Integrand size = 21, antiderivative size = 99 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {a^2 A}{6 x^6}-\frac {a (2 A b+a B)}{5 x^5}-\frac {2 a b B+A \left (b^2+2 a c\right )}{4 x^4}-\frac {b^2 B+2 A b c+2 a B c}{3 x^3}-\frac {c (2 b B+A c)}{2 x^2}-\frac {B c^2}{x} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 99, 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} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {a^2 A}{6 x^6}-\frac {A \left (2 a c+b^2\right )+2 a b B}{4 x^4}-\frac {2 a B c+2 A b c+b^2 B}{3 x^3}-\frac {a (a B+2 A b)}{5 x^5}-\frac {c (A c+2 b B)}{2 x^2}-\frac {B c^2}{x} \]
[In]
[Out]
Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 A}{x^7}+\frac {a (2 A b+a B)}{x^6}+\frac {2 a b B+A \left (b^2+2 a c\right )}{x^5}+\frac {b^2 B+2 A b c+2 a B c}{x^4}+\frac {c (2 b B+A c)}{x^3}+\frac {B c^2}{x^2}\right ) \, dx \\ & = -\frac {a^2 A}{6 x^6}-\frac {a (2 A b+a B)}{5 x^5}-\frac {2 a b B+A \left (b^2+2 a c\right )}{4 x^4}-\frac {b^2 B+2 A b c+2 a B c}{3 x^3}-\frac {c (2 b B+A c)}{2 x^2}-\frac {B c^2}{x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {2 a^2 (5 A+6 B x)+2 a x (5 B x (3 b+4 c x)+3 A (4 b+5 c x))+5 x^2 \left (4 B x \left (b^2+3 b c x+3 c^2 x^2\right )+A \left (3 b^2+8 b c x+6 c^2 x^2\right )\right )}{60 x^6} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {a^{2} A}{6 x^{6}}-\frac {2 A a c +A \,b^{2}+2 a b B}{4 x^{4}}-\frac {a \left (2 A b +B a \right )}{5 x^{5}}-\frac {c \left (A c +2 B b \right )}{2 x^{2}}-\frac {B \,c^{2}}{x}-\frac {2 A b c +2 B a c +B \,b^{2}}{3 x^{3}}\) | \(90\) |
norman | \(\frac {-B \,c^{2} x^{5}+\left (-\frac {1}{2} A \,c^{2}-B b c \right ) x^{4}+\left (-\frac {2}{3} A b c -\frac {2}{3} B a c -\frac {1}{3} B \,b^{2}\right ) x^{3}+\left (-\frac {1}{2} A a c -\frac {1}{4} A \,b^{2}-\frac {1}{2} a b B \right ) x^{2}+\left (-\frac {2}{5} A b a -\frac {1}{5} B \,a^{2}\right ) x -\frac {A \,a^{2}}{6}}{x^{6}}\) | \(93\) |
risch | \(\frac {-B \,c^{2} x^{5}+\left (-\frac {1}{2} A \,c^{2}-B b c \right ) x^{4}+\left (-\frac {2}{3} A b c -\frac {2}{3} B a c -\frac {1}{3} B \,b^{2}\right ) x^{3}+\left (-\frac {1}{2} A a c -\frac {1}{4} A \,b^{2}-\frac {1}{2} a b B \right ) x^{2}+\left (-\frac {2}{5} A b a -\frac {1}{5} B \,a^{2}\right ) x -\frac {A \,a^{2}}{6}}{x^{6}}\) | \(93\) |
gosper | \(-\frac {60 B \,c^{2} x^{5}+30 A \,c^{2} x^{4}+60 x^{4} B b c +40 x^{3} A b c +40 a B c \,x^{3}+20 B \,b^{2} x^{3}+30 a A c \,x^{2}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+24 a A b x +12 a^{2} B x +10 A \,a^{2}}{60 x^{6}}\) | \(102\) |
parallelrisch | \(-\frac {60 B \,c^{2} x^{5}+30 A \,c^{2} x^{4}+60 x^{4} B b c +40 x^{3} A b c +40 a B c \,x^{3}+20 B \,b^{2} x^{3}+30 a A c \,x^{2}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+24 a A b x +12 a^{2} B x +10 A \,a^{2}}{60 x^{6}}\) | \(102\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {60 \, B c^{2} x^{5} + 30 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 20 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 10 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \]
[In]
[Out]
Time = 7.62 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=\frac {- 10 A a^{2} - 60 B c^{2} x^{5} + x^{4} \left (- 30 A c^{2} - 60 B b c\right ) + x^{3} \left (- 40 A b c - 40 B a c - 20 B b^{2}\right ) + x^{2} \left (- 30 A a c - 15 A b^{2} - 30 B a b\right ) + x \left (- 24 A a b - 12 B a^{2}\right )}{60 x^{6}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {60 \, B c^{2} x^{5} + 30 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 20 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 10 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {60 \, B c^{2} x^{5} + 60 \, B b c x^{4} + 30 \, A c^{2} x^{4} + 20 \, B b^{2} x^{3} + 40 \, B a c x^{3} + 40 \, A b c x^{3} + 30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 30 \, A a c x^{2} + 12 \, B a^{2} x + 24 \, A a b x + 10 \, A a^{2}}{60 \, x^{6}} \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.92 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^7} \, dx=-\frac {x^4\,\left (\frac {A\,c^2}{2}+B\,b\,c\right )+\frac {A\,a^2}{6}+x^2\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}+\frac {A\,a\,c}{2}\right )+x^3\,\left (\frac {B\,b^2}{3}+\frac {2\,A\,c\,b}{3}+\frac {2\,B\,a\,c}{3}\right )+x\,\left (\frac {B\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+B\,c^2\,x^5}{x^6} \]
[In]
[Out]