Integrand size = 13, antiderivative size = 104 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=-\frac {a^8}{17 x^{17}}-\frac {8 a^7 b}{15 x^{15}}-\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x} \] Output:
-1/17*a^8/x^17-8/15*a^7*b/x^15-28/13*a^6*b^2/x^13-56/11*a^5*b^3/x^11-70/9* a^4*b^4/x^9-8*a^3*b^5/x^7-28/5*a^2*b^6/x^5-8/3*a*b^7/x^3-b^8/x
Time = 0.01 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=-\frac {a^8}{17 x^{17}}-\frac {8 a^7 b}{15 x^{15}}-\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x} \] Input:
Integrate[(a + b*x^2)^8/x^18,x]
Output:
-1/17*a^8/x^17 - (8*a^7*b)/(15*x^15) - (28*a^6*b^2)/(13*x^13) - (56*a^5*b^ 3)/(11*x^11) - (70*a^4*b^4)/(9*x^9) - (8*a^3*b^5)/x^7 - (28*a^2*b^6)/(5*x^ 5) - (8*a*b^7)/(3*x^3) - b^8/x
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {244, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \int \left (\frac {a^8}{x^{18}}+\frac {8 a^7 b}{x^{16}}+\frac {28 a^6 b^2}{x^{14}}+\frac {56 a^5 b^3}{x^{12}}+\frac {70 a^4 b^4}{x^{10}}+\frac {56 a^3 b^5}{x^8}+\frac {28 a^2 b^6}{x^6}+\frac {8 a b^7}{x^4}+\frac {b^8}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^8}{17 x^{17}}-\frac {8 a^7 b}{15 x^{15}}-\frac {28 a^6 b^2}{13 x^{13}}-\frac {56 a^5 b^3}{11 x^{11}}-\frac {70 a^4 b^4}{9 x^9}-\frac {8 a^3 b^5}{x^7}-\frac {28 a^2 b^6}{5 x^5}-\frac {8 a b^7}{3 x^3}-\frac {b^8}{x}\) |
Input:
Int[(a + b*x^2)^8/x^18,x]
Output:
-1/17*a^8/x^17 - (8*a^7*b)/(15*x^15) - (28*a^6*b^2)/(13*x^13) - (56*a^5*b^ 3)/(11*x^11) - (70*a^4*b^4)/(9*x^9) - (8*a^3*b^5)/x^7 - (28*a^2*b^6)/(5*x^ 5) - (8*a*b^7)/(3*x^3) - b^8/x
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Time = 0.26 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{8}}{17 x^{17}}-\frac {8 a^{7} b}{15 x^{15}}-\frac {28 a^{6} b^{2}}{13 x^{13}}-\frac {56 a^{5} b^{3}}{11 x^{11}}-\frac {70 a^{4} b^{4}}{9 x^{9}}-\frac {8 a^{3} b^{5}}{x^{7}}-\frac {28 a^{2} b^{6}}{5 x^{5}}-\frac {8 a \,b^{7}}{3 x^{3}}-\frac {b^{8}}{x}\) | \(91\) |
norman | \(\frac {-\frac {1}{17} a^{8}-\frac {8}{15} a^{7} b \,x^{2}-\frac {28}{13} a^{6} b^{2} x^{4}-\frac {56}{11} a^{5} b^{3} x^{6}-\frac {70}{9} a^{4} b^{4} x^{8}-8 a^{3} b^{5} x^{10}-\frac {28}{5} a^{2} b^{6} x^{12}-\frac {8}{3} a \,b^{7} x^{14}-b^{8} x^{16}}{x^{17}}\) | \(92\) |
