Integrand size = 23, antiderivative size = 166 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=-\frac {a^4 c}{2 x^2}-\frac {a^4 d}{x}+4 a^3 b c x+2 a^3 b d x^2+\frac {4}{3} a^3 b e x^3+\frac {3}{2} a^2 b^2 c x^4+\frac {6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac {4}{7} a b^3 c x^7+\frac {1}{2} a b^3 d x^8+\frac {4}{9} a b^3 e x^9+\frac {1}{10} b^4 c x^{10}+\frac {1}{11} b^4 d x^{11}+\frac {1}{12} b^4 e x^{12}+a^4 e \log (x) \]
-1/2*a^4*c/x^2-a^4*d/x+4*a^3*b*c*x+2*a^3*b*d*x^2+4/3*a^3*b*e*x^3+3/2*a^2*b ^2*c*x^4+6/5*a^2*b^2*d*x^5+a^2*b^2*e*x^6+4/7*a*b^3*c*x^7+1/2*a*b^3*d*x^8+4 /9*a*b^3*e*x^9+1/10*b^4*c*x^10+1/11*b^4*d*x^11+1/12*b^4*e*x^12+a^4*e*ln(x)
Time = 0.01 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=-\frac {a^4 c}{2 x^2}-\frac {a^4 d}{x}+4 a^3 b c x+2 a^3 b d x^2+\frac {4}{3} a^3 b e x^3+\frac {3}{2} a^2 b^2 c x^4+\frac {6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac {4}{7} a b^3 c x^7+\frac {1}{2} a b^3 d x^8+\frac {4}{9} a b^3 e x^9+\frac {1}{10} b^4 c x^{10}+\frac {1}{11} b^4 d x^{11}+\frac {1}{12} b^4 e x^{12}+a^4 e \log (x) \]
-1/2*(a^4*c)/x^2 - (a^4*d)/x + 4*a^3*b*c*x + 2*a^3*b*d*x^2 + (4*a^3*b*e*x^ 3)/3 + (3*a^2*b^2*c*x^4)/2 + (6*a^2*b^2*d*x^5)/5 + a^2*b^2*e*x^6 + (4*a*b^ 3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (4*a*b^3*e*x^9)/9 + (b^4*c*x^10)/10 + (b^4* d*x^11)/11 + (b^4*e*x^12)/12 + a^4*e*Log[x]
Time = 0.36 (sec) , antiderivative size = 166, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^4 \left (c+d x+e x^2\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \int \left (\frac {a^4 c}{x^3}+\frac {a^4 d}{x^2}+\frac {a^4 e}{x}+4 a^3 b c+4 a^3 b d x+4 a^3 b e x^2+6 a^2 b^2 c x^3+6 a^2 b^2 d x^4+6 a^2 b^2 e x^5+4 a b^3 c x^6+4 a b^3 d x^7+4 a b^3 e x^8+b^4 c x^9+b^4 d x^{10}+b^4 e x^{11}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^4 c}{2 x^2}-\frac {a^4 d}{x}+a^4 e \log (x)+4 a^3 b c x+2 a^3 b d x^2+\frac {4}{3} a^3 b e x^3+\frac {3}{2} a^2 b^2 c x^4+\frac {6}{5} a^2 b^2 d x^5+a^2 b^2 e x^6+\frac {4}{7} a b^3 c x^7+\frac {1}{2} a b^3 d x^8+\frac {4}{9} a b^3 e x^9+\frac {1}{10} b^4 c x^{10}+\frac {1}{11} b^4 d x^{11}+\frac {1}{12} b^4 e x^{12}\) |
-1/2*(a^4*c)/x^2 - (a^4*d)/x + 4*a^3*b*c*x + 2*a^3*b*d*x^2 + (4*a^3*b*e*x^ 3)/3 + (3*a^2*b^2*c*x^4)/2 + (6*a^2*b^2*d*x^5)/5 + a^2*b^2*e*x^6 + (4*a*b^ 3*c*x^7)/7 + (a*b^3*d*x^8)/2 + (4*a*b^3*e*x^9)/9 + (b^4*c*x^10)/10 + (b^4* d*x^11)/11 + (b^4*e*x^12)/12 + a^4*e*Log[x]
3.4.36.3.1 Defintions of rubi rules used
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 = 1.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {a^{4} c}{2 x^{2}}-\frac {a^{4} d}{x}+4 a^{3} b c x +2 a^{3} b d \,x^{2}+\frac {4 a^{3} b e \,x^{3}}{3}+\frac {3 a^{2} b^{2} c \,x^{4}}{2}+\frac {6 a^{2} b^{2} d \,x^{5}}{5}+a^{2} b^{2} e \,x^{6}+\frac {4 a \,b^{3} c \,x^{7}}{7}+\frac {a \,b^{3} d \,x^{8}}{2}+\frac {4 a \,b^{3} e \,x^{9}}{9}+\frac {b^{4} c \,x^{10}}{10}+\frac {b^{4} d \,x^{11}}{11}+\frac {b^{4} e \,x^{12}}{12}+a^{4} e \ln \left (x \right )\) | \(147\) |
