Integrand size = 38, antiderivative size = 149 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {2}{3} a b c x^3+\frac {1}{4} a (2 b d+a g) x^4+\frac {1}{5} a (2 b e+a h) x^5+\frac {1}{6} b^2 c x^6+\frac {1}{7} b (b d+2 a g) x^7+\frac {1}{8} b (b e+2 a h) x^8+\frac {1}{10} b^2 g x^{10}+\frac {1}{11} b^2 h x^{11}+\frac {f \left (a+b x^3\right )^3}{9 b}+a^2 c \log (x) \]
a^2*d*x+1/2*a^2*e*x^2+2/3*a*b*c*x^3+1/4*a*(a*g+2*b*d)*x^4+1/5*a*(a*h+2*b*e )*x^5+1/6*b^2*c*x^6+1/7*b*(2*a*g+b*d)*x^7+1/8*b*(2*a*h+b*e)*x^8+1/10*b^2*g *x^10+1/11*b^2*h*x^11+1/9*f*(b*x^3+a)^3/b+a^2*c*ln(x)
Time = 0.05 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=a^2 d x+\frac {1}{2} a^2 e x^2+\frac {1}{3} a (2 b c+a f) x^3+\frac {1}{4} a (2 b d+a g) x^4+\frac {1}{5} a (2 b e+a h) x^5+\frac {1}{6} b (b c+2 a f) x^6+\frac {1}{7} b (b d+2 a g) x^7+\frac {1}{8} b (b e+2 a h) x^8+\frac {1}{9} b^2 f x^9+\frac {1}{10} b^2 g x^{10}+\frac {1}{11} b^2 h x^{11}+a^2 c \log (x) \]
a^2*d*x + (a^2*e*x^2)/2 + (a*(2*b*c + a*f)*x^3)/3 + (a*(2*b*d + a*g)*x^4)/ 4 + (a*(2*b*e + a*h)*x^5)/5 + (b*(b*c + 2*a*f)*x^6)/6 + (b*(b*d + 2*a*g)*x ^7)/7 + (b*(b*e + 2*a*h)*x^8)/8 + (b^2*f*x^9)/9 + (b^2*g*x^10)/10 + (b^2*h *x^11)/11 + a^2*c*Log[x]
Time = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2018, 2360, 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.
\(\displaystyle \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx\) |
\(\Big \downarrow \) 2018 |
\(\displaystyle \int \frac {\left (b x^3+a\right )^2 \left (h x^5+g x^4+e x^2+d x+c\right )}{x}dx+\frac {f \left (a+b x^3\right )^3}{9 b}\) |
\(\Big \downarrow \) 2360 |
\(\displaystyle \int \left (b^2 h x^{10}+b^2 g x^9+b (b e+2 a h) x^7+b (b d+2 a g) x^6+b^2 c x^5+a (2 b e+a h) x^4+a (2 b d+a g) x^3+2 a b c x^2+a^2 e x+a^2 d+\frac {a^2 c}{x}\right )dx+\frac {f \left (a+b x^3\right )^3}{9 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a^2 c \log (x)+a^2 d x+\frac {1}{2} a^2 e x^2+\frac {2}{3} a b c x^3+\frac {1}{7} b x^7 (2 a g+b d)+\frac {1}{4} a x^4 (a g+2 b d)+\frac {1}{8} b x^8 (2 a h+b e)+\frac {1}{5} a x^5 (a h+2 b e)+\frac {f \left (a+b x^3\right )^3}{9 b}+\frac {1}{6} b^2 c x^6+\frac {1}{10} b^2 g x^{10}+\frac {1}{11} b^2 h x^{11}\) |
a^2*d*x + (a^2*e*x^2)/2 + (2*a*b*c*x^3)/3 + (a*(2*b*d + a*g)*x^4)/4 + (a*( 2*b*e + a*h)*x^5)/5 + (b^2*c*x^6)/6 + (b*(b*d + 2*a*g)*x^7)/7 + (b*(b*e + 2*a*h)*x^8)/8 + (b^2*g*x^10)/10 + (b^2*h*x^11)/11 + (f*(a + b*x^3)^3)/(9*b ) + a^2*c*Log[x]
