Integrand size = 28, antiderivative size = 92 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=a^2 B x+a A b x^2+\frac {1}{3} a (2 b B+a D) x^3+\frac {1}{4} A b^2 x^4+\frac {1}{5} b (b B+2 a D) x^5+\frac {1}{7} b^2 D x^7+\frac {C \left (a+b x^2\right )^3}{6 b}+a^2 A \log (x) \] Output:
a^2*B*x+a*A*b*x^2+1/3*a*(2*B*b+D*a)*x^3+1/4*A*b^2*x^4+1/5*b*(B*b+2*D*a)*x^ 5+1/7*b^2*D*x^7+1/6*C*(b*x^2+a)^3/b+a^2*A*ln(x)
Time = 0.04 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=\frac {1}{420} x \left (70 a^2 (6 B+x (3 C+2 D x))+14 a b x (30 A+x (20 B+3 x (5 C+4 D x)))+b^2 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))\right )+a^2 A \log (x) \] Input:
Integrate[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x,x]
Output:
(x*(70*a^2*(6*B + x*(3*C + 2*D*x)) + 14*a*b*x*(30*A + x*(20*B + 3*x*(5*C + 4*D*x))) + b^2*x^3*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x)))))/420 + a^2*A *Log[x]
Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2018, 2333, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx\) |
\(\Big \downarrow \) 2018 |
\(\displaystyle \int \frac {\left (b x^2+a\right )^2 \left (D x^3+B x+A\right )}{x}dx+\frac {C \left (a+b x^2\right )^3}{6 b}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \int \left (b^2 D x^6+b (b B+2 a D) x^4+A b^2 x^3+a (2 b B+a D) x^2+2 a A b x+a^2 B+\frac {a^2 A}{x}\right )dx+\frac {C \left (a+b x^2\right )^3}{6 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 A \log (x)+a^2 B x+a A b x^2+\frac {1}{5} b x^5 (2 a D+b B)+\frac {1}{3} a x^3 (a D+2 b B)+\frac {C \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} A b^2 x^4+\frac {1}{7} b^2 D x^7\) |
Input:
Int[((a + b*x^2)^2*(A + B*x + C*x^2 + D*x^3))/x,x]
Output:
a^2*B*x + a*A*b*x^2 + (a*(2*b*B + a*D)*x^3)/3 + (A*b^2*x^4)/4 + (b*(b*B + 2*a*D)*x^5)/5 + (b^2*D*x^7)/7 + (C*(a + b*x^2)^3)/(6*b) + a^2*A*Log[x]
Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[Coef f[Px, x, n - m - 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coe ff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a, b, m , n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x , n - m - 1], 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04
method | result | size |
norman | \(\left (\frac {1}{5} B \,b^{2}+\frac {2}{5} a b D\right ) x^{5}+\left (\frac {1}{4} b^{2} A +\frac {1}{2} C a b \right ) x^{4}+\left (a b A +\frac {1}{2} a^{2} C \right ) x^{2}+\left (\frac {2}{3} a b B +\frac {1}{3} D a^{2}\right ) x^{3}+B \,a^{2} x +\frac {b^{2} C \,x^{6}}{6}+\frac {b^{2} D x^{7}}{7}+a^{2} A \ln \left (x \right )\) | \(96\) |
default | \(\frac {b^{2} D x^{7}}{7}+\frac {b^{2} C \,x^{6}}{6}+\frac {b^{2} B \,x^{5}}{5}+\frac {2 D a b \,x^{5}}{5}+\frac {A \,b^{2} x^{4}}{4}+\frac {C a b \,x^{4}}{2}+\frac {2 B a b \,x^{3}}{3}+\frac {D a^{2} x^{3}}{3}+a A b \,x^{2}+\frac {C \,a^{2} x^{2}}{2}+B \,a^{2} x +a^{2} A \ln \left (x \right )\) | \(100\) |
parallelrisch | \(\frac {b^{2} D x^{7}}{7}+\frac {b^{2} C \,x^{6}}{6}+\frac {b^{2} B \,x^{5}}{5}+\frac {2 D a b \,x^{5}}{5}+\frac {A \,b^{2} x^{4}}{4}+\frac {C a b \,x^{4}}{2}+\frac {2 B a b \,x^{3}}{3}+\frac {D a^{2} x^{3}}{3}+a A b \,x^{2}+\frac {C \,a^{2} x^{2}}{2}+B \,a^{2} x +a^{2} A \ln \left (x \right )\) | \(100\) |
Input:
int((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x,x,method=_RETURNVERBOSE)
Output:
(1/5*B*b^2+2/5*a*b*D)*x^5+(1/4*b^2*A+1/2*C*a*b)*x^4+(a*b*A+1/2*a^2*C)*x^2+ (2/3*a*b*B+1/3*D*a^2)*x^3+B*a^2*x+1/6*b^2*C*x^6+1/7*b^2*D*x^7+a^2*A*ln(x)
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=\frac {1}{7} \, D b^{2} x^{7} + \frac {1}{6} \, C b^{2} x^{6} + \frac {1}{5} \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + B a^{2} x + \frac {1}{3} \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac {1}{2} \, {\left (C a^{2} + 2 \, A a b\right )} x^{2} \] Input:
integrate((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x,x, algorithm="fricas")
Output:
1/7*D*b^2*x^7 + 1/6*C*b^2*x^6 + 1/5*(2*D*a*b + B*b^2)*x^5 + 1/4*(2*C*a*b + A*b^2)*x^4 + B*a^2*x + 1/3*(D*a^2 + 2*B*a*b)*x^3 + A*a^2*log(x) + 1/2*(C* a^2 + 2*A*a*b)*x^2
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=A a^{2} \log {\left (x \right )} + B a^{2} x + \frac {C b^{2} x^{6}}{6} + \frac {D b^{2} x^{7}}{7} + x^{5} \left (\frac {B b^{2}}{5} + \frac {2 D a b}{5}\right ) + x^{4} \left (\frac {A b^{2}}{4} + \frac {C a b}{2}\right ) + x^{3} \cdot \left (\frac {2 B a b}{3} + \frac {D a^{2}}{3}\right ) + x^{2} \left (A a b + \frac {C a^{2}}{2}\right ) \] Input:
integrate((b*x**2+a)**2*(D*x**3+C*x**2+B*x+A)/x,x)
Output:
A*a**2*log(x) + B*a**2*x + C*b**2*x**6/6 + D*b**2*x**7/7 + x**5*(B*b**2/5 + 2*D*a*b/5) + x**4*(A*b**2/4 + C*a*b/2) + x**3*(2*B*a*b/3 + D*a**2/3) + x **2*(A*a*b + C*a**2/2)
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=\frac {1}{7} \, D b^{2} x^{7} + \frac {1}{6} \, C b^{2} x^{6} + \frac {1}{5} \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + B a^{2} x + \frac {1}{3} \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} + A a^{2} \log \left (x\right ) + \frac {1}{2} \, {\left (C a^{2} + 2 \, A a b\right )} x^{2} \] Input:
integrate((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x,x, algorithm="maxima")
Output:
1/7*D*b^2*x^7 + 1/6*C*b^2*x^6 + 1/5*(2*D*a*b + B*b^2)*x^5 + 1/4*(2*C*a*b + A*b^2)*x^4 + B*a^2*x + 1/3*(D*a^2 + 2*B*a*b)*x^3 + A*a^2*log(x) + 1/2*(C* a^2 + 2*A*a*b)*x^2
Time = 0.12 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=\frac {1}{7} \, D b^{2} x^{7} + \frac {1}{6} \, C b^{2} x^{6} + \frac {2}{5} \, D a b x^{5} + \frac {1}{5} \, B b^{2} x^{5} + \frac {1}{2} \, C a b x^{4} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{3} \, D a^{2} x^{3} + \frac {2}{3} \, B a b x^{3} + \frac {1}{2} \, C a^{2} x^{2} + A a b x^{2} + B a^{2} x + A a^{2} \log \left ({\left | x \right |}\right ) \] Input:
integrate((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x,x, algorithm="giac")
Output:
1/7*D*b^2*x^7 + 1/6*C*b^2*x^6 + 2/5*D*a*b*x^5 + 1/5*B*b^2*x^5 + 1/2*C*a*b* x^4 + 1/4*A*b^2*x^4 + 1/3*D*a^2*x^3 + 2/3*B*a*b*x^3 + 1/2*C*a^2*x^2 + A*a* b*x^2 + B*a^2*x + A*a^2*log(abs(x))
Time = 2.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=\frac {A\,\left (4\,a^2\,\ln \left (x\right )+b^2\,x^4+4\,a\,b\,x^2\right )}{4}+\frac {B\,x\,\left (15\,a^2+10\,a\,b\,x^2+3\,b^2\,x^4\right )}{15}+\frac {a^2\,x^3\,D}{3}+\frac {b^2\,x^7\,D}{7}+\frac {C\,x^2\,\left (3\,a^2+3\,a\,b\,x^2+b^2\,x^4\right )}{6}+\frac {2\,a\,b\,x^5\,D}{5} \] Input:
int(((a + b*x^2)^2*(A + B*x + C*x^2 + x^3*D))/x,x)
Output:
(A*(4*a^2*log(x) + b^2*x^4 + 4*a*b*x^2))/4 + (B*x*(15*a^2 + 3*b^2*x^4 + 10 *a*b*x^2))/15 + (a^2*x^3*D)/3 + (b^2*x^7*D)/7 + (C*x^2*(3*a^2 + b^2*x^4 + 3*a*b*x^2))/6 + (2*a*b*x^5*D)/5
Time = 0.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx=\mathrm {log}\left (x \right ) a^{3}+a^{2} b \,x^{2}+a^{2} b x +\frac {a^{2} c \,x^{2}}{2}+\frac {a^{2} d \,x^{3}}{3}+\frac {a \,b^{2} x^{4}}{4}+\frac {2 a \,b^{2} x^{3}}{3}+\frac {a b c \,x^{4}}{2}+\frac {2 a b d \,x^{5}}{5}+\frac {b^{3} x^{5}}{5}+\frac {b^{2} c \,x^{6}}{6}+\frac {b^{2} d \,x^{7}}{7} \] Input:
int((b*x^2+a)^2*(D*x^3+C*x^2+B*x+A)/x,x)
Output:
(420*log(x)*a**3 + 420*a**2*b*x**2 + 420*a**2*b*x + 210*a**2*c*x**2 + 140* a**2*d*x**3 + 105*a*b**2*x**4 + 280*a*b**2*x**3 + 210*a*b*c*x**4 + 168*a*b *d*x**5 + 84*b**3*x**5 + 70*b**2*c*x**6 + 60*b**2*d*x**7)/420