Integrand size = 13, antiderivative size = 101 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=-\frac {a^8}{4 x^4}-\frac {4 a^7 b}{x^2}+28 a^5 b^3 x^2+\frac {35}{2} a^4 b^4 x^4+\frac {28}{3} a^3 b^5 x^6+\frac {7}{2} a^2 b^6 x^8+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{12}+28 a^6 b^2 \log (x) \] Output:
-1/4*a^8/x^4-4*a^7*b/x^2+28*a^5*b^3*x^2+35/2*a^4*b^4*x^4+28/3*a^3*b^5*x^6+ 7/2*a^2*b^6*x^8+4/5*a*b^7*x^10+1/12*b^8*x^12+28*a^6*b^2*ln(x)
Time = 0.00 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=-\frac {a^8}{4 x^4}-\frac {4 a^7 b}{x^2}+28 a^5 b^3 x^2+\frac {35}{2} a^4 b^4 x^4+\frac {28}{3} a^3 b^5 x^6+\frac {7}{2} a^2 b^6 x^8+\frac {4}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{12}+28 a^6 b^2 \log (x) \] Input:
Integrate[(a + b*x^2)^8/x^5,x]
Output:
-1/4*a^8/x^4 - (4*a^7*b)/x^2 + 28*a^5*b^3*x^2 + (35*a^4*b^4*x^4)/2 + (28*a ^3*b^5*x^6)/3 + (7*a^2*b^6*x^8)/2 + (4*a*b^7*x^10)/5 + (b^8*x^12)/12 + 28* a^6*b^2*Log[x]
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^6}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (b^8 x^{10}+8 a b^7 x^8+28 a^2 b^6 x^6+56 a^3 b^5 x^4+70 a^4 b^4 x^2+56 a^5 b^3+\frac {28 a^6 b^2}{x^2}+\frac {8 a^7 b}{x^4}+\frac {a^8}{x^6}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {a^8}{2 x^4}-\frac {8 a^7 b}{x^2}+28 a^6 b^2 \log \left (x^2\right )+56 a^5 b^3 x^2+35 a^4 b^4 x^4+\frac {56}{3} a^3 b^5 x^6+7 a^2 b^6 x^8+\frac {8}{5} a b^7 x^{10}+\frac {b^8 x^{12}}{6}\right )\) |
Input:
Int[(a + b*x^2)^8/x^5,x]
Output:
(-1/2*a^8/x^4 - (8*a^7*b)/x^2 + 56*a^5*b^3*x^2 + 35*a^4*b^4*x^4 + (56*a^3* b^5*x^6)/3 + 7*a^2*b^6*x^8 + (8*a*b^7*x^10)/5 + (b^8*x^12)/6 + 28*a^6*b^2* 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 = 0.89
method | result | size |
default | \(-\frac {a^{8}}{4 x^{4}}-\frac {4 a^{7} b}{x^{2}}+28 a^{5} b^{3} x^{2}+\frac {35 a^{4} x^{4} b^{4}}{2}+\frac {28 a^{3} b^{5} x^{6}}{3}+\frac {7 a^{2} b^{6} x^{8}}{2}+\frac {4 a \,b^{7} x^{10}}{5}+\frac {b^{8} x^{12}}{12}+28 a^{6} b^{2} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{4} a^{8}+\frac {1}{12} b^{8} x^{16}+\frac {4}{5} a \,b^{7} x^{14}+\frac {7}{2} a^{2} b^{6} x^{12}+\frac {28}{3} a^{3} b^{5} x^{10}+\frac {35}{2} a^{4} b^{4} x^{8}+28 a^{5} b^{3} x^{6}-4 a^{7} b \,x^{2}}{x^{4}}+28 a^{6} b^{2} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{12}}{12}+\frac {4 a \,b^{7} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{8}}{2}+\frac {28 a^{3} b^{5} x^{6}}{3}+\frac {35 a^{4} x^{4} b^{4}}{2}+28 a^{5} b^{3} x^{2}+\frac {-4 a^{7} b \,x^{2}-\frac {1}{4} a^{8}}{x^{4}}+28 a^{6} b^{2} \ln \left (x \right )\) | \(92\) |
parallelrisch | \(\frac {5 b^{8} x^{16}+48 a \,b^{7} x^{14}+210 a^{2} b^{6} x^{12}+560 a^{3} b^{5} x^{10}+1050 a^{4} b^{4} x^{8}+1680 a^{5} b^{3} x^{6}+1680 a^{6} b^{2} \ln \left (x \right ) x^{4}-240 a^{7} b \,x^{2}-15 a^{8}}{60 x^{4}}\) | \(95\) |
Input:
int((b*x^2+a)^8/x^5,x,method=_RETURNVERBOSE)
Output:
-1/4*a^8/x^4-4*a^7*b/x^2+28*a^5*b^3*x^2+35/2*a^4*x^4*b^4+28/3*a^3*b^5*x^6+ 7/2*a^2*b^6*x^8+4/5*a*b^7*x^10+1/12*b^8*x^12+28*a^6*b^2*ln(x)