3.4.88.3.1 Defintions of rubi rules used
Int[(Px_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[Coef f[Px, x, n - m - 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coe ff[Px, x, n - m - 1]*x^(n - m - 1))*x^m*(a + b*x^n)^p, x] /; FreeQ[{a, b, m , n}, x] && PolyQ[Px, x] && IGtQ[p, 1] && IGtQ[n - m, 0] && NeQ[Coeff[Px, x , n - m - 1], 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
Time = 1.50 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.99
method | result | size |
norman | \(\left (\frac {1}{3} a^{2} f +\frac {2}{3} a b c \right ) x^{3}+\left (\frac {1}{4} a^{2} g +\frac {1}{2} a b d \right ) x^{4}+\left (\frac {1}{5} a^{2} h +\frac {2}{5} a e b \right ) x^{5}+\left (\frac {2}{7} a b g +\frac {1}{7} b^{2} d \right ) x^{7}+\left (\frac {1}{4} a b h +\frac {1}{8} b^{2} e \right ) x^{8}+\left (\frac {1}{3} a f b +\frac {1}{6} b^{2} c \right ) x^{6}+a^{2} d x +\frac {a^{2} e \,x^{2}}{2}+\frac {b^{2} f \,x^{9}}{9}+\frac {b^{2} g \,x^{10}}{10}+\frac {b^{2} h \,x^{11}}{11}+a^{2} c \ln \left (x \right )\) | \(147\) |
default | \(\frac {b^{2} h \,x^{11}}{11}+\frac {b^{2} g \,x^{10}}{10}+\frac {b^{2} f \,x^{9}}{9}+\frac {a b h \,x^{8}}{4}+\frac {b^{2} e \,x^{8}}{8}+\frac {2 a b g \,x^{7}}{7}+\frac {b^{2} d \,x^{7}}{7}+\frac {a b f \,x^{6}}{3}+\frac {b^{2} c \,x^{6}}{6}+\frac {a^{2} h \,x^{5}}{5}+\frac {2 a b e \,x^{5}}{5}+\frac {a^{2} g \,x^{4}}{4}+\frac {a b d \,x^{4}}{2}+\frac {a^{2} f \,x^{3}}{3}+\frac {2 a b c \,x^{3}}{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x +a^{2} c \ln \left (x \right )\) | \(153\) |
risch | \(\frac {b^{2} h \,x^{11}}{11}+\frac {b^{2} g \,x^{10}}{10}+\frac {b^{2} f \,x^{9}}{9}+\frac {a b h \,x^{8}}{4}+\frac {b^{2} e \,x^{8}}{8}+\frac {2 a b g \,x^{7}}{7}+\frac {b^{2} d \,x^{7}}{7}+\frac {a b f \,x^{6}}{3}+\frac {b^{2} c \,x^{6}}{6}+\frac {a^{2} h \,x^{5}}{5}+\frac {2 a b e \,x^{5}}{5}+\frac {a^{2} g \,x^{4}}{4}+\frac {a b d \,x^{4}}{2}+\frac {a^{2} f \,x^{3}}{3}+\frac {2 a b c \,x^{3}}{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x +a^{2} c \ln \left (x \right )\) | \(153\) |
