Integrand size = 30, antiderivative size = 214 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=-\frac {c}{12 a^2 x^{12}}+\frac {2 b c-a d}{9 a^3 x^9}-\frac {3 b^2 c-2 a b d+a^2 e}{6 a^4 x^6}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 \left (a+b x^3\right )}+\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log (x)}{a^6}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log \left (a+b x^3\right )}{3 a^6} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1835, 1634} \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=\frac {2 b c-a d}{9 a^3 x^9}-\frac {c}{12 a^2 x^{12}}-\frac {a^2 e-2 a b d+3 b^2 c}{6 a^4 x^6}-\frac {b \log \left (a+b x^3\right ) \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{3 a^6}+\frac {b \log (x) \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6}+\frac {b \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^5 \left (a+b x^3\right )}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3} \]
[In]
[Out]
Rule 1634
Rule 1835
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {c+d x+e x^2+f x^3}{x^5 (a+b x)^2} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {c}{a^2 x^5}+\frac {-2 b c+a d}{a^3 x^4}+\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x^3}+\frac {-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f}{a^5 x^2}-\frac {b \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^6 x}+\frac {b^2 \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a^5 (a+b x)^2}+\frac {b^2 \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^6 (a+b x)}\right ) \, dx,x,x^3\right ) \\ & = -\frac {c}{12 a^2 x^{12}}+\frac {2 b c-a d}{9 a^3 x^9}-\frac {3 b^2 c-2 a b d+a^2 e}{6 a^4 x^6}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{3 a^5 \left (a+b x^3\right )}+\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log (x)}{a^6}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log \left (a+b x^3\right )}{3 a^6} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=-\frac {\frac {3 a^4 c}{x^{12}}+\frac {4 a^3 (-2 b c+a d)}{x^9}+\frac {6 a^2 \left (3 b^2 c-2 a b d+a^2 e\right )}{x^6}+\frac {12 a \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{x^3}+\frac {12 a b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right )}{a+b x^3}-36 b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log (x)+12 b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right ) \log \left (a+b x^3\right )}{36 a^6} \]
[In]
[Out]
Time = 1.52 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {c}{12 a^{2} x^{12}}-\frac {a d -2 b c}{9 a^{3} x^{9}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{6 a^{4} x^{6}}-\frac {f \,a^{3}-2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c}{3 a^{5} x^{3}}-\frac {b \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) \ln \left (x \right )}{a^{6}}+\frac {b^{2} \left (\frac {\left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{b}-\frac {a \left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right )}{b \left (b \,x^{3}+a \right )}\right )}{3 a^{6}}\) | \(209\) |
norman | \(\frac {-\frac {c}{12 a}-\frac {\left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) x^{9}}{6 a^{4}}-\frac {\left (4 a d -5 b c \right ) x^{3}}{36 a^{2}}-\frac {\left (3 a^{2} e -4 a b d +5 b^{2} c \right ) x^{6}}{18 a^{3}}+\frac {b \left (2 a^{3} b f -3 a^{2} e \,b^{2}+4 a \,b^{3} d -5 b^{4} c \right ) x^{15}}{3 a^{6}}}{x^{12} \left (b \,x^{3}+a \right )}-\frac {b \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) \ln \left (x \right )}{a^{6}}+\frac {b \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) \ln \left (b \,x^{3}+a \right )}{3 a^{6}}\) | \(215\) |
risch | \(\frac {-\frac {c}{12 a}-\frac {\left (4 a d -5 b c \right ) x^{3}}{36 a^{2}}-\frac {\left (3 a^{2} e -4 a b d +5 b^{2} c \right ) x^{6}}{18 a^{3}}-\frac {\left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) x^{9}}{6 a^{4}}-\frac {b \left (2 f \,a^{3}-3 a^{2} b e +4 a \,b^{2} d -5 b^{3} c \right ) x^{12}}{3 a^{5}}}{x^{12} \left (b \,x^{3}+a \right )}-\frac {2 b \ln \left (x \right ) f}{a^{3}}+\frac {3 b^{2} \ln \left (x \right ) e}{a^{4}}-\frac {4 b^{3} \ln \left (x \right ) d}{a^{5}}+\frac {5 b^{4} \ln \left (x \right ) c}{a^{6}}+\frac {2 b \ln \left (-b \,x^{3}-a \right ) f}{3 a^{3}}-\frac {b^{2} \ln \left (-b \,x^{3}-a \right ) e}{a^{4}}+\frac {4 b^{3} \ln \left (-b \,x^{3}-a \right ) d}{3 a^{5}}-\frac {5 b^{4} \ln \left (-b \,x^{3}-a \right ) c}{3 a^{6}}\) | \(256\) |
