Integrand size = 20, antiderivative size = 126 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=-\frac {a^3 c^2}{x}+a^2 \left (3 b c^2+a d^2\right ) x+3 a^2 b c d x^2+a b \left (b c^2+a d^2\right ) x^3+\frac {3}{2} a b^2 c d x^4+\frac {1}{5} b^2 \left (b c^2+3 a d^2\right ) x^5+\frac {1}{3} b^3 c d x^6+\frac {1}{7} b^3 d^2 x^7+2 a^3 c d \log (x) \] Output:
-a^3*c^2/x+a^2*(a*d^2+3*b*c^2)*x+3*a^2*b*c*d*x^2+a*b*(a*d^2+b*c^2)*x^3+3/2 *a*b^2*c*d*x^4+1/5*b^2*(3*a*d^2+b*c^2)*x^5+1/3*b^3*c*d*x^6+1/7*b^3*d^2*x^7 +2*a^3*c*d*ln(x)
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=a^3 \left (-\frac {c^2}{x}+d^2 x\right )+a^2 b x \left (3 c^2+3 c d x+d^2 x^2\right )+\frac {1}{10} a b^2 x^3 \left (10 c^2+15 c d x+6 d^2 x^2\right )+\frac {1}{105} b^3 x^5 \left (21 c^2+35 c d x+15 d^2 x^2\right )+2 a^3 c d \log (x) \] Input:
Integrate[((c + d*x)^2*(a + b*x^2)^3)/x^2,x]
Output:
a^3*(-(c^2/x) + d^2*x) + a^2*b*x*(3*c^2 + 3*c*d*x + d^2*x^2) + (a*b^2*x^3* (10*c^2 + 15*c*d*x + 6*d^2*x^2))/10 + (b^3*x^5*(21*c^2 + 35*c*d*x + 15*d^2 *x^2))/105 + 2*a^3*c*d*Log[x]
Time = 0.47 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {\left (a+b x^2\right )^3 (c+d x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 522 |
\(\displaystyle \int \left (\frac {a^3 c^2}{x^2}+\frac {2 a^3 c d}{x}+a^2 \left (a d^2+3 b c^2\right )+6 a^2 b c d x+b^2 x^4 \left (3 a d^2+b c^2\right )+6 a b^2 c d x^3+3 a b x^2 \left (a d^2+b c^2\right )+2 b^3 c d x^5+b^3 d^2 x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 c^2}{x}+2 a^3 c d \log (x)+a^2 x \left (a d^2+3 b c^2\right )+3 a^2 b c d x^2+\frac {1}{5} b^2 x^5 \left (3 a d^2+b c^2\right )+\frac {3}{2} a b^2 c d x^4+a b x^3 \left (a d^2+b c^2\right )+\frac {1}{3} b^3 c d x^6+\frac {1}{7} b^3 d^2 x^7\) |
Input:
Int[((c + d*x)^2*(a + b*x^2)^3)/x^2,x]
Output:
-((a^3*c^2)/x) + a^2*(3*b*c^2 + a*d^2)*x + 3*a^2*b*c*d*x^2 + a*b*(b*c^2 + a*d^2)*x^3 + (3*a*b^2*c*d*x^4)/2 + (b^2*(b*c^2 + 3*a*d^2)*x^5)/5 + (b^3*c* d*x^6)/3 + (b^3*d^2*x^7)/7 + 2*a^3*c*d*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 = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b^{3} d^{2} x^{7}}{7}+\frac {b^{3} c d \,x^{6}}{3}+\frac {3 a \,b^{2} d^{2} x^{5}}{5}+\frac {b^{3} c^{2} x^{5}}{5}+\frac {3 c d a \,b^{2} x^{4}}{2}+d^{2} x^{3} a^{2} b +a \,b^{2} c^{2} x^{3}+3 x^{2} a^{2} b c d +a^{3} d^{2} x +3 a^{2} b \,c^{2} x +2 a^{3} c d \ln \left (x \right )-\frac {a^{3} c^{2}}{x}\) | \(128\) |
risch | \(\frac {b^{3} d^{2} x^{7}}{7}+\frac {b^{3} c d \,x^{6}}{3}+\frac {3 a \,b^{2} d^{2} x^{5}}{5}+\frac {b^{3} c^{2} x^{5}}{5}+\frac {3 c d a \,b^{2} x^{4}}{2}+d^{2} x^{3} a^{2} b +a \,b^{2} c^{2} x^{3}+3 x^{2} a^{2} b c d +a^{3} d^{2} x +3 a^{2} b \,c^{2} x +2 a^{3} c d \ln \left (x \right )-\frac {a^{3} c^{2}}{x}\) | \(128\) |
