Integrand size = 30, antiderivative size = 109 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {c}{3 a^2 x^3}-\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{3 a^2 b^2 \left (a+b x^3\right )}-\frac {(2 b c-a d) \log (x)}{a^3}+\frac {\left (2 b^3 c-a b^2 d+a^3 f\right ) \log \left (a+b x^3\right )}{3 a^3 b^2} \]
-1/3*c/a^2/x^3+1/3*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/a^2/b^2/(b*x^3+a)-(-a*d+2 *b*c)*ln(x)/a^3+1/3*(a^3*f-a*b^2*d+2*b^3*c)*ln(b*x^3+a)/a^3/b^2
Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {a c}{x^3}+\frac {a \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{b^2 \left (a+b x^3\right )}+3 (-2 b c+a d) \log (x)+\frac {\left (2 b^3 c-a b^2 d+a^3 f\right ) \log \left (a+b x^3\right )}{b^2}}{3 a^3} \]
(-((a*c)/x^3) + (a*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f))/(b^2*(a + b*x^3 )) + 3*(-2*b*c + a*d)*Log[x] + ((2*b^3*c - a*b^2*d + a^3*f)*Log[a + b*x^3] )/b^2)/(3*a^3)
Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2361, 2123, 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.
\(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 2361 |
\(\displaystyle \frac {1}{3} \int \frac {f x^9+e x^6+d x^3+c}{x^6 \left (b x^3+a\right )^2}dx^3\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \frac {1}{3} \int \left (\frac {c}{a^2 x^6}+\frac {f a^3-b^2 d a+2 b^3 c}{a^3 b \left (b x^3+a\right )}+\frac {-f a^3+b e a^2-b^2 d a+b^3 c}{a^2 b \left (b x^3+a\right )^2}+\frac {a d-2 b c}{a^3 x^3}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {\log \left (a+b x^3\right ) \left (a^3 f-a b^2 d+2 b^3 c\right )}{a^3 b^2}-\frac {\log \left (x^3\right ) (2 b c-a d)}{a^3}-\frac {c}{a^2 x^3}-\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{a^2 b^2 \left (a+b x^3\right )}\right )\) |
(-(c/(a^2*x^3)) - (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(a^2*b^2*(a + b*x^3) ) - ((2*b*c - a*d)*Log[x^3])/a^3 + ((2*b^3*c - a*b^2*d + a^3*f)*Log[a + b* x^3])/(a^3*b^2))/3
3.3.56.3.1 Defintions of rubi rules used
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*SubstFor[x^n, Pq, x]*(a + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && PolyQ[Pq, x^n] && IntegerQ[S implify[(m + 1)/n]]
Time = 1.50 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {c}{3 a^{2} x^{3}}+\frac {\left (a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\frac {\left (f \,a^{3}-a \,b^{2} d +2 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b^{2}}+\frac {a \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{b^{2} \left (b \,x^{3}+a \right )}}{3 a^{3}}\) | \(101\) |
norman | \(\frac {-\frac {c}{3 a}+\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -2 b^{3} c \right ) x^{3}}{3 a^{2} b^{2}}}{x^{3} \left (b \,x^{3}+a \right )}+\frac {\left (a d -2 b c \right ) \ln \left (x \right )}{a^{3}}+\frac {\left (f \,a^{3}-a \,b^{2} d +2 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{3} b^{2}}\) | \(107\) |
risch | \(\frac {-\frac {c}{3 a}+\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -2 b^{3} c \right ) x^{3}}{3 a^{2} b^{2}}}{x^{3} \left (b \,x^{3}+a \right )}+\frac {d \ln \left (x \right )}{a^{2}}-\frac {2 b c \ln \left (x \right )}{a^{3}}+\frac {\ln \left (-b \,x^{3}-a \right ) f}{3 b^{2}}-\frac {\ln \left (-b \,x^{3}-a \right ) d}{3 a^{2}}+\frac {2 b \ln \left (-b \,x^{3}-a \right ) c}{3 a^{3}}\) | \(126\) |
