Integrand size = 20, antiderivative size = 88 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {5}{2} a^4 A b x^2+\frac {5}{2} a^3 A b^2 x^4+\frac {5}{3} a^2 A b^3 x^6+\frac {5}{8} a A b^4 x^8+\frac {1}{10} A b^5 x^{10}+\frac {B \left (a+b x^2\right )^6}{12 b}+a^5 A \log (x) \] Output:
5/2*a^4*A*b*x^2+5/2*a^3*A*b^2*x^4+5/3*a^2*A*b^3*x^6+5/8*a*A*b^4*x^8+1/10*A *b^5*x^10+1/12*B*(b*x^2+a)^6/b+a^5*A*ln(x)
Time = 0.02 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{2} a^4 (5 A b+a B) x^2+\frac {5}{4} a^3 b (2 A b+a B) x^4+\frac {5}{3} a^2 b^2 (A b+a B) x^6+\frac {5}{8} a b^3 (A b+2 a B) x^8+\frac {1}{10} b^4 (A b+5 a B) x^{10}+\frac {1}{12} b^5 B x^{12}+a^5 A \log (x) \] Input:
Integrate[((a + b*x^2)^5*(A + B*x^2))/x,x]
Output:
(a^4*(5*A*b + a*B)*x^2)/2 + (5*a^3*b*(2*A*b + a*B)*x^4)/4 + (5*a^2*b^2*(A* b + a*B)*x^6)/3 + (5*a*b^3*(A*b + 2*a*B)*x^8)/8 + (b^4*(A*b + 5*a*B)*x^10) /10 + (b^5*B*x^12)/12 + a^5*A*Log[x]
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {354, 90, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^2+a\right )^5 \left (B x^2+A\right )}{x^2}dx^2\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{2} \left (A \int \frac {\left (b x^2+a\right )^5}{x^2}dx^2+\frac {B \left (a+b x^2\right )^6}{6 b}\right )\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{2} \left (A \int \left (b^5 x^8+5 a b^4 x^6+10 a^2 b^3 x^4+10 a^3 b^2 x^2+5 a^4 b+\frac {a^5}{x^2}\right )dx^2+\frac {B \left (a+b x^2\right )^6}{6 b}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (A \left (a^5 \log \left (x^2\right )+5 a^4 b x^2+5 a^3 b^2 x^4+\frac {10}{3} a^2 b^3 x^6+\frac {5}{4} a b^4 x^8+\frac {b^5 x^{10}}{5}\right )+\frac {B \left (a+b x^2\right )^6}{6 b}\right )\) |
Input:
Int[((a + b*x^2)^5*(A + B*x^2))/x,x]
Output:
((B*(a + b*x^2)^6)/(6*b) + A*(5*a^4*b*x^2 + 5*a^3*b^2*x^4 + (10*a^2*b^3*x^ 6)/3 + (5*a*b^4*x^8)/4 + (b^5*x^10)/5 + a^5*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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Time = 0.34 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.35
method | result | size |
norman | \(\left (\frac {1}{10} b^{5} A +\frac {1}{2} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{8} a \,b^{4} A +\frac {5}{4} a^{2} b^{3} B \right ) x^{8}+\left (\frac {5}{3} a^{2} b^{3} A +\frac {5}{3} a^{3} b^{2} B \right ) x^{6}+\left (\frac {5}{2} a^{3} b^{2} A +\frac {5}{4} a^{4} b B \right ) x^{4}+\left (\frac {5}{2} a^{4} b A +\frac {1}{2} a^{5} B \right ) x^{2}+\frac {b^{5} B \,x^{12}}{12}+a^{5} A \ln \left (x \right )\) | \(119\) |
default | \(\frac {b^{5} B \,x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 a^{2} A \,b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 a^{3} A \,b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 a^{4} A b \,x^{2}}{2}+\frac {B \,a^{5} x^{2}}{2}+a^{5} A \ln \left (x \right )\) | \(124\) |
risch | \(\frac {b^{5} B \,x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 a^{2} A \,b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 a^{3} A \,b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 a^{4} A b \,x^{2}}{2}+\frac {B \,a^{5} x^{2}}{2}+a^{5} A \ln \left (x \right )\) | \(124\) |
parallelrisch | \(\frac {b^{5} B \,x^{12}}{12}+\frac {A \,b^{5} x^{10}}{10}+\frac {B a \,b^{4} x^{10}}{2}+\frac {5 a A \,b^{4} x^{8}}{8}+\frac {5 B \,a^{2} b^{3} x^{8}}{4}+\frac {5 a^{2} A \,b^{3} x^{6}}{3}+\frac {5 B \,a^{3} b^{2} x^{6}}{3}+\frac {5 a^{3} A \,b^{2} x^{4}}{2}+\frac {5 B \,a^{4} b \,x^{4}}{4}+\frac {5 a^{4} A b \,x^{2}}{2}+\frac {B \,a^{5} x^{2}}{2}+a^{5} A \ln \left (x \right )\) | \(124\) |
Input:
int((b*x^2+a)^5*(B*x^2+A)/x,x,method=_RETURNVERBOSE)
Output:
(1/10*b^5*A+1/2*a*b^4*B)*x^10+(5/8*a*b^4*A+5/4*a^2*b^3*B)*x^8+(5/3*a^2*b^3 *A+5/3*a^3*b^2*B)*x^6+(5/2*a^3*b^2*A+5/4*a^4*b*B)*x^4+(5/2*a^4*b*A+1/2*a^5 *B)*x^2+1/12*b^5*B*x^12+a^5*A*ln(x)