parallelrisch | \(-\frac {3 a^{5} b c -180 \ln \left (x \right ) x^{12} a \,b^{5} c -24 \ln \left (b \,x^{3}+a \right ) x^{12} a^{4} b^{2} f +36 \ln \left (b \,x^{3}+a \right ) x^{12} a^{3} b^{3} e -48 \ln \left (b \,x^{3}+a \right ) x^{12} a^{2} b^{4} d +60 \ln \left (b \,x^{3}+a \right ) x^{12} a \,b^{5} c +72 \ln \left (x \right ) x^{15} a^{3} b^{3} f -108 \ln \left (x \right ) x^{15} a^{2} b^{4} e +144 \ln \left (x \right ) x^{15} a \,b^{5} d -24 \ln \left (b \,x^{3}+a \right ) x^{15} a^{3} b^{3} f +36 \ln \left (b \,x^{3}+a \right ) x^{15} a^{2} b^{4} e -48 \ln \left (b \,x^{3}+a \right ) x^{15} a \,b^{5} d +72 \ln \left (x \right ) x^{12} a^{4} b^{2} f -108 \ln \left (x \right ) x^{12} a^{3} b^{3} e +144 \ln \left (x \right ) x^{12} a^{2} b^{4} d +24 x^{12} a^{4} b^{2} f -36 x^{12} a^{3} b^{3} e +48 x^{12} a^{2} b^{4} d -60 x^{12} a \,b^{5} c +12 x^{9} a^{5} b f -18 x^{9} a^{4} b^{2} e +24 x^{9} a^{3} b^{3} d -30 x^{9} a^{2} b^{4} c +6 x^{6} a^{5} b e -8 x^{6} a^{4} b^{2} d +10 x^{6} a^{3} b^{3} c +4 x^{3} a^{5} b d -5 x^{3} a^{4} b^{2} c -180 \ln \left (x \right ) x^{15} b^{6} c +60 \ln \left (b \,x^{3}+a \right ) x^{15} b^{6} c}{36 a^{6} b \,x^{12} \left (b \,x^{3}+a \right )}\) | \(435\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.45 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=\frac {12 \, {\left (5 \, a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{12} + 6 \, {\left (5 \, a^{2} b^{3} c - 4 \, a^{3} b^{2} d + 3 \, a^{4} b e - 2 \, a^{5} f\right )} x^{9} - 2 \, {\left (5 \, a^{3} b^{2} c - 4 \, a^{4} b d + 3 \, a^{5} e\right )} x^{6} - 3 \, a^{5} c + {\left (5 \, a^{4} b c - 4 \, a^{5} d\right )} x^{3} - 12 \, {\left ({\left (5 \, b^{5} c - 4 \, a b^{4} d + 3 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{15} + {\left (5 \, a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{12}\right )} \log \left (b x^{3} + a\right ) + 36 \, {\left ({\left (5 \, b^{5} c - 4 \, a b^{4} d + 3 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} x^{15} + {\left (5 \, a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 2 \, a^{4} b f\right )} x^{12}\right )} \log \left (x\right )}{36 \, {\left (a^{6} b x^{15} + a^{7} x^{12}\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=\frac {12 \, {\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} x^{12} + 6 \, {\left (5 \, a b^{3} c - 4 \, a^{2} b^{2} d + 3 \, a^{3} b e - 2 \, a^{4} f\right )} x^{9} - 2 \, {\left (5 \, a^{2} b^{2} c - 4 \, a^{3} b d + 3 \, a^{4} e\right )} x^{6} - 3 \, a^{4} c + {\left (5 \, a^{3} b c - 4 \, a^{4} d\right )} x^{3}}{36 \, {\left (a^{5} b x^{15} + a^{6} x^{12}\right )}} - \frac {{\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} \log \left (b x^{3} + a\right )}{3 \, a^{6}} + \frac {{\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} \log \left (x^{3}\right )}{3 \, a^{6}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.51 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=\frac {{\left (5 \, b^{4} c - 4 \, a b^{3} d + 3 \, a^{2} b^{2} e - 2 \, a^{3} b f\right )} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {{\left (5 \, b^{5} c - 4 \, a b^{4} d + 3 \, a^{2} b^{3} e - 2 \, a^{3} b^{2} f\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{6} b} + \frac {5 \, b^{5} c x^{3} - 4 \, a b^{4} d x^{3} + 3 \, a^{2} b^{3} e x^{3} - 2 \, a^{3} b^{2} f x^{3} + 6 \, a b^{4} c - 5 \, a^{2} b^{3} d + 4 \, a^{3} b^{2} e - 3 \, a^{4} b f}{3 \, {\left (b x^{3} + a\right )} a^{6}} - \frac {125 \, b^{4} c x^{12} - 100 \, a b^{3} d x^{12} + 75 \, a^{2} b^{2} e x^{12} - 50 \, a^{3} b f x^{12} - 48 \, a b^{3} c x^{9} + 36 \, a^{2} b^{2} d x^{9} - 24 \, a^{3} b e x^{9} + 12 \, a^{4} f x^{9} + 18 \, a^{2} b^{2} c x^{6} - 12 \, a^{3} b d x^{6} + 6 \, a^{4} e x^{6} - 8 \, a^{3} b c x^{3} + 4 \, a^{4} d x^{3} + 3 \, a^{4} c}{36 \, a^{6} x^{12}} \]
[In]
[Out]
Time = 10.33 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x^3+e x^6+f x^9}{x^{13} \left (a+b x^3\right )^2} \, dx=\frac {\ln \left (x\right )\,\left (-2\,f\,a^3\,b+3\,e\,a^2\,b^2-4\,d\,a\,b^3+5\,c\,b^4\right )}{a^6}-\frac {\ln \left (b\,x^3+a\right )\,\left (-2\,f\,a^3\,b+3\,e\,a^2\,b^2-4\,d\,a\,b^3+5\,c\,b^4\right )}{3\,a^6}-\frac {\frac {c}{12\,a}-\frac {x^9\,\left (-2\,f\,a^3+3\,e\,a^2\,b-4\,d\,a\,b^2+5\,c\,b^3\right )}{6\,a^4}+\frac {x^3\,\left (4\,a\,d-5\,b\,c\right )}{36\,a^2}+\frac {x^6\,\left (3\,e\,a^2-4\,d\,a\,b+5\,c\,b^2\right )}{18\,a^3}-\frac {b\,x^{12}\,\left (-2\,f\,a^3+3\,e\,a^2\,b-4\,d\,a\,b^2+5\,c\,b^3\right )}{3\,a^5}}{b\,x^{15}+a\,x^{12}} \]
[In]
[Out]