risch | \(\frac {b^{4} e \,x^{12}}{12}+\frac {b^{4} d \,x^{11}}{11}+\frac {b^{4} c \,x^{10}}{10}+\frac {4 a \,b^{3} e \,x^{9}}{9}+\frac {a \,b^{3} d \,x^{8}}{2}+\frac {4 a \,b^{3} c \,x^{7}}{7}+a^{2} b^{2} e \,x^{6}+\frac {6 a^{2} b^{2} d \,x^{5}}{5}+\frac {3 a^{2} b^{2} c \,x^{4}}{2}+\frac {4 a^{3} b e \,x^{3}}{3}+2 a^{3} b d \,x^{2}+4 a^{3} b c x +\frac {-a^{4} d x -\frac {1}{2} a^{4} c}{x^{2}}+a^{4} e \ln \left (x \right )\) | \(147\) |
norman | \(\frac {a^{2} b^{2} e \,x^{8}-\frac {1}{2} a^{4} c -a^{4} d x +\frac {1}{10} b^{4} c \,x^{12}+\frac {1}{11} b^{4} d \,x^{13}+\frac {1}{12} b^{4} e \,x^{14}+\frac {4}{7} a \,b^{3} c \,x^{9}+\frac {1}{2} a \,b^{3} d \,x^{10}+\frac {4}{9} a \,b^{3} e \,x^{11}+\frac {3}{2} a^{2} b^{2} c \,x^{6}+\frac {6}{5} a^{2} b^{2} d \,x^{7}+4 a^{3} b c \,x^{3}+2 a^{3} b d \,x^{4}+\frac {4}{3} a^{3} b e \,x^{5}}{x^{2}}+a^{4} e \ln \left (x \right )\) | \(149\) |
parallelrisch | \(\frac {1155 b^{4} e \,x^{14}+1260 b^{4} d \,x^{13}+1386 b^{4} c \,x^{12}+6160 a \,b^{3} e \,x^{11}+6930 a \,b^{3} d \,x^{10}+7920 a \,b^{3} c \,x^{9}+13860 a^{2} b^{2} e \,x^{8}+16632 a^{2} b^{2} d \,x^{7}+20790 a^{2} b^{2} c \,x^{6}+18480 a^{3} b e \,x^{5}+27720 a^{3} b d \,x^{4}+13860 e \,a^{4} \ln \left (x \right ) x^{2}+55440 a^{3} b c \,x^{3}-13860 a^{4} d x -6930 a^{4} c}{13860 x^{2}}\) | \(154\) |
-1/2*a^4*c/x^2-a^4*d/x+4*a^3*b*c*x+2*a^3*b*d*x^2+4/3*a^3*b*e*x^3+3/2*a^2*b ^2*c*x^4+6/5*a^2*b^2*d*x^5+a^2*b^2*e*x^6+4/7*a*b^3*c*x^7+1/2*a*b^3*d*x^8+4 /9*a*b^3*e*x^9+1/10*b^4*c*x^10+1/11*b^4*d*x^11+1/12*b^4*e*x^12+a^4*e*ln(x)
Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=\frac {1155 \, b^{4} e x^{14} + 1260 \, b^{4} d x^{13} + 1386 \, b^{4} c x^{12} + 6160 \, a b^{3} e x^{11} + 6930 \, a b^{3} d x^{10} + 7920 \, a b^{3} c x^{9} + 13860 \, a^{2} b^{2} e x^{8} + 16632 \, a^{2} b^{2} d x^{7} + 20790 \, a^{2} b^{2} c x^{6} + 18480 \, a^{3} b e x^{5} + 27720 \, a^{3} b d x^{4} + 55440 \, a^{3} b c x^{3} + 13860 \, a^{4} e x^{2} \log \left (x\right ) - 13860 \, a^{4} d x - 6930 \, a^{4} c}{13860 \, x^{2}} \]
1/13860*(1155*b^4*e*x^14 + 1260*b^4*d*x^13 + 1386*b^4*c*x^12 + 6160*a*b^3* e*x^11 + 6930*a*b^3*d*x^10 + 7920*a*b^3*c*x^9 + 13860*a^2*b^2*e*x^8 + 1663 2*a^2*b^2*d*x^7 + 20790*a^2*b^2*c*x^6 + 18480*a^3*b*e*x^5 + 27720*a^3*b*d* x^4 + 55440*a^3*b*c*x^3 + 13860*a^4*e*x^2*log(x) - 13860*a^4*d*x - 6930*a^ 4*c)/x^2
