Integrand size = 30, antiderivative size = 163 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=-\frac {c}{6 a^3 x^6}+\frac {3 b c-a d}{3 a^4 x^3}+\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{6 a^3 b \left (a+b x^3\right )^2}+\frac {3 b^2 c-2 a b d+a^2 e}{3 a^4 \left (a+b x^3\right )}+\frac {\left (6 b^2 c-3 a b d+a^2 e\right ) \log (x)}{a^5}-\frac {\left (6 b^2 c-3 a b d+a^2 e\right ) \log \left (a+b x^3\right )}{3 a^5} \]
-1/6*c/a^3/x^6+1/3*(-a*d+3*b*c)/a^4/x^3+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c) /a^3/b/(b*x^3+a)^2+1/3*(a^2*e-2*a*b*d+3*b^2*c)/a^4/(b*x^3+a)+(a^2*e-3*a*b* d+6*b^2*c)*ln(x)/a^5-1/3*(a^2*e-3*a*b*d+6*b^2*c)*ln(b*x^3+a)/a^5
Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {-\frac {a^2 c}{x^6}-\frac {2 a (-3 b c+a d)}{x^3}+\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b \left (a+b x^3\right )^2}+\frac {2 a \left (3 b^2 c-2 a b d+a^2 e\right )}{a+b x^3}+6 \left (6 b^2 c-3 a b d+a^2 e\right ) \log (x)-2 \left (6 b^2 c-3 a b d+a^2 e\right ) \log \left (a+b x^3\right )}{6 a^5} \]
(-((a^2*c)/x^6) - (2*a*(-3*b*c + a*d))/x^3 + (a^2*(b^3*c - a*b^2*d + a^2*b *e - a^3*f))/(b*(a + b*x^3)^2) + (2*a*(3*b^2*c - 2*a*b*d + a^2*e))/(a + b* x^3) + 6*(6*b^2*c - 3*a*b*d + a^2*e)*Log[x] - 2*(6*b^2*c - 3*a*b*d + a^2*e )*Log[a + b*x^3])/(6*a^5)
Time = 0.43 (sec) , antiderivative size = 161, 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.
| \(\displaystyle \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2361 |
| \(\displaystyle \frac {1}{3} \int \frac {f x^9+e x^6+d x^3+c}{x^9 \left (b x^3+a\right )^3}dx^3\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{3} \int \left (\frac {c}{a^3 x^9}-\frac {b \left (e a^2-3 b d a+6 b^2 c\right )}{a^5 \left (b x^3+a\right )}-\frac {b \left (e a^2-2 b d a+3 b^2 c\right )}{a^4 \left (b x^3+a\right )^2}+\frac {e a^2-3 b d a+6 b^2 c}{a^5 x^3}+\frac {f a^3-b e a^2+b^2 d a-b^3 c}{a^3 \left (b x^3+a\right )^3}+\frac {a d-3 b c}{a^4 x^6}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {3 b c-a d}{a^4 x^3}-\frac {c}{2 a^3 x^6}+\frac {\log \left (x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}-\frac {\log \left (a+b x^3\right ) \left (a^2 e-3 a b d+6 b^2 c\right )}{a^5}+\frac {a^2 e-2 a b d+3 b^2 c}{a^4 \left (a+b x^3\right )}+\frac {a^3 (-f)+a^2 b e-a b^2 d+b^3 c}{2 a^3 b \left (a+b x^3\right )^2}\right )\) |
(-1/2*c/(a^3*x^6) + (3*b*c - a*d)/(a^4*x^3) + (b^3*c - a*b^2*d + a^2*b*e - a^3*f)/(2*a^3*b*(a + b*x^3)^2) + (3*b^2*c - 2*a*b*d + a^2*e)/(a^4*(a + b* x^3)) + ((6*b^2*c - 3*a*b*d + a^2*e)*Log[x^3])/a^5 - ((6*b^2*c - 3*a*b*d + a^2*e)*Log[a + b*x^3])/a^5)/3
