Integrand size = 20, antiderivative size = 114 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=2 a^3 c d x+\frac {3}{2} a^2 b c^2 x^2+2 a^2 b c d x^3+\frac {3}{4} a b^2 c^2 x^4+\frac {6}{5} a b^2 c d x^5+\frac {1}{6} b^3 c^2 x^6+\frac {2}{7} b^3 c d x^7+\frac {d^2 \left (a+b x^2\right )^4}{8 b}+a^3 c^2 \log (x) \] Output:
2*a^3*c*d*x+3/2*a^2*b*c^2*x^2+2*a^2*b*c*d*x^3+3/4*a*b^2*c^2*x^4+6/5*a*b^2* c*d*x^5+1/6*b^3*c^2*x^6+2/7*b^3*c*d*x^7+1/8*d^2*(b*x^2+a)^4/b+a^3*c^2*ln(x )
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=\frac {1}{840} x \left (420 a^3 d (4 c+d x)+210 a^2 b x \left (6 c^2+8 c d x+3 d^2 x^2\right )+42 a b^2 x^3 \left (15 c^2+24 c d x+10 d^2 x^2\right )+5 b^3 x^5 \left (28 c^2+48 c d x+21 d^2 x^2\right )\right )+a^3 c^2 \log (x) \] Input:
Integrate[((c + d*x)^2*(a + b*x^2)^3)/x,x]
Output:
(x*(420*a^3*d*(4*c + d*x) + 210*a^2*b*x*(6*c^2 + 8*c*d*x + 3*d^2*x^2) + 42 *a*b^2*x^3*(15*c^2 + 24*c*d*x + 10*d^2*x^2) + 5*b^3*x^5*(28*c^2 + 48*c*d*x + 21*d^2*x^2)))/840 + a^3*c^2*Log[x]
Time = 0.46 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^3 (c+d x)^2}{x} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {a^3 c^2}{x}+2 a^3 c d+a^2 x \left (a d^2+3 b c^2\right )+6 a^2 b c d x^2+b^2 x^5 \left (3 a d^2+b c^2\right )+6 a b^2 c d x^4+3 a b x^3 \left (a d^2+b c^2\right )+2 b^3 c d x^6+b^3 d^2 x^7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^3 c^2 \log (x)+2 a^3 c d x+\frac {1}{2} a^2 x^2 \left (a d^2+3 b c^2\right )+2 a^2 b c d x^3+\frac {1}{6} b^2 x^6 \left (3 a d^2+b c^2\right )+\frac {6}{5} a b^2 c d x^5+\frac {3}{4} a b x^4 \left (a d^2+b c^2\right )+\frac {2}{7} b^3 c d x^7+\frac {1}{8} b^3 d^2 x^8\) |
Input:
Int[((c + d*x)^2*(a + b*x^2)^3)/x,x]
Output:
2*a^3*c*d*x + (a^2*(3*b*c^2 + a*d^2)*x^2)/2 + 2*a^2*b*c*d*x^3 + (3*a*b*(b* c^2 + a*d^2)*x^4)/4 + (6*a*b^2*c*d*x^5)/5 + (b^2*(b*c^2 + 3*a*d^2)*x^6)/6 + (2*b^3*c*d*x^7)/7 + (b^3*d^2*x^8)/8 + a^3*c^2*Log[x]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.13
method | result | size |
norman | \(\left (\frac {1}{2} a^{3} d^{2}+\frac {3}{2} a^{2} b \,c^{2}\right ) x^{2}+\left (\frac {1}{2} a \,b^{2} d^{2}+\frac {1}{6} b^{3} c^{2}\right ) x^{6}+\left (\frac {3}{4} a^{2} b \,d^{2}+\frac {3}{4} a \,c^{2} b^{2}\right ) x^{4}+\frac {b^{3} d^{2} x^{8}}{8}+2 a^{3} c d x +\frac {2 d \,x^{7} c \,b^{3}}{7}+\frac {6 a \,b^{2} c d \,x^{5}}{5}+2 a^{2} b c d \,x^{3}+a^{3} c^{2} \ln \left (x \right )\) | \(129\) |
default | \(\frac {b^{3} d^{2} x^{8}}{8}+\frac {2 d \,x^{7} c \,b^{3}}{7}+\frac {a \,b^{2} d^{2} x^{6}}{2}+\frac {b^{3} c^{2} x^{6}}{6}+\frac {6 a \,b^{2} c d \,x^{5}}{5}+\frac {3 a^{2} b \,d^{2} x^{4}}{4}+\frac {3 a \,b^{2} c^{2} x^{4}}{4}+2 a^{2} b c d \,x^{3}+\frac {a^{3} d^{2} x^{2}}{2}+\frac {3 a^{2} b \,c^{2} x^{2}}{2}+2 a^{3} c d x +a^{3} c^{2} \ln \left (x \right )\) | \(132\) |