Time = 0.17 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=a^{4} e \log {\left (x \right )} + 4 a^{3} b c x + 2 a^{3} b d x^{2} + \frac {4 a^{3} b e x^{3}}{3} + \frac {3 a^{2} b^{2} c x^{4}}{2} + \frac {6 a^{2} b^{2} d x^{5}}{5} + a^{2} b^{2} e x^{6} + \frac {4 a b^{3} c x^{7}}{7} + \frac {a b^{3} d x^{8}}{2} + \frac {4 a b^{3} e x^{9}}{9} + \frac {b^{4} c x^{10}}{10} + \frac {b^{4} d x^{11}}{11} + \frac {b^{4} e x^{12}}{12} + \frac {- a^{4} c - 2 a^{4} d x}{2 x^{2}} \]
a**4*e*log(x) + 4*a**3*b*c*x + 2*a**3*b*d*x**2 + 4*a**3*b*e*x**3/3 + 3*a** 2*b**2*c*x**4/2 + 6*a**2*b**2*d*x**5/5 + a**2*b**2*e*x**6 + 4*a*b**3*c*x** 7/7 + a*b**3*d*x**8/2 + 4*a*b**3*e*x**9/9 + b**4*c*x**10/10 + b**4*d*x**11 /11 + b**4*e*x**12/12 + (-a**4*c - 2*a**4*d*x)/(2*x**2)
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=\frac {1}{12} \, b^{4} e x^{12} + \frac {1}{11} \, b^{4} d x^{11} + \frac {1}{10} \, b^{4} c x^{10} + \frac {4}{9} \, a b^{3} e x^{9} + \frac {1}{2} \, a b^{3} d x^{8} + \frac {4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} e x^{6} + \frac {6}{5} \, a^{2} b^{2} d x^{5} + \frac {3}{2} \, a^{2} b^{2} c x^{4} + \frac {4}{3} \, a^{3} b e x^{3} + 2 \, a^{3} b d x^{2} + 4 \, a^{3} b c x + a^{4} e \log \left (x\right ) - \frac {2 \, a^{4} d x + a^{4} c}{2 \, x^{2}} \]
1/12*b^4*e*x^12 + 1/11*b^4*d*x^11 + 1/10*b^4*c*x^10 + 4/9*a*b^3*e*x^9 + 1/ 2*a*b^3*d*x^8 + 4/7*a*b^3*c*x^7 + a^2*b^2*e*x^6 + 6/5*a^2*b^2*d*x^5 + 3/2* a^2*b^2*c*x^4 + 4/3*a^3*b*e*x^3 + 2*a^3*b*d*x^2 + 4*a^3*b*c*x + a^4*e*log( x) - 1/2*(2*a^4*d*x + a^4*c)/x^2
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.89 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=\frac {1}{12} \, b^{4} e x^{12} + \frac {1}{11} \, b^{4} d x^{11} + \frac {1}{10} \, b^{4} c x^{10} + \frac {4}{9} \, a b^{3} e x^{9} + \frac {1}{2} \, a b^{3} d x^{8} + \frac {4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} e x^{6} + \frac {6}{5} \, a^{2} b^{2} d x^{5} + \frac {3}{2} \, a^{2} b^{2} c x^{4} + \frac {4}{3} \, a^{3} b e x^{3} + 2 \, a^{3} b d x^{2} + 4 \, a^{3} b c x + a^{4} e \log \left ({\left | x \right |}\right ) - \frac {2 \, a^{4} d x + a^{4} c}{2 \, x^{2}} \]
1/12*b^4*e*x^12 + 1/11*b^4*d*x^11 + 1/10*b^4*c*x^10 + 4/9*a*b^3*e*x^9 + 1/ 2*a*b^3*d*x^8 + 4/7*a*b^3*c*x^7 + a^2*b^2*e*x^6 + 6/5*a^2*b^2*d*x^5 + 3/2* a^2*b^2*c*x^4 + 4/3*a^3*b*e*x^3 + 2*a^3*b*d*x^2 + 4*a^3*b*c*x + a^4*e*log( abs(x)) - 1/2*(2*a^4*d*x + a^4*c)/x^2
Time = 9.05 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x^3} \, dx=\frac {b^4\,c\,x^{10}}{10}-\frac {\frac {a^4\,c}{2}+a^4\,d\,x}{x^2}+\frac {b^4\,d\,x^{11}}{11}+\frac {b^4\,e\,x^{12}}{12}+a^4\,e\,\ln \left (x\right )+\frac {3\,a^2\,b^2\,c\,x^4}{2}+\frac {6\,a^2\,b^2\,d\,x^5}{5}+a^2\,b^2\,e\,x^6+4\,a^3\,b\,c\,x+\frac {4\,a\,b^3\,c\,x^7}{7}+2\,a^3\,b\,d\,x^2+\frac {a\,b^3\,d\,x^8}{2}+\frac {4\,a^3\,b\,e\,x^3}{3}+\frac {4\,a\,b^3\,e\,x^9}{9} \]