Integrand size = 23, antiderivative size = 166 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {4}{3} a^3 b c x^3+a^3 b d x^4+\frac {4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac {6}{7} a^2 b^2 d x^7+\frac {3}{4} a^2 b^2 e x^8+\frac {4}{9} a b^3 c x^9+\frac {2}{5} a b^3 d x^{10}+\frac {4}{11} a b^3 e x^{11}+\frac {1}{12} b^4 c x^{12}+\frac {1}{13} b^4 d x^{13}+\frac {1}{14} b^4 e x^{14}+a^4 c \log (x) \]
a^4*d*x+1/2*a^4*e*x^2+4/3*a^3*b*c*x^3+a^3*b*d*x^4+4/5*a^3*b*e*x^5+a^2*b^2* c*x^6+6/7*a^2*b^2*d*x^7+3/4*a^2*b^2*e*x^8+4/9*a*b^3*c*x^9+2/5*a*b^3*d*x^10 +4/11*a*b^3*e*x^11+1/12*b^4*c*x^12+1/13*b^4*d*x^13+1/14*b^4*e*x^14+a^4*c*l n(x)
Time = 0.01 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=a^4 d x+\frac {1}{2} a^4 e x^2+\frac {4}{3} a^3 b c x^3+a^3 b d x^4+\frac {4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac {6}{7} a^2 b^2 d x^7+\frac {3}{4} a^2 b^2 e x^8+\frac {4}{9} a b^3 c x^9+\frac {2}{5} a b^3 d x^{10}+\frac {4}{11} a b^3 e x^{11}+\frac {1}{12} b^4 c x^{12}+\frac {1}{13} b^4 d x^{13}+\frac {1}{14} b^4 e x^{14}+a^4 c \log (x) \]
a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (4*a^3*b*e*x^5 )/5 + a^2*b^2*c*x^6 + (6*a^2*b^2*d*x^7)/7 + (3*a^2*b^2*e*x^8)/4 + (4*a*b^3 *c*x^9)/9 + (2*a*b^3*d*x^10)/5 + (4*a*b^3*e*x^11)/11 + (b^4*c*x^12)/12 + ( b^4*d*x^13)/13 + (b^4*e*x^14)/14 + a^4*c*Log[x]
Time = 0.33 (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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^4 \left (c+d x+e x^2\right )}{x} \, dx\) |
\(\Big \downarrow \) 2159 |
\(\displaystyle \int \left (\frac {a^4 c}{x}+a^4 d+a^4 e x+4 a^3 b c x^2+4 a^3 b d x^3+4 a^3 b e x^4+6 a^2 b^2 c x^5+6 a^2 b^2 d x^6+6 a^2 b^2 e x^7+4 a b^3 c x^8+4 a b^3 d x^9+4 a b^3 e x^{10}+b^4 c x^{11}+b^4 d x^{12}+b^4 e x^{13}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^4 c \log (x)+a^4 d x+\frac {1}{2} a^4 e x^2+\frac {4}{3} a^3 b c x^3+a^3 b d x^4+\frac {4}{5} a^3 b e x^5+a^2 b^2 c x^6+\frac {6}{7} a^2 b^2 d x^7+\frac {3}{4} a^2 b^2 e x^8+\frac {4}{9} a b^3 c x^9+\frac {2}{5} a b^3 d x^{10}+\frac {4}{11} a b^3 e x^{11}+\frac {1}{12} b^4 c x^{12}+\frac {1}{13} b^4 d x^{13}+\frac {1}{14} b^4 e x^{14}\) |
a^4*d*x + (a^4*e*x^2)/2 + (4*a^3*b*c*x^3)/3 + a^3*b*d*x^4 + (4*a^3*b*e*x^5 )/5 + a^2*b^2*c*x^6 + (6*a^2*b^2*d*x^7)/7 + (3*a^2*b^2*e*x^8)/4 + (4*a*b^3 *c*x^9)/9 + (2*a*b^3*d*x^10)/5 + (4*a*b^3*e*x^11)/11 + (b^4*c*x^12)/12 + ( b^4*d*x^13)/13 + (b^4*e*x^14)/14 + a^4*c*Log[x]
3.4.34.3.1 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 = 1.49 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.87
method | result | size |
