{1680 x^{16}}\) | \(95\) |
Input:
int((b*x^2+a)^8/x^17,x,method=_RETURNVERBOSE)
Output:
-1/16*a^8/x^16-4/7*a^7*b/x^14-7/3*a^6*b^2/x^12-28/5*a^5*b^3/x^10-35/4*a^4* b^4/x^8-28/3*a^3*b^5/x^6-7*a^2*b^6/x^4-4*a*b^7/x^2+b^8*ln(x)
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=\frac {1680 \, b^{8} x^{16} \log \left (x\right ) - 6720 \, a b^{7} x^{14} - 11760 \, a^{2} b^{6} x^{12} - 15680 \, a^{3} b^{5} x^{10} - 14700 \, a^{4} b^{4} x^{8} - 9408 \, a^{5} b^{3} x^{6} - 3920 \, a^{6} b^{2} x^{4} - 960 \, a^{7} b x^{2} - 105 \, a^{8}}{1680 \, x^{16}} \] Input:
integrate((b*x^2+a)^8/x^17,x, algorithm="fricas")
Output:
1/1680*(1680*b^8*x^16*log(x) - 6720*a*b^7*x^14 - 11760*a^2*b^6*x^12 - 1568 0*a^3*b^5*x^10 - 14700*a^4*b^4*x^8 - 9408*a^5*b^3*x^6 - 3920*a^6*b^2*x^4 - 960*a^7*b*x^2 - 105*a^8)/x^16
Time = 0.48 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=b^{8} \log {\left (x \right )} + \frac {- 105 a^{8} - 960 a^{7} b x^{2} - 3920 a^{6} b^{2} x^{4} - 9408 a^{5} b^{3} x^{6} - 14700 a^{4} b^{4} x^{8} - 15680 a^{3} b^{5} x^{10} - 11760 a^{2} b^{6} x^{12} - 6720 a b^{7} x^{14}}{1680 x^{16}} \] Input:
integrate((b*x**2+a)**8/x**17,x)
Output:
b**8*log(x) + (-105*a**8 - 960*a**7*b*x**2 - 3920*a**6*b**2*x**4 - 9408*a* *5*b**3*x**6 - 14700*a**4*b**4*x**8 - 15680*a**3*b**5*x**10 - 11760*a**2*b **6*x**12 - 6720*a*b**7*x**14)/(1680*x**16)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=\frac {1}{2} \, b^{8} \log \left (x^{2}\right ) - \frac {6720 \, a b^{7} x^{14} + 11760 \, a^{2} b^{6} x^{12} + 15680 \, a^{3} b^{5} x^{10} + 14700 \, a^{4} b^{4} x^{8} + 9408 \, a^{5} b^{3} x^{6} + 3920 \, a^{6} b^{2} x^{4} + 960 \, a^{7} b x^{2} + 105 \, a^{8}}{1680 \, x^{16}} \] Input:
integrate((b*x^2+a)^8/x^17,x, algorithm="maxima")
Output:
1/2*b^8*log(x^2) - 1/1680*(6720*a*b^7*x^14 + 11760*a^2*b^6*x^12 + 15680*a^ 3*b^5*x^10 + 14700*a^4*b^4*x^8 + 9408*a^5*b^3*x^6 + 3920*a^6*b^2*x^4 + 960 *a^7*b*x^2 + 105*a^8)/x^16
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=\frac {1}{2} \, b^{8} \log \left (x^{2}\right ) - \frac {2283 \, b^{8} x^{16} + 6720 \, a b^{7} x^{14} + 11760 \, a^{2} b^{6} x^{12} + 15680 \, a^{3} b^{5} x^{10} + 14700 \, a^{4} b^{4} x^{8} + 9408 \, a^{5} b^{3} x^{6} + 3920 \, a^{6} b^{2} x^{4} + 960 \, a^{7} b x^{2} + 105 \, a^{8}}{1680 \, x^{16}} \] Input:
integrate((b*x^2+a)^8/x^17,x, algorithm="giac")
Output:
1/2*b^8*log(x^2) - 1/1680*(2283*b^8*x^16 + 6720*a*b^7*x^14 + 11760*a^2*b^6 *x^12 + 15680*a^3*b^5*x^10 + 14700*a^4*b^4*x^8 + 9408*a^5*b^3*x^6 + 3920*a ^6*b^2*x^4 + 960*a^7*b*x^2 + 105*a^8)/x^16
Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=b^8\,\ln \left (x\right )-\frac {\frac {a^8}{16}+\frac {4\,a^7\,b\,x^2}{7}+\frac {7\,a^6\,b^2\,x^4}{3}+\frac {28\,a^5\,b^3\,x^6}{5}+\frac {35\,a^4\,b^4\,x^8}{4}+\frac {28\,a^3\,b^5\,x^{10}}{3}+7\,a^2\,b^6\,x^{12}+4\,a\,b^7\,x^{14}}{x^{16}} \] Input:
int((a + b*x^2)^8/x^17,x)
Output:
b^8*log(x) - (a^8/16 + (4*a^7*b*x^2)/7 + 4*a*b^7*x^14 + (7*a^6*b^2*x^4)/3 + (28*a^5*b^3*x^6)/5 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/3 + 7*a^2*b^ 6*x^12)/x^16
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^8}{x^{17}} \, dx=\frac {1680 \,\mathrm {log}\left (x \right ) b^{8} x^{16}-105 a^{8}-960 a^{7} b \,x^{2}-3920 a^{6} b^{2} x^{4}-9408 a^{5} b^{3} x^{6}-14700 a^{4} b^{4} x^{8}-15680 a^{3} b^{5} x^{10}-11760 a^{2} b^{6} x^{12}-6720 a \,b^{7} x^{14}}{1680 x^{16}} \] Input:
int((b*x^2+a)^8/x^17,x)
Output:
(1680*log(x)*b**8*x**16 - 105*a**8 - 960*a**7*b*x**2 - 3920*a**6*b**2*x**4 - 9408*a**5*b**3*x**6 - 14700*a**4*b**4*x**8 - 15680*a**3*b**5*x**10 - 11 760*a**2*b**6*x**12 - 6720*a*b**7*x**14)/(1680*x**16)