Integrand size = 23, antiderivative size = 125 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=-\frac {a^3 c}{x}+a^3 e x+\frac {3}{2} a^2 b c x^2+a^2 b d x^3+\frac {3}{4} a^2 b e x^4+\frac {3}{5} a b^2 c x^5+\frac {1}{2} a b^2 d x^6+\frac {3}{7} a b^2 e x^7+\frac {1}{8} b^3 c x^8+\frac {1}{9} b^3 d x^9+\frac {1}{10} b^3 e x^{10}+a^3 d \log (x) \] Output:
-a^3*c/x+a^3*e*x+3/2*a^2*b*c*x^2+a^2*b*d*x^3+3/4*a^2*b*e*x^4+3/5*a*b^2*c*x ^5+1/2*a*b^2*d*x^6+3/7*a*b^2*e*x^7+1/8*b^3*c*x^8+1/9*b^3*d*x^9+1/10*b^3*e* x^10+a^3*d*ln(x)
Time = 0.01 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=-\frac {a^3 c}{x}+a^3 e x+\frac {3}{2} a^2 b c x^2+a^2 b d x^3+\frac {3}{4} a^2 b e x^4+\frac {3}{5} a b^2 c x^5+\frac {1}{2} a b^2 d x^6+\frac {3}{7} a b^2 e x^7+\frac {1}{8} b^3 c x^8+\frac {1}{9} b^3 d x^9+\frac {1}{10} b^3 e x^{10}+a^3 d \log (x) \] Input:
Integrate[((c + d*x + e*x^2)*(a + b*x^3)^3)/x^2,x]
Output:
-((a^3*c)/x) + a^3*e*x + (3*a^2*b*c*x^2)/2 + a^2*b*d*x^3 + (3*a^2*b*e*x^4) /4 + (3*a*b^2*c*x^5)/5 + (a*b^2*d*x^6)/2 + (3*a*b^2*e*x^7)/7 + (b^3*c*x^8) /8 + (b^3*d*x^9)/9 + (b^3*e*x^10)/10 + a^3*d*Log[x]
Time = 0.51 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2159, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^3 \left (c+d x+e x^2\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \int \left (\frac {a^3 c}{x^2}+\frac {a^3 d}{x}+a^3 e+3 a^2 b c x+3 a^2 b d x^2+3 a^2 b e x^3+3 a b^2 c x^4+3 a b^2 d x^5+3 a b^2 e x^6+b^3 c x^7+b^3 d x^8+b^3 e x^9\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 c}{x}+a^3 d \log (x)+a^3 e x+\frac {3}{2} a^2 b c x^2+a^2 b d x^3+\frac {3}{4} a^2 b e x^4+\frac {3}{5} a b^2 c x^5+\frac {1}{2} a b^2 d x^6+\frac {3}{7} a b^2 e x^7+\frac {1}{8} b^3 c x^8+\frac {1}{9} b^3 d x^9+\frac {1}{10} b^3 e x^{10}\) |
Input:
Int[((c + d*x + e*x^2)*(a + b*x^3)^3)/x^2,x]
Output:
-((a^3*c)/x) + a^3*e*x + (3*a^2*b*c*x^2)/2 + a^2*b*d*x^3 + (3*a^2*b*e*x^4) /4 + (3*a*b^2*c*x^5)/5 + (a*b^2*d*x^6)/2 + (3*a*b^2*e*x^7)/7 + (b^3*c*x^8) /8 + (b^3*d*x^9)/9 + (b^3*e*x^10)/10 + a^3*d*Log[x]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88
method | result | size |
default | \(-\frac {a^{3} c}{x}+a^{3} e x +\frac {3 a^{2} b c \,x^{2}}{2}+a^{2} b d \,x^{3}+\frac {3 a^{2} b e \,x^{4}}{4}+\frac {3 a \,b^{2} c \,x^{5}}{5}+\frac {a \,b^{2} d \,x^{6}}{2}+\frac {3 a \,b^{2} e \,x^{7}}{7}+\frac {b^{3} c \,x^{8}}{8}+\frac {b^{3} d \,x^{9}}{9}+\frac {b^{3} e \,x^{10}}{10}+a^{3} d \ln \left (x \right )\) | \(110\) |
