Integrand size = 21, antiderivative size = 160 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=-\frac {a^3 A}{7 x^7}-\frac {a^2 (3 A b+a B)}{6 x^6}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{5 x^5}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{4 x^4}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{3 x^3}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{2 x^2}-\frac {c^2 (3 b B+A c)}{x}+B c^3 \log (x) \]
[Out]
Time = 0.07 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {779} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=-\frac {a^3 A}{7 x^7}-\frac {a^2 (a B+3 A b)}{6 x^6}-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{5 x^5}-\frac {3 c \left (a B c+A b c+b^2 B\right )}{2 x^2}-\frac {3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{3 x^3}-\frac {A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{4 x^4}-\frac {c^2 (A c+3 b B)}{x}+B c^3 \log (x) \]
[In]
[Out]
Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^3 A}{x^8}+\frac {a^2 (3 A b+a B)}{x^7}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^6}+\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^5}+\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^4}+\frac {3 c \left (b^2 B+A b c+a B c\right )}{x^3}+\frac {c^2 (3 b B+A c)}{x^2}+\frac {B c^3}{x}\right ) \, dx \\ & = -\frac {a^3 A}{7 x^7}-\frac {a^2 (3 A b+a B)}{6 x^6}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{5 x^5}-\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{4 x^4}-\frac {b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{3 x^3}-\frac {3 c \left (b^2 B+A b c+a B c\right )}{2 x^2}-\frac {c^2 (3 b B+A c)}{x}+B c^3 \log (x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=-\frac {10 a^3 (6 A+7 B x)+21 a^2 x (3 B x (4 b+5 c x)+2 A (5 b+6 c x))+21 a x^2 \left (5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )+2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+35 x^3 \left (2 b B x \left (2 b^2+9 b c x+18 c^2 x^2\right )+3 A \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )\right )-420 B c^3 x^7 \log (x)}{420 x^7} \]
[In]
[Out]
Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {a^{2} \left (3 A b +B a \right )}{6 x^{6}}-\frac {6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}}{4 x^{4}}-\frac {3 a \left (A a c +A \,b^{2}+a b B \right )}{5 x^{5}}+B \,c^{3} \ln \left (x \right )-\frac {3 c \left (A b c +B a c +B \,b^{2}\right )}{2 x^{2}}-\frac {a^{3} A}{7 x^{7}}-\frac {c^{2} \left (A c +3 B b \right )}{x}-\frac {3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}}{3 x^{3}}\) | \(152\) |
norman | \(\frac {\left (-\frac {1}{2} A \,a^{2} b -\frac {1}{6} B \,a^{3}\right ) x +\left (-\frac {3}{5} A \,a^{2} c -\frac {3}{5} A a \,b^{2}-\frac {3}{5} B b \,a^{2}\right ) x^{2}+\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B a \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-A a \,c^{2}-A \,b^{2} c -2 B a b c -\frac {1}{3} B \,b^{3}\right ) x^{4}+\left (-\frac {3}{2} A a b c -\frac {1}{4} A \,b^{3}-\frac {3}{4} B \,a^{2} c -\frac {3}{4} B a \,b^{2}\right ) x^{3}+\left (-A \,c^{3}-3 B b \,c^{2}\right ) x^{6}-\frac {A \,a^{3}}{7}}{x^{7}}+B \,c^{3} \ln \left (x \right )\) | \(168\) |
risch | \(\frac {\left (-\frac {1}{2} A \,a^{2} b -\frac {1}{6} B \,a^{3}\right ) x +\left (-\frac {3}{5} A \,a^{2} c -\frac {3}{5} A a \,b^{2}-\frac {3}{5} B b \,a^{2}\right ) x^{2}+\left (-\frac {3}{2} A b \,c^{2}-\frac {3}{2} B a \,c^{2}-\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-A a \,c^{2}-A \,b^{2} c -2 B a b c -\frac {1}{3} B \,b^{3}\right ) x^{4}+\left (-\frac {3}{2} A a b c -\frac {1}{4} A \,b^{3}-\frac {3}{4} B \,a^{2} c -\frac {3}{4} B a \,b^{2}\right ) x^{3}+\left (-A \,c^{3}-3 B b \,c^{2}\right ) x^{6}-\frac {A \,a^{3}}{7}}{x^{7}}+B \,c^{3} \ln \left (x \right )\) | \(168\) |
parallelrisch | \(-\frac {-420 B \,c^{3} \ln \left (x \right ) x^{7}+420 A \,c^{3} x^{6}+1260 B b \,c^{2} x^{6}+630 A b \,c^{2} x^{5}+630 a B \,c^{2} x^{5}+630 B \,b^{2} c \,x^{5}+420 a A \,c^{2} x^{4}+420 A \,b^{2} c \,x^{4}+840 B a b c \,x^{4}+140 x^{4} B \,b^{3}+630 A a b c \,x^{3}+105 A \,b^{3} x^{3}+315 a^{2} B c \,x^{3}+315 B a \,b^{2} x^{3}+252 a^{2} A c \,x^{2}+252 A a \,b^{2} x^{2}+252 B \,a^{2} b \,x^{2}+210 A \,a^{2} b x +70 a^{3} B x +60 A \,a^{3}}{420 x^{7}}\) | \(194\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=\frac {420 \, B c^{3} x^{7} \log \left (x\right ) - 420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} - 630 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} - 140 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 60 \, A a^{3} - 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} - 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B c^{3} \log \left (x\right ) - \frac {420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 140 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B c^{3} \log \left ({\left | x \right |}\right ) - \frac {420 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 630 \, {\left (B b^{2} c + B a c^{2} + A b c^{2}\right )} x^{5} + 140 \, {\left (B b^{3} + 6 \, B a b c + 3 \, A b^{2} c + 3 \, A a c^{2}\right )} x^{4} + 60 \, A a^{3} + 105 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} x^{3} + 252 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 70 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]
[In]
[Out]
Time = 10.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^8} \, dx=B\,c^3\,\ln \left (x\right )-\frac {x^3\,\left (\frac {3\,B\,c\,a^2}{4}+\frac {3\,B\,a\,b^2}{4}+\frac {3\,A\,c\,a\,b}{2}+\frac {A\,b^3}{4}\right )+x^4\,\left (\frac {B\,b^3}{3}+A\,b^2\,c+2\,B\,a\,b\,c+A\,a\,c^2\right )+x\,\left (\frac {B\,a^3}{6}+\frac {A\,b\,a^2}{2}\right )+\frac {A\,a^3}{7}+x^6\,\left (A\,c^3+3\,B\,b\,c^2\right )+x^2\,\left (\frac {3\,B\,a^2\,b}{5}+\frac {3\,A\,c\,a^2}{5}+\frac {3\,A\,a\,b^2}{5}\right )+x^5\,\left (\frac {3\,B\,b^2\,c}{2}+\frac {3\,A\,b\,c^2}{2}+\frac {3\,B\,a\,c^2}{2}\right )}{x^7} \]
[In]
[Out]