3.3.83.3.1 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 = 1.52 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(-\frac {c}{6 a^{3} x^{6}}-\frac {a d -3 b c}{3 a^{4} x^{3}}+\frac {\left (a^{2} e -3 a b d +6 b^{2} c \right ) \ln \left (x \right )}{a^{5}}+\frac {\left (-a^{2} e +3 a b d -6 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )-\frac {a^{2} \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{2 b \left (b \,x^{3}+a \right )^{2}}+\frac {a \left (a^{2} e -2 a b d +3 b^{2} c \right )}{b \,x^{3}+a}}{3 a^{5}}\) | \(153\) |
| norman | \(\frac {-\frac {c}{6 a}-\frac {\left (a d -2 b c \right ) x^{3}}{3 a^{2}}+\frac {\left (f \,a^{3}-2 a^{2} b e +6 a \,b^{2} d -12 b^{3} c \right ) x^{9}}{3 a^{4}}+\frac {b \left (f \,a^{3}-3 a^{2} b e +9 a \,b^{2} d -18 b^{3} c \right ) x^{12}}{6 a^{5}}}{x^{6} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (a^{2} e -3 a b d +6 b^{2} c \right ) \ln \left (x \right )}{a^{5}}-\frac {\left (a^{2} e -3 a b d +6 b^{2} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{5}}\) | \(160\) |
| risch | \(\frac {\frac {b \left (a^{2} e -3 a b d +6 b^{2} c \right ) x^{9}}{3 a^{4}}-\frac {\left (f \,a^{3}-3 a^{2} b e +9 a \,b^{2} d -18 b^{3} c \right ) x^{6}}{6 a^{3} b}-\frac {\left (a d -2 b c \right ) x^{3}}{3 a^{2}}-\frac {c}{6 a}}{x^{6} \left (b \,x^{3}+a \right )^{2}}+\frac {e \ln \left (x \right )}{a^{3}}-\frac {3 \ln \left (x \right ) b d}{a^{4}}+\frac {6 \ln \left (x \right ) b^{2} c}{a^{5}}-\frac {e \ln \left (b \,x^{3}+a \right )}{3 a^{3}}+\frac {\ln \left (b \,x^{3}+a \right ) b d}{a^{4}}-\frac {2 \ln \left (b \,x^{3}+a \right ) b^{2} c}{a^{5}}\) | \(173\) |
| parallelrisch | \(\frac {-4 a^{3} b e \,x^{9}+2 a^{4} f \,x^{9}+12 a^{2} b^{2} d \,x^{9}-18 b^{4} c \,x^{12}+36 \ln \left (x \right ) x^{6} a^{2} b^{2} c +6 \ln \left (b \,x^{3}+a \right ) x^{6} a^{3} b d -12 \ln \left (b \,x^{3}+a \right ) x^{6} a^{2} b^{2} c -a^{4} c -2 a^{4} d \,x^{3}+4 a^{3} b c \,x^{3}-24 a \,b^{3} c \,x^{9}-2 \ln \left (b \,x^{3}+a \right ) x^{12} a^{2} b^{2} e +6 \ln \left (b \,x^{3}+a \right ) x^{12} a \,b^{3} d +6 \ln \left (x \right ) x^{12} a^{2} b^{2} e -18 \ln \left (x \right ) x^{12} a \,b^{3} d -12 \ln \left (b \,x^{3}+a \right ) x^{12} b^{4} c +6 \ln \left (x \right ) x^{6} a^{4} e -2 \ln \left (b \,x^{3}+a \right ) x^{6} a^{4} e +x^{12} a^{3} b f -3 x^{12} a^{2} b^{2} e +9 x^{12} a \,b^{3} d +36 \ln \left (x \right ) x^{12} b^{4} c +12 \ln \left (x \right ) x^{9} a^{3} b e -36 \ln \left (x \right ) x^{9} a^{2} b^{2} d +72 \ln \left (x \right ) x^{9} a \,b^{3} c -4 \ln \left (b \,x^{3}+a \right ) x^{9} a^{3} b e +12 \ln \left (b \,x^{3}+a \right ) x^{9} a^{2} b^{2} d -24 \ln \left (b \,x^{3}+a \right ) x^{9} a \,b^{3} c -18 \ln \left (x \right ) x^{6} a^{3} b d}{6 a^{5} x^{6} \left (b \,x^{3}+a \right )^{2}}\) | \(403\) |
-1/6*c/a^3/x^6-1/3*(a*d-3*b*c)/a^4/x^3+(a^2*e-3*a*b*d+6*b^2*c)*ln(x)/a^5+1 /3/a^5*((-a^2*e+3*a*b*d-6*b^2*c)*ln(b*x^3+a)-1/2*a^2*(a^3*f-a^2*b*e+a*b^2* d-b^3*c)/b/(b*x^3+a)^2+a*(a^2*e-2*a*b*d+3*b^2*c)/(b*x^3+a))
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (153) = 306\).
