Integrand size = 13, antiderivative size = 100 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=4 a^7 b x^2+7 a^6 b^2 x^4+\frac {28}{3} a^5 b^3 x^6+\frac {35}{4} a^4 b^4 x^8+\frac {28}{5} a^3 b^5 x^{10}+\frac {7}{3} a^2 b^6 x^{12}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{16}+a^8 \log (x) \] Output:
4*a^7*b*x^2+7*a^6*b^2*x^4+28/3*a^5*b^3*x^6+35/4*a^4*b^4*x^8+28/5*a^3*b^5*x ^10+7/3*a^2*b^6*x^12+4/7*a*b^7*x^14+1/16*b^8*x^16+a^8*ln(x)
Time = 0.00 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=4 a^7 b x^2+7 a^6 b^2 x^4+\frac {28}{3} a^5 b^3 x^6+\frac {35}{4} a^4 b^4 x^8+\frac {28}{5} a^3 b^5 x^{10}+\frac {7}{3} a^2 b^6 x^{12}+\frac {4}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{16}+a^8 \log (x) \] Input:
Integrate[(a + b*x^2)^8/x,x]
Output:
4*a^7*b*x^2 + 7*a^6*b^2*x^4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (2 8*a^3*b^5*x^10)/5 + (7*a^2*b^6*x^12)/3 + (4*a*b^7*x^14)/7 + (b^8*x^16)/16 + a^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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^8}{x} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^8}{x^2}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \int \left (b^8 x^{14}+8 a b^7 x^{12}+28 a^2 b^6 x^{10}+56 a^3 b^5 x^8+70 a^4 b^4 x^6+56 a^5 b^3 x^4+28 a^6 b^2 x^2+8 a^7 b+\frac {a^8}{x^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (a^8 \log \left (x^2\right )+8 a^7 b x^2+14 a^6 b^2 x^4+\frac {56}{3} a^5 b^3 x^6+\frac {35}{2} a^4 b^4 x^8+\frac {56}{5} a^3 b^5 x^{10}+\frac {14}{3} a^2 b^6 x^{12}+\frac {8}{7} a b^7 x^{14}+\frac {b^8 x^{16}}{8}\right )\) |
Input:
Int[(a + b*x^2)^8/x,x]
Output:
(8*a^7*b*x^2 + 14*a^6*b^2*x^4 + (56*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/2 + (56*a^3*b^5*x^10)/5 + (14*a^2*b^6*x^12)/3 + (8*a*b^7*x^14)/7 + (b^8*x^16)/ 8 + a^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 = 0.38 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.89
method | result | size |
default | \(4 a^{7} b \,x^{2}+7 a^{6} b^{2} x^{4}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {b^{8} x^{16}}{16}+a^{8} \ln \left (x \right )\) | \(89\) |
norman | \(4 a^{7} b \,x^{2}+7 a^{6} b^{2} x^{4}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {b^{8} x^{16}}{16}+a^{8} \ln \left (x \right )\) | \(89\) |
parallelrisch | \(4 a^{7} b \,x^{2}+7 a^{6} b^{2} x^{4}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {b^{8} x^{16}}{16}+a^{8} \ln \left (x \right )\) | \(89\) |
risch | \(4 a^{7} b \,x^{2}+\frac {176 a^{8}}{105}+a^{8} \ln \left (x \right )+\frac {b^{8} x^{16}}{16}+7 a^{6} b^{2} x^{4}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {4 a \,b^{7} x^{14}}{7}\) | \(94\) |
Input:
int((b*x^2+a)^8/x,x,method=_RETURNVERBOSE)
Output:
4*a^7*b*x^2+7*a^6*b^2*x^4+28/3*a^5*b^3*x^6+35/4*a^4*b^4*x^8+28/5*a^3*b^5*x ^10+7/3*a^2*b^6*x^12+4/7*a*b^7*x^14+1/16*b^8*x^16+a^8*ln(x)
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=\frac {1}{16} \, b^{8} x^{16} + \frac {4}{7} \, a b^{7} x^{14} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {28}{5} \, a^{3} b^{5} x^{10} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8} \log \left (x\right ) \] Input:
integrate((b*x^2+a)^8/x,x, algorithm="fricas")
Output:
1/16*b^8*x^16 + 4/7*a*b^7*x^14 + 7/3*a^2*b^6*x^12 + 28/5*a^3*b^5*x^10 + 35 /4*a^4*b^4*x^8 + 28/3*a^5*b^3*x^6 + 7*a^6*b^2*x^4 + 4*a^7*b*x^2 + a^8*log( x)
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=a^{8} \log {\left (x \right )} + 4 a^{7} b x^{2} + 7 a^{6} b^{2} x^{4} + \frac {28 a^{5} b^{3} x^{6}}{3} + \frac {35 a^{4} b^{4} x^{8}}{4} + \frac {28 a^{3} b^{5} x^{10}}{5} + \frac {7 a^{2} b^{6} x^{12}}{3} + \frac {4 a b^{7} x^{14}}{7} + \frac {b^{8} x^{16}}{16} \] Input:
integrate((b*x**2+a)**8/x,x)
Output:
a**8*log(x) + 4*a**7*b*x**2 + 7*a**6*b**2*x**4 + 28*a**5*b**3*x**6/3 + 35* a**4*b**4*x**8/4 + 28*a**3*b**5*x**10/5 + 7*a**2*b**6*x**12/3 + 4*a*b**7*x **14/7 + b**8*x**16/16
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=\frac {1}{16} \, b^{8} x^{16} + \frac {4}{7} \, a b^{7} x^{14} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {28}{5} \, a^{3} b^{5} x^{10} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac {1}{2} \, a^{8} \log \left (x^{2}\right ) \] Input:
integrate((b*x^2+a)^8/x,x, algorithm="maxima")
Output:
1/16*b^8*x^16 + 4/7*a*b^7*x^14 + 7/3*a^2*b^6*x^12 + 28/5*a^3*b^5*x^10 + 35 /4*a^4*b^4*x^8 + 28/3*a^5*b^3*x^6 + 7*a^6*b^2*x^4 + 4*a^7*b*x^2 + 1/2*a^8* log(x^2)
Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=\frac {1}{16} \, b^{8} x^{16} + \frac {4}{7} \, a b^{7} x^{14} + \frac {7}{3} \, a^{2} b^{6} x^{12} + \frac {28}{5} \, a^{3} b^{5} x^{10} + \frac {35}{4} \, a^{4} b^{4} x^{8} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 7 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + \frac {1}{2} \, a^{8} \log \left (x^{2}\right ) \] Input:
integrate((b*x^2+a)^8/x,x, algorithm="giac")
Output:
1/16*b^8*x^16 + 4/7*a*b^7*x^14 + 7/3*a^2*b^6*x^12 + 28/5*a^3*b^5*x^10 + 35 /4*a^4*b^4*x^8 + 28/3*a^5*b^3*x^6 + 7*a^6*b^2*x^4 + 4*a^7*b*x^2 + 1/2*a^8* log(x^2)
Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=a^8\,\ln \left (x\right )+\frac {b^8\,x^{16}}{16}+4\,a^7\,b\,x^2+\frac {4\,a\,b^7\,x^{14}}{7}+7\,a^6\,b^2\,x^4+\frac {28\,a^5\,b^3\,x^6}{3}+\frac {35\,a^4\,b^4\,x^8}{4}+\frac {28\,a^3\,b^5\,x^{10}}{5}+\frac {7\,a^2\,b^6\,x^{12}}{3} \] Input:
int((a + b*x^2)^8/x,x)
Output:
a^8*log(x) + (b^8*x^16)/16 + 4*a^7*b*x^2 + (4*a*b^7*x^14)/7 + 7*a^6*b^2*x^ 4 + (28*a^5*b^3*x^6)/3 + (35*a^4*b^4*x^8)/4 + (28*a^3*b^5*x^10)/5 + (7*a^2 *b^6*x^12)/3
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^2\right )^8}{x} \, dx=\mathrm {log}\left (x \right ) a^{8}+4 a^{7} b \,x^{2}+7 a^{6} b^{2} x^{4}+\frac {28 a^{5} b^{3} x^{6}}{3}+\frac {35 a^{4} b^{4} x^{8}}{4}+\frac {28 a^{3} b^{5} x^{10}}{5}+\frac {7 a^{2} b^{6} x^{12}}{3}+\frac {4 a \,b^{7} x^{14}}{7}+\frac {b^{8} x^{16}}{16} \] Input:
int((b*x^2+a)^8/x,x)
Output:
(1680*log(x)*a**8 + 6720*a**7*b*x**2 + 11760*a**6*b**2*x**4 + 15680*a**5*b **3*x**6 + 14700*a**4*b**4*x**8 + 9408*a**3*b**5*x**10 + 3920*a**2*b**6*x* *12 + 960*a*b**7*x**14 + 105*b**8*x**16)/1680