risch | \(\frac {b^{3} d^{2} x^{8}}{8}+\frac {2 d \,x^{7} c \,b^{3}}{7}+\frac {a \,b^{2} d^{2} x^{6}}{2}+\frac {b^{3} c^{2} x^{6}}{6}+\frac {6 a \,b^{2} c d \,x^{5}}{5}+\frac {3 a^{2} b \,d^{2} x^{4}}{4}+\frac {3 a \,b^{2} c^{2} x^{4}}{4}+2 a^{2} b c d \,x^{3}+\frac {a^{3} d^{2} x^{2}}{2}+\frac {3 a^{2} b \,c^{2} x^{2}}{2}+2 a^{3} c d x +a^{3} c^{2} \ln \left (x \right )\) | \(132\) |
parallelrisch | \(\frac {b^{3} d^{2} x^{8}}{8}+\frac {2 d \,x^{7} c \,b^{3}}{7}+\frac {a \,b^{2} d^{2} x^{6}}{2}+\frac {b^{3} c^{2} x^{6}}{6}+\frac {6 a \,b^{2} c d \,x^{5}}{5}+\frac {3 a^{2} b \,d^{2} x^{4}}{4}+\frac {3 a \,b^{2} c^{2} x^{4}}{4}+2 a^{2} b c d \,x^{3}+\frac {a^{3} d^{2} x^{2}}{2}+\frac {3 a^{2} b \,c^{2} x^{2}}{2}+2 a^{3} c d x +a^{3} c^{2} \ln \left (x \right )\) | \(132\) |
Input:
int((d*x+c)^2*(b*x^2+a)^3/x,x,method=_RETURNVERBOSE)
Output:
(1/2*a^3*d^2+3/2*a^2*b*c^2)*x^2+(1/2*a*b^2*d^2+1/6*b^3*c^2)*x^6+(3/4*a^2*b *d^2+3/4*a*c^2*b^2)*x^4+1/8*b^3*d^2*x^8+2*a^3*c*d*x+2/7*d*x^7*c*b^3+6/5*a* b^2*c*d*x^5+2*a^2*b*c*d*x^3+a^3*c^2*ln(x)
Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=\frac {1}{8} \, b^{3} d^{2} x^{8} + \frac {2}{7} \, b^{3} c d x^{7} + \frac {6}{5} \, a b^{2} c d x^{5} + 2 \, a^{2} b c d x^{3} + \frac {1}{6} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{6} + 2 \, a^{3} c d x + a^{3} c^{2} \log \left (x\right ) + \frac {3}{4} \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{4} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{2} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^3/x,x, algorithm="fricas")
Output:
1/8*b^3*d^2*x^8 + 2/7*b^3*c*d*x^7 + 6/5*a*b^2*c*d*x^5 + 2*a^2*b*c*d*x^3 + 1/6*(b^3*c^2 + 3*a*b^2*d^2)*x^6 + 2*a^3*c*d*x + a^3*c^2*log(x) + 3/4*(a*b^ 2*c^2 + a^2*b*d^2)*x^4 + 1/2*(3*a^2*b*c^2 + a^3*d^2)*x^2
Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.28 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=a^{3} c^{2} \log {\left (x \right )} + 2 a^{3} c d x + 2 a^{2} b c d x^{3} + \frac {6 a b^{2} c d x^{5}}{5} + \frac {2 b^{3} c d x^{7}}{7} + \frac {b^{3} d^{2} x^{8}}{8} + x^{6} \left (\frac {a b^{2} d^{2}}{2} + \frac {b^{3} c^{2}}{6}\right ) + x^{4} \cdot \left (\frac {3 a^{2} b d^{2}}{4} + \frac {3 a b^{2} c^{2}}{4}\right ) + x^{2} \left (\frac {a^{3} d^{2}}{2} + \frac {3 a^{2} b c^{2}}{2}\right ) \] Input:
integrate((d*x+c)**2*(b*x**2+a)**3/x,x)
Output:
a**3*c**2*log(x) + 2*a**3*c*d*x + 2*a**2*b*c*d*x**3 + 6*a*b**2*c*d*x**5/5 + 2*b**3*c*d*x**7/7 + b**3*d**2*x**8/8 + x**6*(a*b**2*d**2/2 + b**3*c**2/6 ) + x**4*(3*a**2*b*d**2/4 + 3*a*b**2*c**2/4) + x**2*(a**3*d**2/2 + 3*a**2* b*c**2/2)
Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.11 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=\frac {1}{8} \, b^{3} d^{2} x^{8} + \frac {2}{7} \, b^{3} c d x^{7} + \frac {6}{5} \, a b^{2} c d x^{5} + 2 \, a^{2} b c d x^{3} + \frac {1}{6} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{6} + 2 \, a^{3} c d x + a^{3} c^{2} \log \left (x\right ) + \frac {3}{4} \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{4} + \frac {1}{2} \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{2} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^3/x,x, algorithm="maxima")
Output:
1/8*b^3*d^2*x^8 + 2/7*b^3*c*d*x^7 + 6/5*a*b^2*c*d*x^5 + 2*a^2*b*c*d*x^3 + 1/6*(b^3*c^2 + 3*a*b^2*d^2)*x^6 + 2*a^3*c*d*x + a^3*c^2*log(x) + 3/4*(a*b^ 2*c^2 + a^2*b*d^2)*x^4 + 1/2*(3*a^2*b*c^2 + a^3*d^2)*x^2
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=\frac {1}{8} \, b^{3} d^{2} x^{8} + \frac {2}{7} \, b^{3} c d x^{7} + \frac {1}{6} \, b^{3} c^{2} x^{6} + \frac {1}{2} \, a b^{2} d^{2} x^{6} + \frac {6}{5} \, a b^{2} c d x^{5} + \frac {3}{4} \, a b^{2} c^{2} x^{4} + \frac {3}{4} \, a^{2} b d^{2} x^{4} + 2 \, a^{2} b c d x^{3} + \frac {3}{2} \, a^{2} b c^{2} x^{2} + \frac {1}{2} \, a^{3} d^{2} x^{2} + 2 \, a^{3} c d x + a^{3} c^{2} \log \left ({\left | x \right |}\right ) \] Input:
integrate((d*x+c)^2*(b*x^2+a)^3/x,x, algorithm="giac")
Output:
1/8*b^3*d^2*x^8 + 2/7*b^3*c*d*x^7 + 1/6*b^3*c^2*x^6 + 1/2*a*b^2*d^2*x^6 + 6/5*a*b^2*c*d*x^5 + 3/4*a*b^2*c^2*x^4 + 3/4*a^2*b*d^2*x^4 + 2*a^2*b*c*d*x^ 3 + 3/2*a^2*b*c^2*x^2 + 1/2*a^3*d^2*x^2 + 2*a^3*c*d*x + a^3*c^2*log(abs(x) )
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.08 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=x^2\,\left (\frac {a^3\,d^2}{2}+\frac {3\,b\,a^2\,c^2}{2}\right )+x^6\,\left (\frac {b^3\,c^2}{6}+\frac {a\,b^2\,d^2}{2}\right )+\frac {b^3\,d^2\,x^8}{8}+a^3\,c^2\,\ln \left (x\right )+2\,a^3\,c\,d\,x+\frac {3\,a\,b\,x^4\,\left (b\,c^2+a\,d^2\right )}{4}+\frac {2\,b^3\,c\,d\,x^7}{7}+2\,a^2\,b\,c\,d\,x^3+\frac {6\,a\,b^2\,c\,d\,x^5}{5} \] Input:
int(((a + b*x^2)^3*(c + d*x)^2)/x,x)
Output:
x^2*((a^3*d^2)/2 + (3*a^2*b*c^2)/2) + x^6*((b^3*c^2)/6 + (a*b^2*d^2)/2) + (b^3*d^2*x^8)/8 + a^3*c^2*log(x) + 2*a^3*c*d*x + (3*a*b*x^4*(a*d^2 + b*c^2 ))/4 + (2*b^3*c*d*x^7)/7 + 2*a^2*b*c*d*x^3 + (6*a*b^2*c*d*x^5)/5
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x} \, dx=\mathrm {log}\left (x \right ) a^{3} c^{2}+2 a^{3} c d x +\frac {a^{3} d^{2} x^{2}}{2}+\frac {3 a^{2} b \,c^{2} x^{2}}{2}+2 a^{2} b c d \,x^{3}+\frac {3 a^{2} b \,d^{2} x^{4}}{4}+\frac {3 a \,b^{2} c^{2} x^{4}}{4}+\frac {6 a \,b^{2} c d \,x^{5}}{5}+\frac {a \,b^{2} d^{2} x^{6}}{2}+\frac {b^{3} c^{2} x^{6}}{6}+\frac {2 b^{3} c d \,x^{7}}{7}+\frac {b^{3} d^{2} x^{8}}{8} \] Input:
int((d*x+c)^2*(b*x^2+a)^3/x,x)
Output:
(840*log(x)*a**3*c**2 + 1680*a**3*c*d*x + 420*a**3*d**2*x**2 + 1260*a**2*b *c**2*x**2 + 1680*a**2*b*c*d*x**3 + 630*a**2*b*d**2*x**4 + 630*a*b**2*c**2 *x**4 + 1008*a*b**2*c*d*x**5 + 420*a*b**2*d**2*x**6 + 140*b**3*c**2*x**6 + 240*b**3*c*d*x**7 + 105*b**3*d**2*x**8)/840