Integrand size = 13, antiderivative size = 99 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=-\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+\frac {b^8 x^2}{2}+8 a b^7 \log (x) \] Output:
-1/14*a^8/x^14-2/3*a^7*b/x^12-14/5*a^6*b^2/x^10-7*a^5*b^3/x^8-35/3*a^4*b^4 /x^6-14*a^3*b^5/x^4-14*a^2*b^6/x^2+1/2*b^8*x^2+8*a*b^7*ln(x)
Time = 0.00 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=-\frac {a^8}{14 x^{14}}-\frac {2 a^7 b}{3 x^{12}}-\frac {14 a^6 b^2}{5 x^{10}}-\frac {7 a^5 b^3}{x^8}-\frac {35 a^4 b^4}{3 x^6}-\frac {14 a^3 b^5}{x^4}-\frac {14 a^2 b^6}{x^2}+\frac {b^8 x^2}{2}+8 a b^7 \log (x) \] Input:
Integrate[(a + b*x^2)^8/x^15,x]
Output:
-1/14*a^8/x^14 - (2*a^7*b)/(3*x^12) - (14*a^6*b^2)/(5*x^10) - (7*a^5*b^3)/ x^8 - (35*a^4*b^4)/(3*x^6) - (14*a^3*b^5)/x^4 - (14*a^2*b^6)/x^2 + (b^8*x^ 2)/2 + 8*a*b^7*Log[x]
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.03, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^{16}}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (\frac {a^8}{x^{16}}+\frac {8 b a^7}{x^{14}}+\frac {28 b^2 a^6}{x^{12}}+\frac {56 b^3 a^5}{x^{10}}+\frac {70 b^4 a^4}{x^8}+\frac {56 b^5 a^3}{x^6}+\frac {28 b^6 a^2}{x^4}+\frac {8 b^7 a}{x^2}+b^8\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^8}{7 x^{14}}-\frac {4 a^7 b}{3 x^{12}}-\frac {28 a^6 b^2}{5 x^{10}}-\frac {14 a^5 b^3}{x^8}-\frac {70 a^4 b^4}{3 x^6}-\frac {28 a^3 b^5}{x^4}-\frac {28 a^2 b^6}{x^2}+8 a b^7 \log \left (x^2\right )+b^8 x^2\right )\) |
Input:
Int[(a + b*x^2)^8/x^15,x]
Output:
(-1/7*a^8/x^14 - (4*a^7*b)/(3*x^12) - (28*a^6*b^2)/(5*x^10) - (14*a^5*b^3) /x^8 - (70*a^4*b^4)/(3*x^6) - (28*a^3*b^5)/x^4 - (28*a^2*b^6)/x^2 + b^8*x^ 2 + 8*a*b^7*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 = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {a^{8}}{14 x^{14}}-\frac {2 a^{7} b}{3 x^{12}}-\frac {14 a^{6} b^{2}}{5 x^{10}}-\frac {7 a^{5} b^{3}}{x^{8}}-\frac {35 a^{4} b^{4}}{3 x^{6}}-\frac {14 a^{3} b^{5}}{x^{4}}-\frac {14 a^{2} b^{6}}{x^{2}}+\frac {b^{8} x^{2}}{2}+8 a \,b^{7} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{14} a^{8}+\frac {1}{2} b^{8} x^{16}-14 a^{2} b^{6} x^{12}-14 a^{3} b^{5} x^{10}-\frac {35}{3} a^{4} b^{4} x^{8}-7 a^{5} b^{3} x^{6}-\frac {14}{5} a^{6} b^{2} x^{4}-\frac {2}{3} a^{7} b \,x^{2}}{x^{14}}+8 a \,b^{7} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{2}}{2}+\frac {-14 a^{2} b^{6} x^{12}-14 a^{3} b^{5} x^{10}-\frac {35}{3} a^{4} b^{4} x^{8}-7 a^{5} b^{3} x^{6}-\frac {14}{5} a^{6} b^{2} x^{4}-\frac {2}{3} a^{7} b \,x^{2}-\frac {1}{14} a^{8}}{x^{14}}+8 a \,b^{7} \ln \left (x \right )\) | \(92\) |
parallelrisch | \(\frac {105 b^{8} x^{16}+1680 a \,b^{7} \ln \left (x \right ) x^{14}-2940 a^{2} b^{6} x^{12}-2940 a^{3} b^{5} x^{10}-2450 a^{4} b^{4} x^{8}-1470 a^{5} b^{3} x^{6}-588 a^{6} b^{2} x^{4}-140 a^{7} b \,x^{2}-15 a^{8}}{210 x^{14}}\) | \(95\) |
Input:
int((b*x^2+a)^8/x^15,x,method=_RETURNVERBOSE)
Output:
-1/14*a^8/x^14-2/3*a^7*b/x^12-14/5*a^6*b^2/x^10-7*a^5*b^3/x^8-35/3*a^4*b^4 /x^6-14*a^3*b^5/x^4-14*a^2*b^6/x^2+1/2*b^8*x^2+8*a*b^7*ln(x)