risch | \(-\frac {a^{3} c}{x}+a^{3} e x +\frac {3 a^{2} b c \,x^{2}}{2}+a^{2} b d \,x^{3}+\frac {3 a^{2} b e \,x^{4}}{4}+\frac {3 a \,b^{2} c \,x^{5}}{5}+\frac {a \,b^{2} d \,x^{6}}{2}+\frac {3 a \,b^{2} e \,x^{7}}{7}+\frac {b^{3} c \,x^{8}}{8}+\frac {b^{3} d \,x^{9}}{9}+\frac {b^{3} e \,x^{10}}{10}+a^{3} d \ln \left (x \right )\) | \(110\) |
norman | \(\frac {a^{3} e \,x^{2}+a^{2} b d \,x^{4}-c \,a^{3}+\frac {1}{8} b^{3} c \,x^{9}+\frac {1}{9} b^{3} d \,x^{10}+\frac {1}{10} b^{3} e \,x^{11}+\frac {3}{5} a \,b^{2} c \,x^{6}+\frac {1}{2} a \,b^{2} d \,x^{7}+\frac {3}{7} a \,b^{2} e \,x^{8}+\frac {3}{2} a^{2} b c \,x^{3}+\frac {3}{4} a^{2} b e \,x^{5}}{x}+a^{3} d \ln \left (x \right )\) | \(114\) |
parallelrisch | \(\frac {252 b^{3} e \,x^{11}+280 b^{3} d \,x^{10}+315 b^{3} c \,x^{9}+1080 a \,b^{2} e \,x^{8}+1260 a \,b^{2} d \,x^{7}+1512 a \,b^{2} c \,x^{6}+1890 a^{2} b e \,x^{5}+2520 a^{2} b d \,x^{4}+3780 a^{2} b c \,x^{3}+2520 a^{3} d \ln \left (x \right ) x +2520 a^{3} e \,x^{2}-2520 c \,a^{3}}{2520 x}\) | \(118\) |
Input:
int((e*x^2+d*x+c)*(b*x^3+a)^3/x^2,x,method=_RETURNVERBOSE)
Output:
-a^3*c/x+a^3*e*x+3/2*a^2*b*c*x^2+a^2*b*d*x^3+3/4*a^2*b*e*x^4+3/5*a*b^2*c*x ^5+1/2*a*b^2*d*x^6+3/7*a*b^2*e*x^7+1/8*b^3*c*x^8+1/9*b^3*d*x^9+1/10*b^3*e* x^10+a^3*d*ln(x)
Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=\frac {252 \, b^{3} e x^{11} + 280 \, b^{3} d x^{10} + 315 \, b^{3} c x^{9} + 1080 \, a b^{2} e x^{8} + 1260 \, a b^{2} d x^{7} + 1512 \, a b^{2} c x^{6} + 1890 \, a^{2} b e x^{5} + 2520 \, a^{2} b d x^{4} + 3780 \, a^{2} b c x^{3} + 2520 \, a^{3} e x^{2} + 2520 \, a^{3} d x \log \left (x\right ) - 2520 \, a^{3} c}{2520 \, x} \] Input:
integrate((e*x^2+d*x+c)*(b*x^3+a)^3/x^2,x, algorithm="fricas")
Output:
1/2520*(252*b^3*e*x^11 + 280*b^3*d*x^10 + 315*b^3*c*x^9 + 1080*a*b^2*e*x^8 + 1260*a*b^2*d*x^7 + 1512*a*b^2*c*x^6 + 1890*a^2*b*e*x^5 + 2520*a^2*b*d*x ^4 + 3780*a^2*b*c*x^3 + 2520*a^3*e*x^2 + 2520*a^3*d*x*log(x) - 2520*a^3*c) /x
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=- \frac {a^{3} c}{x} + a^{3} d \log {\left (x \right )} + a^{3} e x + \frac {3 a^{2} b c x^{2}}{2} + a^{2} b d x^{3} + \frac {3 a^{2} b e x^{4}}{4} + \frac {3 a b^{2} c x^{5}}{5} + \frac {a b^{2} d x^{6}}{2} + \frac {3 a b^{2} e x^{7}}{7} + \frac {b^{3} c x^{8}}{8} + \frac {b^{3} d x^{9}}{9} + \frac {b^{3} e x^{10}}{10} \] Input:
integrate((e*x**2+d*x+c)*(b*x**3+a)**3/x**2,x)
Output:
-a**3*c/x + a**3*d*log(x) + a**3*e*x + 3*a**2*b*c*x**2/2 + a**2*b*d*x**3 + 3*a**2*b*e*x**4/4 + 3*a*b**2*c*x**5/5 + a*b**2*d*x**6/2 + 3*a*b**2*e*x**7 /7 + b**3*c*x**8/8 + b**3*d*x**9/9 + b**3*e*x**10/10
Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=\frac {1}{10} \, b^{3} e x^{10} + \frac {1}{9} \, b^{3} d x^{9} + \frac {1}{8} \, b^{3} c x^{8} + \frac {3}{7} \, a b^{2} e x^{7} + \frac {1}{2} \, a b^{2} d x^{6} + \frac {3}{5} \, a b^{2} c x^{5} + \frac {3}{4} \, a^{2} b e x^{4} + a^{2} b d x^{3} + \frac {3}{2} \, a^{2} b c x^{2} + a^{3} e x + a^{3} d \log \left (x\right ) - \frac {a^{3} c}{x} \] Input:
integrate((e*x^2+d*x+c)*(b*x^3+a)^3/x^2,x, algorithm="maxima")
Output:
1/10*b^3*e*x^10 + 1/9*b^3*d*x^9 + 1/8*b^3*c*x^8 + 3/7*a*b^2*e*x^7 + 1/2*a* b^2*d*x^6 + 3/5*a*b^2*c*x^5 + 3/4*a^2*b*e*x^4 + a^2*b*d*x^3 + 3/2*a^2*b*c* x^2 + a^3*e*x + a^3*d*log(x) - a^3*c/x
Time = 0.12 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=\frac {1}{10} \, b^{3} e x^{10} + \frac {1}{9} \, b^{3} d x^{9} + \frac {1}{8} \, b^{3} c x^{8} + \frac {3}{7} \, a b^{2} e x^{7} + \frac {1}{2} \, a b^{2} d x^{6} + \frac {3}{5} \, a b^{2} c x^{5} + \frac {3}{4} \, a^{2} b e x^{4} + a^{2} b d x^{3} + \frac {3}{2} \, a^{2} b c x^{2} + a^{3} e x + a^{3} d \log \left ({\left | x \right |}\right ) - \frac {a^{3} c}{x} \] Input:
integrate((e*x^2+d*x+c)*(b*x^3+a)^3/x^2,x, algorithm="giac")
Output:
1/10*b^3*e*x^10 + 1/9*b^3*d*x^9 + 1/8*b^3*c*x^8 + 3/7*a*b^2*e*x^7 + 1/2*a* b^2*d*x^6 + 3/5*a*b^2*c*x^5 + 3/4*a^2*b*e*x^4 + a^2*b*d*x^3 + 3/2*a^2*b*c* x^2 + a^3*e*x + a^3*d*log(abs(x)) - a^3*c/x
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=\frac {b^3\,c\,x^8}{8}-\frac {a^3\,c}{x}+\frac {b^3\,d\,x^9}{9}+\frac {b^3\,e\,x^{10}}{10}+a^3\,d\,\ln \left (x\right )+a^3\,e\,x+\frac {3\,a^2\,b\,c\,x^2}{2}+\frac {3\,a\,b^2\,c\,x^5}{5}+a^2\,b\,d\,x^3+\frac {a\,b^2\,d\,x^6}{2}+\frac {3\,a^2\,b\,e\,x^4}{4}+\frac {3\,a\,b^2\,e\,x^7}{7} \] Input:
int(((a + b*x^3)^3*(c + d*x + e*x^2))/x^2,x)
Output:
(b^3*c*x^8)/8 - (a^3*c)/x + (b^3*d*x^9)/9 + (b^3*e*x^10)/10 + a^3*d*log(x) + a^3*e*x + (3*a^2*b*c*x^2)/2 + (3*a*b^2*c*x^5)/5 + a^2*b*d*x^3 + (a*b^2* d*x^6)/2 + (3*a^2*b*e*x^4)/4 + (3*a*b^2*e*x^7)/7
Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^3}{x^2} \, dx=\frac {2520 \,\mathrm {log}\left (x \right ) a^{3} d x -2520 a^{3} c +2520 a^{3} e \,x^{2}+3780 a^{2} b c \,x^{3}+2520 a^{2} b d \,x^{4}+1890 a^{2} b e \,x^{5}+1512 a \,b^{2} c \,x^{6}+1260 a \,b^{2} d \,x^{7}+1080 a \,b^{2} e \,x^{8}+315 b^{3} c \,x^{9}+280 b^{3} d \,x^{10}+252 b^{3} e \,x^{11}}{2520 x} \] Input:
int((e*x^2+d*x+c)*(b*x^3+a)^3/x^2,x)
Output:
(2520*log(x)*a**3*d*x - 2520*a**3*c + 2520*a**3*e*x**2 + 3780*a**2*b*c*x** 3 + 2520*a**2*b*d*x**4 + 1890*a**2*b*e*x**5 + 1512*a*b**2*c*x**6 + 1260*a* b**2*d*x**7 + 1080*a*b**2*e*x**8 + 315*b**3*c*x**9 + 280*b**3*d*x**10 + 25 2*b**3*e*x**11)/(2520*x)