default | \(a^{4} d x +\frac {a^{4} e \,x^{2}}{2}+\frac {4 a^{3} b c \,x^{3}}{3}+a^{3} b d \,x^{4}+\frac {4 a^{3} b e \,x^{5}}{5}+a^{2} b^{2} c \,x^{6}+\frac {6 a^{2} b^{2} d \,x^{7}}{7}+\frac {3 a^{2} b^{2} e \,x^{8}}{4}+\frac {4 a \,b^{3} c \,x^{9}}{9}+\frac {2 a \,b^{3} d \,x^{10}}{5}+\frac {4 a \,b^{3} e \,x^{11}}{11}+\frac {b^{4} c \,x^{12}}{12}+\frac {b^{4} d \,x^{13}}{13}+\frac {b^{4} e \,x^{14}}{14}+a^{4} c \ln \left (x \right )\) | \(145\) |
norman | \(a^{4} d x +\frac {a^{4} e \,x^{2}}{2}+\frac {4 a^{3} b c \,x^{3}}{3}+a^{3} b d \,x^{4}+\frac {4 a^{3} b e \,x^{5}}{5}+a^{2} b^{2} c \,x^{6}+\frac {6 a^{2} b^{2} d \,x^{7}}{7}+\frac {3 a^{2} b^{2} e \,x^{8}}{4}+\frac {4 a \,b^{3} c \,x^{9}}{9}+\frac {2 a \,b^{3} d \,x^{10}}{5}+\frac {4 a \,b^{3} e \,x^{11}}{11}+\frac {b^{4} c \,x^{12}}{12}+\frac {b^{4} d \,x^{13}}{13}+\frac {b^{4} e \,x^{14}}{14}+a^{4} c \ln \left (x \right )\) | \(145\) |
risch | \(a^{4} d x +\frac {a^{4} e \,x^{2}}{2}+\frac {4 a^{3} b c \,x^{3}}{3}+a^{3} b d \,x^{4}+\frac {4 a^{3} b e \,x^{5}}{5}+a^{2} b^{2} c \,x^{6}+\frac {6 a^{2} b^{2} d \,x^{7}}{7}+\frac {3 a^{2} b^{2} e \,x^{8}}{4}+\frac {4 a \,b^{3} c \,x^{9}}{9}+\frac {2 a \,b^{3} d \,x^{10}}{5}+\frac {4 a \,b^{3} e \,x^{11}}{11}+\frac {b^{4} c \,x^{12}}{12}+\frac {b^{4} d \,x^{13}}{13}+\frac {b^{4} e \,x^{14}}{14}+a^{4} c \ln \left (x \right )\) | \(145\) |
parallelrisch | \(a^{4} d x +\frac {a^{4} e \,x^{2}}{2}+\frac {4 a^{3} b c \,x^{3}}{3}+a^{3} b d \,x^{4}+\frac {4 a^{3} b e \,x^{5}}{5}+a^{2} b^{2} c \,x^{6}+\frac {6 a^{2} b^{2} d \,x^{7}}{7}+\frac {3 a^{2} b^{2} e \,x^{8}}{4}+\frac {4 a \,b^{3} c \,x^{9}}{9}+\frac {2 a \,b^{3} d \,x^{10}}{5}+\frac {4 a \,b^{3} e \,x^{11}}{11}+\frac {b^{4} c \,x^{12}}{12}+\frac {b^{4} d \,x^{13}}{13}+\frac {b^{4} e \,x^{14}}{14}+a^{4} c \ln \left (x \right )\) | \(145\) |
a^4*d*x+1/2*a^4*e*x^2+4/3*a^3*b*c*x^3+a^3*b*d*x^4+4/5*a^3*b*e*x^5+a^2*b^2* c*x^6+6/7*a^2*b^2*d*x^7+3/4*a^2*b^2*e*x^8+4/9*a*b^3*c*x^9+2/5*a*b^3*d*x^10 +4/11*a*b^3*e*x^11+1/12*b^4*c*x^12+1/13*b^4*d*x^13+1/14*b^4*e*x^14+a^4*c*l n(x)
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=\frac {1}{14} \, b^{4} e x^{14} + \frac {1}{13} \, b^{4} d x^{13} + \frac {1}{12} \, b^{4} c x^{12} + \frac {4}{11} \, a b^{3} e x^{11} + \frac {2}{5} \, a b^{3} d x^{10} + \frac {4}{9} \, a b^{3} c x^{9} + \frac {3}{4} \, a^{2} b^{2} e x^{8} + \frac {6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac {4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac {4}{3} \, a^{3} b c x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} c \log \left (x\right ) \]
1/14*b^4*e*x^14 + 1/13*b^4*d*x^13 + 1/12*b^4*c*x^12 + 4/11*a*b^3*e*x^11 + 2/5*a*b^3*d*x^10 + 4/9*a*b^3*c*x^9 + 3/4*a^2*b^2*e*x^8 + 6/7*a^2*b^2*d*x^7 + a^2*b^2*c*x^6 + 4/5*a^3*b*e*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2*a ^4*e*x^2 + a^4*d*x + a^4*c*log(x)
Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.05 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=a^{4} c \log {\left (x \right )} + a^{4} d x + \frac {a^{4} e x^{2}}{2} + \frac {4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac {4 a^{3} b e x^{5}}{5} + a^{2} b^{2} c x^{6} + \frac {6 a^{2} b^{2} d x^{7}}{7} + \frac {3 a^{2} b^{2} e x^{8}}{4} + \frac {4 a b^{3} c x^{9}}{9} + \frac {2 a b^{3} d x^{10}}{5} + \frac {4 a b^{3} e x^{11}}{11} + \frac {b^{4} c x^{12}}{12} + \frac {b^{4} d x^{13}}{13} + \frac {b^{4} e x^{14}}{14} \]
a**4*c*log(x) + a**4*d*x + a**4*e*x**2/2 + 4*a**3*b*c*x**3/3 + a**3*b*d*x* *4 + 4*a**3*b*e*x**5/5 + a**2*b**2*c*x**6 + 6*a**2*b**2*d*x**7/7 + 3*a**2* b**2*e*x**8/4 + 4*a*b**3*c*x**9/9 + 2*a*b**3*d*x**10/5 + 4*a*b**3*e*x**11/ 11 + b**4*c*x**12/12 + b**4*d*x**13/13 + b**4*e*x**14/14
Time = 0.21 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=\frac {1}{14} \, b^{4} e x^{14} + \frac {1}{13} \, b^{4} d x^{13} + \frac {1}{12} \, b^{4} c x^{12} + \frac {4}{11} \, a b^{3} e x^{11} + \frac {2}{5} \, a b^{3} d x^{10} + \frac {4}{9} \, a b^{3} c x^{9} + \frac {3}{4} \, a^{2} b^{2} e x^{8} + \frac {6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac {4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac {4}{3} \, a^{3} b c x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} c \log \left (x\right ) \]
1/14*b^4*e*x^14 + 1/13*b^4*d*x^13 + 1/12*b^4*c*x^12 + 4/11*a*b^3*e*x^11 + 2/5*a*b^3*d*x^10 + 4/9*a*b^3*c*x^9 + 3/4*a^2*b^2*e*x^8 + 6/7*a^2*b^2*d*x^7 + a^2*b^2*c*x^6 + 4/5*a^3*b*e*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2*a ^4*e*x^2 + a^4*d*x + a^4*c*log(x)
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=\frac {1}{14} \, b^{4} e x^{14} + \frac {1}{13} \, b^{4} d x^{13} + \frac {1}{12} \, b^{4} c x^{12} + \frac {4}{11} \, a b^{3} e x^{11} + \frac {2}{5} \, a b^{3} d x^{10} + \frac {4}{9} \, a b^{3} c x^{9} + \frac {3}{4} \, a^{2} b^{2} e x^{8} + \frac {6}{7} \, a^{2} b^{2} d x^{7} + a^{2} b^{2} c x^{6} + \frac {4}{5} \, a^{3} b e x^{5} + a^{3} b d x^{4} + \frac {4}{3} \, a^{3} b c x^{3} + \frac {1}{2} \, a^{4} e x^{2} + a^{4} d x + a^{4} c \log \left ({\left | x \right |}\right ) \]
1/14*b^4*e*x^14 + 1/13*b^4*d*x^13 + 1/12*b^4*c*x^12 + 4/11*a*b^3*e*x^11 + 2/5*a*b^3*d*x^10 + 4/9*a*b^3*c*x^9 + 3/4*a^2*b^2*e*x^8 + 6/7*a^2*b^2*d*x^7 + a^2*b^2*c*x^6 + 4/5*a^3*b*e*x^5 + a^3*b*d*x^4 + 4/3*a^3*b*c*x^3 + 1/2*a ^4*e*x^2 + a^4*d*x + a^4*c*log(abs(x))
Time = 0.16 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.87 \[ \int \frac {\left (c+d x+e x^2\right ) \left (a+b x^3\right )^4}{x} \, dx=\frac {b^4\,c\,x^{12}}{12}+\frac {a^4\,e\,x^2}{2}+\frac {b^4\,d\,x^{13}}{13}+\frac {b^4\,e\,x^{14}}{14}+a^4\,c\,\ln \left (x\right )+a^4\,d\,x+a^2\,b^2\,c\,x^6+\frac {6\,a^2\,b^2\,d\,x^7}{7}+\frac {3\,a^2\,b^2\,e\,x^8}{4}+\frac {4\,a^3\,b\,c\,x^3}{3}+\frac {4\,a\,b^3\,c\,x^9}{9}+a^3\,b\,d\,x^4+\frac {2\,a\,b^3\,d\,x^{10}}{5}+\frac {4\,a^3\,b\,e\,x^5}{5}+\frac {4\,a\,b^3\,e\,x^{11}}{11} \]