Time = 0.28 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.94 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} + {\left (18 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 3 \, a^{4} b e - a^{5} f\right )} x^{6} - a^{4} b c + 2 \, {\left (2 \, a^{3} b^{2} c - a^{4} b d\right )} x^{3} - 2 \, {\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \, {\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} + {\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left ({\left (6 \, b^{5} c - 3 \, a b^{4} d + a^{2} b^{3} e\right )} x^{12} + 2 \, {\left (6 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{9} + {\left (6 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d + a^{4} b e\right )} x^{6}\right )} \log \left (x\right )}{6 \, {\left (a^{5} b^{3} x^{12} + 2 \, a^{6} b^{2} x^{9} + a^{7} b x^{6}\right )}} \]
1/6*(2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (18*a^2*b^3*c - 9*a^3*b ^2*d + 3*a^4*b*e - a^5*f)*x^6 - a^4*b*c + 2*(2*a^3*b^2*c - a^4*b*d)*x^3 - 2*((6*b^5*c - 3*a*b^4*d + a^2*b^3*e)*x^12 + 2*(6*a*b^4*c - 3*a^2*b^3*d + a ^3*b^2*e)*x^9 + (6*a^2*b^3*c - 3*a^3*b^2*d + a^4*b*e)*x^6)*log(b*x^3 + a) + 6*((6*b^5*c - 3*a*b^4*d + a^2*b^3*e)*x^12 + 2*(6*a*b^4*c - 3*a^2*b^3*d + a^3*b^2*e)*x^9 + (6*a^2*b^3*c - 3*a^3*b^2*d + a^4*b*e)*x^6)*log(x))/(a^5* b^3*x^12 + 2*a^6*b^2*x^9 + a^7*b*x^6)
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {2 \, {\left (6 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e\right )} x^{9} + {\left (18 \, a b^{3} c - 9 \, a^{2} b^{2} d + 3 \, a^{3} b e - a^{4} f\right )} x^{6} - a^{3} b c + 2 \, {\left (2 \, a^{2} b^{2} c - a^{3} b d\right )} x^{3}}{6 \, {\left (a^{4} b^{3} x^{12} + 2 \, a^{5} b^{2} x^{9} + a^{6} b x^{6}\right )}} - \frac {{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (b x^{3} + a\right )}{3 \, a^{5}} + \frac {{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left (x^{3}\right )}{3 \, a^{5}} \]
1/6*(2*(6*b^4*c - 3*a*b^3*d + a^2*b^2*e)*x^9 + (18*a*b^3*c - 9*a^2*b^2*d + 3*a^3*b*e - a^4*f)*x^6 - a^3*b*c + 2*(2*a^2*b^2*c - a^3*b*d)*x^3)/(a^4*b^ 3*x^12 + 2*a^5*b^2*x^9 + a^6*b*x^6) - 1/3*(6*b^2*c - 3*a*b*d + a^2*e)*log( b*x^3 + a)/a^5 + 1/3*(6*b^2*c - 3*a*b*d + a^2*e)*log(x^3)/a^5
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.13 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {{\left (6 \, b^{2} c - 3 \, a b d + a^{2} e\right )} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {{\left (6 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5} b} + \frac {12 \, b^{4} c x^{9} - 6 \, a b^{3} d x^{9} + 2 \, a^{2} b^{2} e x^{9} + 18 \, a b^{3} c x^{6} - 9 \, a^{2} b^{2} d x^{6} + 3 \, a^{3} b e x^{6} - a^{4} f x^{6} + 4 \, a^{2} b^{2} c x^{3} - 2 \, a^{3} b d x^{3} - a^{3} b c}{6 \, {\left (b x^{6} + a x^{3}\right )}^{2} a^{4} b} \]
(6*b^2*c - 3*a*b*d + a^2*e)*log(abs(x))/a^5 - 1/3*(6*b^3*c - 3*a*b^2*d + a ^2*b*e)*log(abs(b*x^3 + a))/(a^5*b) + 1/6*(12*b^4*c*x^9 - 6*a*b^3*d*x^9 + 2*a^2*b^2*e*x^9 + 18*a*b^3*c*x^6 - 9*a^2*b^2*d*x^6 + 3*a^3*b*e*x^6 - a^4*f *x^6 + 4*a^2*b^2*c*x^3 - 2*a^3*b*d*x^3 - a^3*b*c)/((b*x^6 + a*x^3)^2*a^4*b )
Time = 9.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^7 \left (a+b x^3\right )^3} \, dx=\frac {\ln \left (x\right )\,\left (e\,a^2-3\,d\,a\,b+6\,c\,b^2\right )}{a^5}-\frac {\ln \left (b\,x^3+a\right )\,\left (e\,a^2-3\,d\,a\,b+6\,c\,b^2\right )}{3\,a^5}-\frac {\frac {c}{6\,a}+\frac {x^3\,\left (a\,d-2\,b\,c\right )}{3\,a^2}-\frac {b\,x^9\,\left (e\,a^2-3\,d\,a\,b+6\,c\,b^2\right )}{3\,a^4}-\frac {x^6\,\left (-f\,a^3+3\,e\,a^2\,b-9\,d\,a\,b^2+18\,c\,b^3\right )}{6\,a^3\,b}}{a^2\,x^6+2\,a\,b\,x^9+b^2\,x^{12}} \]