risch | \(\frac {-\frac {1}{17} a^{8}-\frac {8}{15} a^{7} b \,x^{2}-\frac {28}{13} a^{6} b^{2} x^{4}-\frac {56}{11} a^{5} b^{3} x^{6}-\frac {70}{9} a^{4} b^{4} x^{8}-8 a^{3} b^{5} x^{10}-\frac {28}{5} a^{2} b^{6} x^{12}-\frac {8}{3} a \,b^{7} x^{14}-b^{8} x^{16}}{x^{17}}\) | \(92\) |
gosper | \(-\frac {109395 b^{8} x^{16}+291720 a \,b^{7} x^{14}+612612 a^{2} b^{6} x^{12}+875160 a^{3} b^{5} x^{10}+850850 a^{4} b^{4} x^{8}+556920 a^{5} b^{3} x^{6}+235620 a^{6} b^{2} x^{4}+58344 a^{7} b \,x^{2}+6435 a^{8}}{109395 x^{17}}\) | \(93\) |
parallelrisch | \(\frac {-109395 b^{8} x^{16}-291720 a \,b^{7} x^{14}-612612 a^{2} b^{6} x^{12}-875160 a^{3} b^{5} x^{10}-850850 a^{4} b^{4} x^{8}-556920 a^{5} b^{3} x^{6}-235620 a^{6} b^{2} x^{4}-58344 a^{7} b \,x^{2}-6435 a^{8}}{109395 x^{17}}\) | \(93\) |
orering | \(-\frac {109395 b^{8} x^{16}+291720 a \,b^{7} x^{14}+612612 a^{2} b^{6} x^{12}+875160 a^{3} b^{5} x^{10}+850850 a^{4} b^{4} x^{8}+556920 a^{5} b^{3} x^{6}+235620 a^{6} b^{2} x^{4}+58344 a^{7} b \,x^{2}+6435 a^{8}}{109395 x^{17}}\) | \(93\) |
Input:
int((b*x^2+a)^8/x^18,x,method=_RETURNVERBOSE)
Output:
-1/17*a^8/x^17-8/15*a^7*b/x^15-28/13*a^6*b^2/x^13-56/11*a^5*b^3/x^11-70/9* a^4*b^4/x^9-8*a^3*b^5/x^7-28/5*a^2*b^6/x^5-8/3*a*b^7/x^3-b^8/x
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=-\frac {109395 \, b^{8} x^{16} + 291720 \, a b^{7} x^{14} + 612612 \, a^{2} b^{6} x^{12} + 875160 \, a^{3} b^{5} x^{10} + 850850 \, a^{4} b^{4} x^{8} + 556920 \, a^{5} b^{3} x^{6} + 235620 \, a^{6} b^{2} x^{4} + 58344 \, a^{7} b x^{2} + 6435 \, a^{8}}{109395 \, x^{17}} \] Input:
integrate((b*x^2+a)^8/x^18,x, algorithm="fricas")
Output:
-1/109395*(109395*b^8*x^16 + 291720*a*b^7*x^14 + 612612*a^2*b^6*x^12 + 875 160*a^3*b^5*x^10 + 850850*a^4*b^4*x^8 + 556920*a^5*b^3*x^6 + 235620*a^6*b^ 2*x^4 + 58344*a^7*b*x^2 + 6435*a^8)/x^17
Time = 0.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=\frac {- 6435 a^{8} - 58344 a^{7} b x^{2} - 235620 a^{6} b^{2} x^{4} - 556920 a^{5} b^{3} x^{6} - 850850 a^{4} b^{4} x^{8} - 875160 a^{3} b^{5} x^{10} - 612612 a^{2} b^{6} x^{12} - 291720 a b^{7} x^{14} - 109395 b^{8} x^{16}}{109395 x^{17}} \] Input:
integrate((b*x**2+a)**8/x**18,x)
Output:
(-6435*a**8 - 58344*a**7*b*x**2 - 235620*a**6*b**2*x**4 - 556920*a**5*b**3 *x**6 - 850850*a**4*b**4*x**8 - 875160*a**3*b**5*x**10 - 612612*a**2*b**6* x**12 - 291720*a*b**7*x**14 - 109395*b**8*x**16)/(109395*x**17)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=-\frac {109395 \, b^{8} x^{16} + 291720 \, a b^{7} x^{14} + 612612 \, a^{2} b^{6} x^{12} + 875160 \, a^{3} b^{5} x^{10} + 850850 \, a^{4} b^{4} x^{8} + 556920 \, a^{5} b^{3} x^{6} + 235620 \, a^{6} b^{2} x^{4} + 58344 \, a^{7} b x^{2} + 6435 \, a^{8}}{109395 \, x^{17}} \] Input:
integrate((b*x^2+a)^8/x^18,x, algorithm="maxima")
Output:
-1/109395*(109395*b^8*x^16 + 291720*a*b^7*x^14 + 612612*a^2*b^6*x^12 + 875 160*a^3*b^5*x^10 + 850850*a^4*b^4*x^8 + 556920*a^5*b^3*x^6 + 235620*a^6*b^ 2*x^4 + 58344*a^7*b*x^2 + 6435*a^8)/x^17
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=-\frac {109395 \, b^{8} x^{16} + 291720 \, a b^{7} x^{14} + 612612 \, a^{2} b^{6} x^{12} + 875160 \, a^{3} b^{5} x^{10} + 850850 \, a^{4} b^{4} x^{8} + 556920 \, a^{5} b^{3} x^{6} + 235620 \, a^{6} b^{2} x^{4} + 58344 \, a^{7} b x^{2} + 6435 \, a^{8}}{109395 \, x^{17}} \] Input:
integrate((b*x^2+a)^8/x^18,x, algorithm="giac")
Output:
-1/109395*(109395*b^8*x^16 + 291720*a*b^7*x^14 + 612612*a^2*b^6*x^12 + 875 160*a^3*b^5*x^10 + 850850*a^4*b^4*x^8 + 556920*a^5*b^3*x^6 + 235620*a^6*b^ 2*x^4 + 58344*a^7*b*x^2 + 6435*a^8)/x^17
Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=-\frac {\frac {a^8}{17}+\frac {8\,a^7\,b\,x^2}{15}+\frac {28\,a^6\,b^2\,x^4}{13}+\frac {56\,a^5\,b^3\,x^6}{11}+\frac {70\,a^4\,b^4\,x^8}{9}+8\,a^3\,b^5\,x^{10}+\frac {28\,a^2\,b^6\,x^{12}}{5}+\frac {8\,a\,b^7\,x^{14}}{3}+b^8\,x^{16}}{x^{17}} \] Input:
int((a + b*x^2)^8/x^18,x)
Output:
-(a^8/17 + b^8*x^16 + (8*a^7*b*x^2)/15 + (8*a*b^7*x^14)/3 + (28*a^6*b^2*x^ 4)/13 + (56*a^5*b^3*x^6)/11 + (70*a^4*b^4*x^8)/9 + 8*a^3*b^5*x^10 + (28*a^ 2*b^6*x^12)/5)/x^17
Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x^{18}} \, dx=\frac {-109395 b^{8} x^{16}-291720 a \,b^{7} x^{14}-612612 a^{2} b^{6} x^{12}-875160 a^{3} b^{5} x^{10}-850850 a^{4} b^{4} x^{8}-556920 a^{5} b^{3} x^{6}-235620 a^{6} b^{2} x^{4}-58344 a^{7} b \,x^{2}-6435 a^{8}}{109395 x^{17}} \] Input:
int((b*x^2+a)^8/x^18,x)
Output:
( - 6435*a**8 - 58344*a**7*b*x**2 - 235620*a**6*b**2*x**4 - 556920*a**5*b* *3*x**6 - 850850*a**4*b**4*x**8 - 875160*a**3*b**5*x**10 - 612612*a**2*b** 6*x**12 - 291720*a*b**7*x**14 - 109395*b**8*x**16)/(109395*x**17)