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=\frac {5 \, b^{8} x^{16} + 48 \, a b^{7} x^{14} + 210 \, a^{2} b^{6} x^{12} + 560 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} + 1680 \, a^{6} b^{2} x^{4} \log \left (x\right ) - 240 \, a^{7} b x^{2} - 15 \, a^{8}}{60 \, x^{4}} \] Input:
integrate((b*x^2+a)^8/x^5,x, algorithm="fricas")
Output:
1/60*(5*b^8*x^16 + 48*a*b^7*x^14 + 210*a^2*b^6*x^12 + 560*a^3*b^5*x^10 + 1 050*a^4*b^4*x^8 + 1680*a^5*b^3*x^6 + 1680*a^6*b^2*x^4*log(x) - 240*a^7*b*x ^2 - 15*a^8)/x^4
Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=28 a^{6} b^{2} \log {\left (x \right )} + 28 a^{5} b^{3} x^{2} + \frac {35 a^{4} b^{4} x^{4}}{2} + \frac {28 a^{3} b^{5} x^{6}}{3} + \frac {7 a^{2} b^{6} x^{8}}{2} + \frac {4 a b^{7} x^{10}}{5} + \frac {b^{8} x^{12}}{12} + \frac {- a^{8} - 16 a^{7} b x^{2}}{4 x^{4}} \] Input:
integrate((b*x**2+a)**8/x**5,x)
Output:
28*a**6*b**2*log(x) + 28*a**5*b**3*x**2 + 35*a**4*b**4*x**4/2 + 28*a**3*b* *5*x**6/3 + 7*a**2*b**6*x**8/2 + 4*a*b**7*x**10/5 + b**8*x**12/12 + (-a**8 - 16*a**7*b*x**2)/(4*x**4)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=\frac {1}{12} \, b^{8} x^{12} + \frac {4}{5} \, a b^{7} x^{10} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {35}{2} \, a^{4} b^{4} x^{4} + 28 \, a^{5} b^{3} x^{2} + 14 \, a^{6} b^{2} \log \left (x^{2}\right ) - \frac {16 \, a^{7} b x^{2} + a^{8}}{4 \, x^{4}} \] Input:
integrate((b*x^2+a)^8/x^5,x, algorithm="maxima")
Output:
1/12*b^8*x^12 + 4/5*a*b^7*x^10 + 7/2*a^2*b^6*x^8 + 28/3*a^3*b^5*x^6 + 35/2 *a^4*b^4*x^4 + 28*a^5*b^3*x^2 + 14*a^6*b^2*log(x^2) - 1/4*(16*a^7*b*x^2 + a^8)/x^4
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=\frac {1}{12} \, b^{8} x^{12} + \frac {4}{5} \, a b^{7} x^{10} + \frac {7}{2} \, a^{2} b^{6} x^{8} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {35}{2} \, a^{4} b^{4} x^{4} + 28 \, a^{5} b^{3} x^{2} + 14 \, a^{6} b^{2} \log \left (x^{2}\right ) - \frac {84 \, a^{6} b^{2} x^{4} + 16 \, a^{7} b x^{2} + a^{8}}{4 \, x^{4}} \] Input:
integrate((b*x^2+a)^8/x^5,x, algorithm="giac")
Output:
1/12*b^8*x^12 + 4/5*a*b^7*x^10 + 7/2*a^2*b^6*x^8 + 28/3*a^3*b^5*x^6 + 35/2 *a^4*b^4*x^4 + 28*a^5*b^3*x^2 + 14*a^6*b^2*log(x^2) - 1/4*(84*a^6*b^2*x^4 + 16*a^7*b*x^2 + a^8)/x^4
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=\frac {b^8\,x^{12}}{12}-\frac {\frac {a^8}{4}+4\,b\,a^7\,x^2}{x^4}+\frac {4\,a\,b^7\,x^{10}}{5}+28\,a^5\,b^3\,x^2+\frac {35\,a^4\,b^4\,x^4}{2}+\frac {28\,a^3\,b^5\,x^6}{3}+\frac {7\,a^2\,b^6\,x^8}{2}+28\,a^6\,b^2\,\ln \left (x\right ) \] Input:
int((a + b*x^2)^8/x^5,x)
Output:
(b^8*x^12)/12 - (a^8/4 + 4*a^7*b*x^2)/x^4 + (4*a*b^7*x^10)/5 + 28*a^5*b^3* x^2 + (35*a^4*b^4*x^4)/2 + (28*a^3*b^5*x^6)/3 + (7*a^2*b^6*x^8)/2 + 28*a^6 *b^2*log(x)
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^2\right )^8}{x^5} \, dx=\frac {1680 \,\mathrm {log}\left (x \right ) a^{6} b^{2} x^{4}-15 a^{8}-240 a^{7} b \,x^{2}+1680 a^{5} b^{3} x^{6}+1050 a^{4} b^{4} x^{8}+560 a^{3} b^{5} x^{10}+210 a^{2} b^{6} x^{12}+48 a \,b^{7} x^{14}+5 b^{8} x^{16}}{60 x^{4}} \] Input:
int((b*x^2+a)^8/x^5,x)
Output:
(1680*log(x)*a**6*b**2*x**4 - 15*a**8 - 240*a**7*b*x**2 + 1680*a**5*b**3*x **6 + 1050*a**4*b**4*x**8 + 560*a**3*b**5*x**10 + 210*a**2*b**6*x**12 + 48 *a*b**7*x**14 + 5*b**8*x**16)/(60*x**4)