parallelrisch | \(\frac {3 \ln \left (x \right ) x^{6} a \,b^{3} d -6 \ln \left (x \right ) x^{6} b^{4} c +\ln \left (b \,x^{3}+a \right ) x^{6} a^{3} b f -\ln \left (b \,x^{3}+a \right ) x^{6} a \,b^{3} d +2 \ln \left (b \,x^{3}+a \right ) x^{6} b^{4} c +3 \ln \left (x \right ) x^{3} a^{2} b^{2} d -6 \ln \left (x \right ) x^{3} a \,b^{3} c +\ln \left (b \,x^{3}+a \right ) x^{3} a^{4} f -\ln \left (b \,x^{3}+a \right ) x^{3} a^{2} b^{2} d +2 \ln \left (b \,x^{3}+a \right ) x^{3} a \,b^{3} c +a^{4} f \,x^{3}-a^{3} b e \,x^{3}+a^{2} b^{2} d \,x^{3}-2 a \,b^{3} c \,x^{3}-a^{2} b^{2} c}{3 a^{3} b^{2} x^{3} \left (b \,x^{3}+a \right )}\) | \(225\) |
-1/3*c/a^2/x^3+(a*d-2*b*c)/a^3*ln(x)+1/3/a^3*((a^3*f-a*b^2*d+2*b^3*c)/b^2* ln(b*x^3+a)+a*(a^3*f-a^2*b*e+a*b^2*d-b^3*c)/b^2/(b*x^3+a))
Time = 0.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.58 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {a^{2} b^{2} c + {\left (2 \, a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x^{3} - {\left ({\left (2 \, b^{4} c - a b^{3} d + a^{3} b f\right )} x^{6} + {\left (2 \, a b^{3} c - a^{2} b^{2} d + a^{4} f\right )} x^{3}\right )} \log \left (b x^{3} + a\right ) + 3 \, {\left ({\left (2 \, b^{4} c - a b^{3} d\right )} x^{6} + {\left (2 \, a b^{3} c - a^{2} b^{2} d\right )} x^{3}\right )} \log \left (x\right )}{3 \, {\left (a^{3} b^{3} x^{6} + a^{4} b^{2} x^{3}\right )}} \]
-1/3*(a^2*b^2*c + (2*a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x^3 - ((2*b^4* c - a*b^3*d + a^3*b*f)*x^6 + (2*a*b^3*c - a^2*b^2*d + a^4*f)*x^3)*log(b*x^ 3 + a) + 3*((2*b^4*c - a*b^3*d)*x^6 + (2*a*b^3*c - a^2*b^2*d)*x^3)*log(x)) /(a^3*b^3*x^6 + a^4*b^2*x^3)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {a b^{2} c + {\left (2 \, b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3}}{3 \, {\left (a^{2} b^{3} x^{6} + a^{3} b^{2} x^{3}\right )}} - \frac {{\left (2 \, b c - a d\right )} \log \left (x^{3}\right )}{3 \, a^{3}} + \frac {{\left (2 \, b^{3} c - a b^{2} d + a^{3} f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{3} b^{2}} \]
-1/3*(a*b^2*c + (2*b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3)/(a^2*b^3*x^6 + a^3*b^2*x^3) - 1/3*(2*b*c - a*d)*log(x^3)/a^3 + 1/3*(2*b^3*c - a*b^2*d + a ^3*f)*log(b*x^3 + a)/(a^3*b^2)
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=-\frac {{\left (2 \, b c - a d\right )} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {{\left (2 \, b^{3} c - a b^{2} d + a^{3} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3} b^{2}} - \frac {a^{2} b f x^{6} + 4 \, b^{3} c x^{3} - 2 \, a b^{2} d x^{3} + 2 \, a^{2} b e x^{3} - a^{3} f x^{3} + 2 \, a b^{2} c}{6 \, {\left (b x^{6} + a x^{3}\right )} a^{2} b^{2}} \]
-(2*b*c - a*d)*log(abs(x))/a^3 + 1/3*(2*b^3*c - a*b^2*d + a^3*f)*log(abs(b *x^3 + a))/(a^3*b^2) - 1/6*(a^2*b*f*x^6 + 4*b^3*c*x^3 - 2*a*b^2*d*x^3 + 2* a^2*b*e*x^3 - a^3*f*x^3 + 2*a*b^2*c)/((b*x^6 + a*x^3)*a^2*b^2)
Time = 9.79 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^4 \left (a+b x^3\right )^2} \, dx=\frac {\ln \left (x\right )\,\left (a\,d-2\,b\,c\right )}{a^3}-\frac {\frac {c}{3\,a}+\frac {x^3\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+2\,c\,b^3\right )}{3\,a^2\,b^2}}{b\,x^6+a\,x^3}+\frac {\ln \left (b\,x^3+a\right )\,\left (f\,a^3-d\,a\,b^2+2\,c\,b^3\right )}{3\,a^3\,b^2} \]