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=\frac {105 \, b^{8} x^{16} + 1680 \, a b^{7} x^{14} \log \left (x\right ) - 2940 \, a^{2} b^{6} x^{12} - 2940 \, a^{3} b^{5} x^{10} - 2450 \, a^{4} b^{4} x^{8} - 1470 \, a^{5} b^{3} x^{6} - 588 \, a^{6} b^{2} x^{4} - 140 \, a^{7} b x^{2} - 15 \, a^{8}}{210 \, x^{14}} \] Input:
integrate((b*x^2+a)^8/x^15,x, algorithm="fricas")
Output:
1/210*(105*b^8*x^16 + 1680*a*b^7*x^14*log(x) - 2940*a^2*b^6*x^12 - 2940*a^ 3*b^5*x^10 - 2450*a^4*b^4*x^8 - 1470*a^5*b^3*x^6 - 588*a^6*b^2*x^4 - 140*a ^7*b*x^2 - 15*a^8)/x^14
Time = 0.43 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=8 a b^{7} \log {\left (x \right )} + \frac {b^{8} x^{2}}{2} + \frac {- 15 a^{8} - 140 a^{7} b x^{2} - 588 a^{6} b^{2} x^{4} - 1470 a^{5} b^{3} x^{6} - 2450 a^{4} b^{4} x^{8} - 2940 a^{3} b^{5} x^{10} - 2940 a^{2} b^{6} x^{12}}{210 x^{14}} \] Input:
integrate((b*x**2+a)**8/x**15,x)
Output:
8*a*b**7*log(x) + b**8*x**2/2 + (-15*a**8 - 140*a**7*b*x**2 - 588*a**6*b** 2*x**4 - 1470*a**5*b**3*x**6 - 2450*a**4*b**4*x**8 - 2940*a**3*b**5*x**10 - 2940*a**2*b**6*x**12)/(210*x**14)
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=\frac {1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac {2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \] Input:
integrate((b*x^2+a)^8/x^15,x, algorithm="maxima")
Output:
1/2*b^8*x^2 + 4*a*b^7*log(x^2) - 1/210*(2940*a^2*b^6*x^12 + 2940*a^3*b^5*x ^10 + 2450*a^4*b^4*x^8 + 1470*a^5*b^3*x^6 + 588*a^6*b^2*x^4 + 140*a^7*b*x^ 2 + 15*a^8)/x^14
Time = 0.13 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=\frac {1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac {2178 \, a b^{7} x^{14} + 2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \] Input:
integrate((b*x^2+a)^8/x^15,x, algorithm="giac")
Output:
1/2*b^8*x^2 + 4*a*b^7*log(x^2) - 1/210*(2178*a*b^7*x^14 + 2940*a^2*b^6*x^1 2 + 2940*a^3*b^5*x^10 + 2450*a^4*b^4*x^8 + 1470*a^5*b^3*x^6 + 588*a^6*b^2* x^4 + 140*a^7*b*x^2 + 15*a^8)/x^14
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=-\frac {\frac {a^8}{14}-\frac {b^8\,x^{16}}{2}+\frac {2\,a^7\,b\,x^2}{3}+\frac {14\,a^6\,b^2\,x^4}{5}+7\,a^5\,b^3\,x^6+\frac {35\,a^4\,b^4\,x^8}{3}+14\,a^3\,b^5\,x^{10}+14\,a^2\,b^6\,x^{12}-8\,a\,b^7\,x^{14}\,\ln \left (x\right )}{x^{14}} \] Input:
int((a + b*x^2)^8/x^15,x)
Output:
-(a^8/14 - (b^8*x^16)/2 + (2*a^7*b*x^2)/3 + (14*a^6*b^2*x^4)/5 + 7*a^5*b^3 *x^6 + (35*a^4*b^4*x^8)/3 + 14*a^3*b^5*x^10 + 14*a^2*b^6*x^12 - 8*a*b^7*x^ 14*log(x))/x^14
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^2\right )^8}{x^{15}} \, dx=\frac {1680 \,\mathrm {log}\left (x \right ) a \,b^{7} x^{14}-15 a^{8}-140 a^{7} b \,x^{2}-588 a^{6} b^{2} x^{4}-1470 a^{5} b^{3} x^{6}-2450 a^{4} b^{4} x^{8}-2940 a^{3} b^{5} x^{10}-2940 a^{2} b^{6} x^{12}+105 b^{8} x^{16}}{210 x^{14}} \] Input:
int((b*x^2+a)^8/x^15,x)
Output:
(1680*log(x)*a*b**7*x**14 - 15*a**8 - 140*a**7*b*x**2 - 588*a**6*b**2*x**4 - 1470*a**5*b**3*x**6 - 2450*a**4*b**4*x**8 - 2940*a**3*b**5*x**10 - 2940 *a**2*b**6*x**12 + 105*b**8*x**16)/(210*x**14)