norman | \(\frac {\left (\frac {3}{5} a \,b^{2} d^{2}+\frac {1}{5} b^{3} c^{2}\right ) x^{6}+\left (a^{2} b \,d^{2}+a \,c^{2} b^{2}\right ) x^{4}+\left (a^{3} d^{2}+3 a^{2} b \,c^{2}\right ) x^{2}-c^{2} a^{3}+\frac {b^{3} d^{2} x^{8}}{7}+\frac {d \,x^{7} c \,b^{3}}{3}+\frac {3 a \,b^{2} c d \,x^{5}}{2}+3 a^{2} b c d \,x^{3}}{x}+2 a^{3} c d \ln \left (x \right )\) | \(131\) |
parallelrisch | \(\frac {30 b^{3} d^{2} x^{8}+70 d \,x^{7} c \,b^{3}+126 a \,b^{2} d^{2} x^{6}+42 b^{3} c^{2} x^{6}+315 a \,b^{2} c d \,x^{5}+210 a^{2} b \,d^{2} x^{4}+210 a \,b^{2} c^{2} x^{4}+630 a^{2} b c d \,x^{3}+420 a^{3} d c \ln \left (x \right ) x +210 a^{3} d^{2} x^{2}+630 a^{2} b \,c^{2} x^{2}-210 c^{2} a^{3}}{210 x}\) | \(138\) |
Input:
int((d*x+c)^2*(b*x^2+a)^3/x^2,x,method=_RETURNVERBOSE)
Output:
1/7*b^3*d^2*x^7+1/3*b^3*c*d*x^6+3/5*a*b^2*d^2*x^5+1/5*b^3*c^2*x^5+3/2*c*d* a*b^2*x^4+d^2*x^3*a^2*b+a*b^2*c^2*x^3+3*x^2*a^2*b*c*d+a^3*d^2*x+3*a^2*b*c^ 2*x+2*a^3*c*d*ln(x)-a^3*c^2/x
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=\frac {30 \, b^{3} d^{2} x^{8} + 70 \, b^{3} c d x^{7} + 315 \, a b^{2} c d x^{5} + 630 \, a^{2} b c d x^{3} + 42 \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{6} + 420 \, a^{3} c d x \log \left (x\right ) - 210 \, a^{3} c^{2} + 210 \, {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{4} + 210 \, {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x^{2}}{210 \, x} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^3/x^2,x, algorithm="fricas")
Output:
1/210*(30*b^3*d^2*x^8 + 70*b^3*c*d*x^7 + 315*a*b^2*c*d*x^5 + 630*a^2*b*c*d *x^3 + 42*(b^3*c^2 + 3*a*b^2*d^2)*x^6 + 420*a^3*c*d*x*log(x) - 210*a^3*c^2 + 210*(a*b^2*c^2 + a^2*b*d^2)*x^4 + 210*(3*a^2*b*c^2 + a^3*d^2)*x^2)/x
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=- \frac {a^{3} c^{2}}{x} + 2 a^{3} c d \log {\left (x \right )} + 3 a^{2} b c d x^{2} + \frac {3 a b^{2} c d x^{4}}{2} + \frac {b^{3} c d x^{6}}{3} + \frac {b^{3} d^{2} x^{7}}{7} + x^{5} \cdot \left (\frac {3 a b^{2} d^{2}}{5} + \frac {b^{3} c^{2}}{5}\right ) + x^{3} \left (a^{2} b d^{2} + a b^{2} c^{2}\right ) + x \left (a^{3} d^{2} + 3 a^{2} b c^{2}\right ) \] Input:
integrate((d*x+c)**2*(b*x**2+a)**3/x**2,x)
Output:
-a**3*c**2/x + 2*a**3*c*d*log(x) + 3*a**2*b*c*d*x**2 + 3*a*b**2*c*d*x**4/2 + b**3*c*d*x**6/3 + b**3*d**2*x**7/7 + x**5*(3*a*b**2*d**2/5 + b**3*c**2/ 5) + x**3*(a**2*b*d**2 + a*b**2*c**2) + x*(a**3*d**2 + 3*a**2*b*c**2)
Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=\frac {1}{7} \, b^{3} d^{2} x^{7} + \frac {1}{3} \, b^{3} c d x^{6} + \frac {3}{2} \, a b^{2} c d x^{4} + 3 \, a^{2} b c d x^{2} + \frac {1}{5} \, {\left (b^{3} c^{2} + 3 \, a b^{2} d^{2}\right )} x^{5} + 2 \, a^{3} c d \log \left (x\right ) - \frac {a^{3} c^{2}}{x} + {\left (a b^{2} c^{2} + a^{2} b d^{2}\right )} x^{3} + {\left (3 \, a^{2} b c^{2} + a^{3} d^{2}\right )} x \] Input:
integrate((d*x+c)^2*(b*x^2+a)^3/x^2,x, algorithm="maxima")
Output:
1/7*b^3*d^2*x^7 + 1/3*b^3*c*d*x^6 + 3/2*a*b^2*c*d*x^4 + 3*a^2*b*c*d*x^2 + 1/5*(b^3*c^2 + 3*a*b^2*d^2)*x^5 + 2*a^3*c*d*log(x) - a^3*c^2/x + (a*b^2*c^ 2 + a^2*b*d^2)*x^3 + (3*a^2*b*c^2 + a^3*d^2)*x
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=\frac {1}{7} \, b^{3} d^{2} x^{7} + \frac {1}{3} \, b^{3} c d x^{6} + \frac {1}{5} \, b^{3} c^{2} x^{5} + \frac {3}{5} \, a b^{2} d^{2} x^{5} + \frac {3}{2} \, a b^{2} c d x^{4} + a b^{2} c^{2} x^{3} + a^{2} b d^{2} x^{3} + 3 \, a^{2} b c d x^{2} + 3 \, a^{2} b c^{2} x + a^{3} d^{2} x + 2 \, a^{3} c d \log \left ({\left | x \right |}\right ) - \frac {a^{3} c^{2}}{x} \] Input:
integrate((d*x+c)^2*(b*x^2+a)^3/x^2,x, algorithm="giac")
Output:
1/7*b^3*d^2*x^7 + 1/3*b^3*c*d*x^6 + 1/5*b^3*c^2*x^5 + 3/5*a*b^2*d^2*x^5 + 3/2*a*b^2*c*d*x^4 + a*b^2*c^2*x^3 + a^2*b*d^2*x^3 + 3*a^2*b*c*d*x^2 + 3*a^ 2*b*c^2*x + a^3*d^2*x + 2*a^3*c*d*log(abs(x)) - a^3*c^2/x
Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.97 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=x\,\left (a^3\,d^2+3\,b\,a^2\,c^2\right )+x^5\,\left (\frac {b^3\,c^2}{5}+\frac {3\,a\,b^2\,d^2}{5}\right )-\frac {a^3\,c^2}{x}+\frac {b^3\,d^2\,x^7}{7}+a\,b\,x^3\,\left (b\,c^2+a\,d^2\right )+\frac {b^3\,c\,d\,x^6}{3}+2\,a^3\,c\,d\,\ln \left (x\right )+3\,a^2\,b\,c\,d\,x^2+\frac {3\,a\,b^2\,c\,d\,x^4}{2} \] Input:
int(((a + b*x^2)^3*(c + d*x)^2)/x^2,x)
Output:
x*(a^3*d^2 + 3*a^2*b*c^2) + x^5*((b^3*c^2)/5 + (3*a*b^2*d^2)/5) - (a^3*c^2 )/x + (b^3*d^2*x^7)/7 + a*b*x^3*(a*d^2 + b*c^2) + (b^3*c*d*x^6)/3 + 2*a^3* c*d*log(x) + 3*a^2*b*c*d*x^2 + (3*a*b^2*c*d*x^4)/2
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^2 \left (a+b x^2\right )^3}{x^2} \, dx=\frac {420 \,\mathrm {log}\left (x \right ) a^{3} c d x -210 a^{3} c^{2}+210 a^{3} d^{2} x^{2}+630 a^{2} b \,c^{2} x^{2}+630 a^{2} b c d \,x^{3}+210 a^{2} b \,d^{2} x^{4}+210 a \,b^{2} c^{2} x^{4}+315 a \,b^{2} c d \,x^{5}+126 a \,b^{2} d^{2} x^{6}+42 b^{3} c^{2} x^{6}+70 b^{3} c d \,x^{7}+30 b^{3} d^{2} x^{8}}{210 x} \] Input:
int((d*x+c)^2*(b*x^2+a)^3/x^2,x)
Output:
(420*log(x)*a**3*c*d*x - 210*a**3*c**2 + 210*a**3*d**2*x**2 + 630*a**2*b*c **2*x**2 + 630*a**2*b*c*d*x**3 + 210*a**2*b*d**2*x**4 + 210*a*b**2*c**2*x* *4 + 315*a*b**2*c*d*x**5 + 126*a*b**2*d**2*x**6 + 42*b**3*c**2*x**6 + 70*b **3*c*d*x**7 + 30*b**3*d**2*x**8)/(210*x)