parallelrisch | \(\frac {b^{2} h \,x^{11}}{11}+\frac {b^{2} g \,x^{10}}{10}+\frac {b^{2} f \,x^{9}}{9}+\frac {a b h \,x^{8}}{4}+\frac {b^{2} e \,x^{8}}{8}+\frac {2 a b g \,x^{7}}{7}+\frac {b^{2} d \,x^{7}}{7}+\frac {a b f \,x^{6}}{3}+\frac {b^{2} c \,x^{6}}{6}+\frac {a^{2} h \,x^{5}}{5}+\frac {2 a b e \,x^{5}}{5}+\frac {a^{2} g \,x^{4}}{4}+\frac {a b d \,x^{4}}{2}+\frac {a^{2} f \,x^{3}}{3}+\frac {2 a b c \,x^{3}}{3}+\frac {a^{2} e \,x^{2}}{2}+a^{2} d x +a^{2} c \ln \left (x \right )\) | \(153\) |
(1/3*a^2*f+2/3*a*b*c)*x^3+(1/4*a^2*g+1/2*a*b*d)*x^4+(1/5*a^2*h+2/5*a*e*b)* x^5+(2/7*a*b*g+1/7*b^2*d)*x^7+(1/4*a*b*h+1/8*b^2*e)*x^8+(1/3*a*f*b+1/6*b^2 *c)*x^6+a^2*d*x+1/2*a^2*e*x^2+1/9*b^2*f*x^9+1/10*b^2*g*x^10+1/11*b^2*h*x^1 1+a^2*c*ln(x)
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=\frac {1}{11} \, b^{2} h x^{11} + \frac {1}{10} \, b^{2} g x^{10} + \frac {1}{9} \, b^{2} f x^{9} + \frac {1}{8} \, {\left (b^{2} e + 2 \, a b h\right )} x^{8} + \frac {1}{7} \, {\left (b^{2} d + 2 \, a b g\right )} x^{7} + \frac {1}{6} \, {\left (b^{2} c + 2 \, a b f\right )} x^{6} + \frac {1}{5} \, {\left (2 \, a b e + a^{2} h\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{4} \, {\left (2 \, a b d + a^{2} g\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b c + a^{2} f\right )} x^{3} + a^{2} c \log \left (x\right ) \]
1/11*b^2*h*x^11 + 1/10*b^2*g*x^10 + 1/9*b^2*f*x^9 + 1/8*(b^2*e + 2*a*b*h)* x^8 + 1/7*(b^2*d + 2*a*b*g)*x^7 + 1/6*(b^2*c + 2*a*b*f)*x^6 + 1/5*(2*a*b*e + a^2*h)*x^5 + 1/2*a^2*e*x^2 + 1/4*(2*a*b*d + a^2*g)*x^4 + a^2*d*x + 1/3* (2*a*b*c + a^2*f)*x^3 + a^2*c*log(x)
Time = 0.13 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=a^{2} c \log {\left (x \right )} + a^{2} d x + \frac {a^{2} e x^{2}}{2} + \frac {b^{2} f x^{9}}{9} + \frac {b^{2} g x^{10}}{10} + \frac {b^{2} h x^{11}}{11} + x^{8} \left (\frac {a b h}{4} + \frac {b^{2} e}{8}\right ) + x^{7} \cdot \left (\frac {2 a b g}{7} + \frac {b^{2} d}{7}\right ) + x^{6} \left (\frac {a b f}{3} + \frac {b^{2} c}{6}\right ) + x^{5} \left (\frac {a^{2} h}{5} + \frac {2 a b e}{5}\right ) + x^{4} \left (\frac {a^{2} g}{4} + \frac {a b d}{2}\right ) + x^{3} \left (\frac {a^{2} f}{3} + \frac {2 a b c}{3}\right ) \]
a**2*c*log(x) + a**2*d*x + a**2*e*x**2/2 + b**2*f*x**9/9 + b**2*g*x**10/10 + b**2*h*x**11/11 + x**8*(a*b*h/4 + b**2*e/8) + x**7*(2*a*b*g/7 + b**2*d/ 7) + x**6*(a*b*f/3 + b**2*c/6) + x**5*(a**2*h/5 + 2*a*b*e/5) + x**4*(a**2* g/4 + a*b*d/2) + x**3*(a**2*f/3 + 2*a*b*c/3)
Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=\frac {1}{11} \, b^{2} h x^{11} + \frac {1}{10} \, b^{2} g x^{10} + \frac {1}{9} \, b^{2} f x^{9} + \frac {1}{8} \, {\left (b^{2} e + 2 \, a b h\right )} x^{8} + \frac {1}{7} \, {\left (b^{2} d + 2 \, a b g\right )} x^{7} + \frac {1}{6} \, {\left (b^{2} c + 2 \, a b f\right )} x^{6} + \frac {1}{5} \, {\left (2 \, a b e + a^{2} h\right )} x^{5} + \frac {1}{2} \, a^{2} e x^{2} + \frac {1}{4} \, {\left (2 \, a b d + a^{2} g\right )} x^{4} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b c + a^{2} f\right )} x^{3} + a^{2} c \log \left (x\right ) \]
1/11*b^2*h*x^11 + 1/10*b^2*g*x^10 + 1/9*b^2*f*x^9 + 1/8*(b^2*e + 2*a*b*h)* x^8 + 1/7*(b^2*d + 2*a*b*g)*x^7 + 1/6*(b^2*c + 2*a*b*f)*x^6 + 1/5*(2*a*b*e + a^2*h)*x^5 + 1/2*a^2*e*x^2 + 1/4*(2*a*b*d + a^2*g)*x^4 + a^2*d*x + 1/3* (2*a*b*c + a^2*f)*x^3 + a^2*c*log(x)
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=\frac {1}{11} \, b^{2} h x^{11} + \frac {1}{10} \, b^{2} g x^{10} + \frac {1}{9} \, b^{2} f x^{9} + \frac {1}{8} \, b^{2} e x^{8} + \frac {1}{4} \, a b h x^{8} + \frac {1}{7} \, b^{2} d x^{7} + \frac {2}{7} \, a b g x^{7} + \frac {1}{6} \, b^{2} c x^{6} + \frac {1}{3} \, a b f x^{6} + \frac {2}{5} \, a b e x^{5} + \frac {1}{5} \, a^{2} h x^{5} + \frac {1}{2} \, a b d x^{4} + \frac {1}{4} \, a^{2} g x^{4} + \frac {2}{3} \, a b c x^{3} + \frac {1}{3} \, a^{2} f x^{3} + \frac {1}{2} \, a^{2} e x^{2} + a^{2} d x + a^{2} c \log \left ({\left | x \right |}\right ) \]
1/11*b^2*h*x^11 + 1/10*b^2*g*x^10 + 1/9*b^2*f*x^9 + 1/8*b^2*e*x^8 + 1/4*a* b*h*x^8 + 1/7*b^2*d*x^7 + 2/7*a*b*g*x^7 + 1/6*b^2*c*x^6 + 1/3*a*b*f*x^6 + 2/5*a*b*e*x^5 + 1/5*a^2*h*x^5 + 1/2*a*b*d*x^4 + 1/4*a^2*g*x^4 + 2/3*a*b*c* x^3 + 1/3*a^2*f*x^3 + 1/2*a^2*e*x^2 + a^2*d*x + a^2*c*log(abs(x))
Time = 10.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^3\right )^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{x} \, dx=x^3\,\left (\frac {f\,a^2}{3}+\frac {2\,b\,c\,a}{3}\right )+x^6\,\left (\frac {c\,b^2}{6}+\frac {a\,f\,b}{3}\right )+x^4\,\left (\frac {g\,a^2}{4}+\frac {b\,d\,a}{2}\right )+x^7\,\left (\frac {d\,b^2}{7}+\frac {2\,a\,g\,b}{7}\right )+x^5\,\left (\frac {h\,a^2}{5}+\frac {2\,b\,e\,a}{5}\right )+x^8\,\left (\frac {e\,b^2}{8}+\frac {a\,h\,b}{4}\right )+\frac {a^2\,e\,x^2}{2}+\frac {b^2\,f\,x^9}{9}+\frac {b^2\,g\,x^{10}}{10}+\frac {b^2\,h\,x^{11}}{11}+a^2\,c\,\ln \left (x\right )+a^2\,d\,x \]