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + A a^{5} \log \left (x\right ) + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \] Input:
integrate((b*x^2+a)^5*(B*x^2+A)/x,x, algorithm="fricas")
Output:
1/12*B*b^5*x^12 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/8*(2*B*a^2*b^3 + A*a*b ^4)*x^8 + 5/3*(B*a^3*b^2 + A*a^2*b^3)*x^6 + A*a^5*log(x) + 5/4*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 1/2*(B*a^5 + 5*A*a^4*b)*x^2
Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=A a^{5} \log {\left (x \right )} + \frac {B b^{5} x^{12}}{12} + x^{10} \left (\frac {A b^{5}}{10} + \frac {B a b^{4}}{2}\right ) + x^{8} \cdot \left (\frac {5 A a b^{4}}{8} + \frac {5 B a^{2} b^{3}}{4}\right ) + x^{6} \cdot \left (\frac {5 A a^{2} b^{3}}{3} + \frac {5 B a^{3} b^{2}}{3}\right ) + x^{4} \cdot \left (\frac {5 A a^{3} b^{2}}{2} + \frac {5 B a^{4} b}{4}\right ) + x^{2} \cdot \left (\frac {5 A a^{4} b}{2} + \frac {B a^{5}}{2}\right ) \] Input:
integrate((b*x**2+a)**5*(B*x**2+A)/x,x)
Output:
A*a**5*log(x) + B*b**5*x**12/12 + x**10*(A*b**5/10 + B*a*b**4/2) + x**8*(5 *A*a*b**4/8 + 5*B*a**2*b**3/4) + x**6*(5*A*a**2*b**3/3 + 5*B*a**3*b**2/3) + x**4*(5*A*a**3*b**2/2 + 5*B*a**4*b/4) + x**2*(5*A*a**4*b/2 + B*a**5/2)
Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{10} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac {5}{8} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + \frac {1}{2} \, A a^{5} \log \left (x^{2}\right ) + \frac {5}{4} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + \frac {1}{2} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \] Input:
integrate((b*x^2+a)^5*(B*x^2+A)/x,x, algorithm="maxima")
Output:
1/12*B*b^5*x^12 + 1/10*(5*B*a*b^4 + A*b^5)*x^10 + 5/8*(2*B*a^2*b^3 + A*a*b ^4)*x^8 + 5/3*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 1/2*A*a^5*log(x^2) + 5/4*(B*a^ 4*b + 2*A*a^3*b^2)*x^4 + 1/2*(B*a^5 + 5*A*a^4*b)*x^2
Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\frac {1}{12} \, B b^{5} x^{12} + \frac {1}{2} \, B a b^{4} x^{10} + \frac {1}{10} \, A b^{5} x^{10} + \frac {5}{4} \, B a^{2} b^{3} x^{8} + \frac {5}{8} \, A a b^{4} x^{8} + \frac {5}{3} \, B a^{3} b^{2} x^{6} + \frac {5}{3} \, A a^{2} b^{3} x^{6} + \frac {5}{4} \, B a^{4} b x^{4} + \frac {5}{2} \, A a^{3} b^{2} x^{4} + \frac {1}{2} \, B a^{5} x^{2} + \frac {5}{2} \, A a^{4} b x^{2} + \frac {1}{2} \, A a^{5} \log \left (x^{2}\right ) \] Input:
integrate((b*x^2+a)^5*(B*x^2+A)/x,x, algorithm="giac")
Output:
1/12*B*b^5*x^12 + 1/2*B*a*b^4*x^10 + 1/10*A*b^5*x^10 + 5/4*B*a^2*b^3*x^8 + 5/8*A*a*b^4*x^8 + 5/3*B*a^3*b^2*x^6 + 5/3*A*a^2*b^3*x^6 + 5/4*B*a^4*b*x^4 + 5/2*A*a^3*b^2*x^4 + 1/2*B*a^5*x^2 + 5/2*A*a^4*b*x^2 + 1/2*A*a^5*log(x^2 )
Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=x^2\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )+x^{10}\,\left (\frac {A\,b^5}{10}+\frac {B\,a\,b^4}{2}\right )+\frac {B\,b^5\,x^{12}}{12}+A\,a^5\,\ln \left (x\right )+\frac {5\,a^2\,b^2\,x^6\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^4\,\left (2\,A\,b+B\,a\right )}{4}+\frac {5\,a\,b^3\,x^8\,\left (A\,b+2\,B\,a\right )}{8} \] Input:
int(((A + B*x^2)*(a + b*x^2)^5)/x,x)
Output:
x^2*((B*a^5)/2 + (5*A*a^4*b)/2) + x^10*((A*b^5)/10 + (B*a*b^4)/2) + (B*b^5 *x^12)/12 + A*a^5*log(x) + (5*a^2*b^2*x^6*(A*b + B*a))/3 + (5*a^3*b*x^4*(2 *A*b + B*a))/4 + (5*a*b^3*x^8*(A*b + 2*B*a))/8
Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x} \, dx=\mathrm {log}\left (x \right ) a^{6}+3 a^{5} b \,x^{2}+\frac {15 a^{4} b^{2} x^{4}}{4}+\frac {10 a^{3} b^{3} x^{6}}{3}+\frac {15 a^{2} b^{4} x^{8}}{8}+\frac {3 a \,b^{5} x^{10}}{5}+\frac {b^{6} x^{12}}{12} \] Input:
int((b*x^2+a)^5*(B*x^2+A)/x,x)
Output:
(120*log(x)*a**6 + 360*a**5*b*x**2 + 450*a**4*b**2*x**4 + 400*a**3*b**3*x* *6 + 225*a**2*b**4*x**8 + 72*a*b**5*x**10 + 10*b**6*x**12)/120