Integrand size = 21, antiderivative size = 155 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=-\frac {a^3 A}{3 x^3}-\frac {a^2 (3 A b+a B)}{2 x^2}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x+\frac {3}{2} c \left (b^2 B+A b c+a B c\right ) x^2+\frac {1}{3} c^2 (3 b B+A c) x^3+\frac {1}{4} B c^3 x^4+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) \log (x) \]
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Time = 0.10 (sec) , antiderivative size = 155, 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} \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=-\frac {a^3 A}{3 x^3}-\frac {a^2 (a B+3 A b)}{2 x^2}+\frac {3}{2} c x^2 \left (a B c+A b c+b^2 B\right )-\frac {3 a \left (A \left (a c+b^2\right )+a b B\right )}{x}+x \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\log (x) \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{3} c^2 x^3 (A c+3 b B)+\frac {1}{4} B c^3 x^4 \]
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Rule 779
Rubi steps \begin{align*} \text {integral}& = \int \left (b^3 B \left (1+\frac {3 c \left (2 a b B+A \left (b^2+a c\right )\right )}{b^3 B}\right )+\frac {a^3 A}{x^4}+\frac {a^2 (3 A b+a B)}{x^3}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^2}+\frac {3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x}+3 c \left (b^2 B+A b c+a B c\right ) x+c^2 (3 b B+A c) x^2+B c^3 x^3\right ) \, dx \\ & = -\frac {a^3 A}{3 x^3}-\frac {a^2 (3 A b+a B)}{2 x^2}-\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x+\frac {3}{2} c \left (b^2 B+A b c+a B c\right ) x^2+\frac {1}{3} c^2 (3 b B+A c) x^3+\frac {1}{4} B c^3 x^4+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) \log (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=\frac {-2 a^3 (2 A+3 B x)+x^4 \left (12 b^3 B+18 b^2 c (2 A+B x)+6 b c^2 x (3 A+2 B x)+c^3 x^2 (4 A+3 B x)\right )-18 a^2 x (2 b B x+A (b+2 c x))+18 a x^2 \left (B c x^2 (4 b+c x)-2 A \left (b^2-c^2 x^2\right )\right )+12 \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^3 \log (x)}{12 x^3} \]
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Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {B \,c^{3} x^{4}}{4}+\frac {A \,c^{3} x^{3}}{3}+B b \,c^{2} x^{3}+\frac {3 A b \,c^{2} x^{2}}{2}+\frac {3 a B \,c^{2} x^{2}}{2}+\frac {3 B \,b^{2} c \,x^{2}}{2}+3 a A \,c^{2} x +3 A \,b^{2} c x +6 B a b c x +B \,b^{3} x +\left (6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}\right ) \ln \left (x \right )-\frac {a^{2} \left (3 A b +B a \right )}{2 x^{2}}-\frac {3 a \left (A a c +A \,b^{2}+a b B \right )}{x}-\frac {a^{3} A}{3 x^{3}}\) | \(162\) |
norman | \(\frac {\left (\frac {1}{3} A \,c^{3}+B b \,c^{2}\right ) x^{6}+\left (-\frac {3}{2} A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) x +\left (\frac {3}{2} A b \,c^{2}+\frac {3}{2} B a \,c^{2}+\frac {3}{2} B \,b^{2} c \right ) x^{5}+\left (-3 A \,a^{2} c -3 A a \,b^{2}-3 B b \,a^{2}\right ) x^{2}+\left (3 A a \,c^{2}+3 A \,b^{2} c +6 B a b c +B \,b^{3}\right ) x^{4}-\frac {A \,a^{3}}{3}+\frac {B \,c^{3} x^{7}}{4}}{x^{3}}+\left (6 A a b c +A \,b^{3}+3 B \,a^{2} c +3 B a \,b^{2}\right ) \ln \left (x \right )\) | \(166\) |
risch | \(\frac {B \,c^{3} x^{4}}{4}+\frac {A \,c^{3} x^{3}}{3}+B b \,c^{2} x^{3}+\frac {3 A b \,c^{2} x^{2}}{2}+\frac {3 a B \,c^{2} x^{2}}{2}+\frac {3 B \,b^{2} c \,x^{2}}{2}+3 a A \,c^{2} x +3 A \,b^{2} c x +6 B a b c x +B \,b^{3} x +\frac {\left (-3 A \,a^{2} c -3 A a \,b^{2}-3 B b \,a^{2}\right ) x^{2}+\left (-\frac {3}{2} A \,a^{2} b -\frac {1}{2} B \,a^{3}\right ) x -\frac {A \,a^{3}}{3}}{x^{3}}+6 A \ln \left (x \right ) a b c +A \,b^{3} \ln \left (x \right )+3 B \ln \left (x \right ) a^{2} c +3 B \ln \left (x \right ) a \,b^{2}\) | \(174\) |
parallelrisch | \(\frac {3 B \,c^{3} x^{7}+4 A \,c^{3} x^{6}+12 B b \,c^{2} x^{6}+18 A b \,c^{2} x^{5}+18 a B \,c^{2} x^{5}+18 B \,b^{2} c \,x^{5}+72 A \ln \left (x \right ) x^{3} a b c +12 A \ln \left (x \right ) x^{3} b^{3}+36 a A \,c^{2} x^{4}+36 A \,b^{2} c \,x^{4}+36 B \ln \left (x \right ) x^{3} a^{2} c +36 B \ln \left (x \right ) x^{3} a \,b^{2}+72 B a b c \,x^{4}+12 x^{4} B \,b^{3}-36 a^{2} A c \,x^{2}-36 A a \,b^{2} x^{2}-36 B \,a^{2} b \,x^{2}-18 A \,a^{2} b x -6 a^{3} B x -4 A \,a^{3}}{12 x^{3}}\) | \(200\) |
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Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=\frac {3 \, B c^{3} x^{7} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 12 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} \log \left (x\right ) - 4 \, A a^{3} - 36 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{12 \, x^{3}} \]
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Time = 0.62 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=\frac {B c^{3} x^{4}}{4} + x^{3} \left (\frac {A c^{3}}{3} + B b c^{2}\right ) + x^{2} \cdot \left (\frac {3 A b c^{2}}{2} + \frac {3 B a c^{2}}{2} + \frac {3 B b^{2} c}{2}\right ) + x \left (3 A a c^{2} + 3 A b^{2} c + 6 B a b c + B b^{3}\right ) + \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right ) \log {\left (x \right )} + \frac {- 2 A a^{3} + x^{2} \left (- 18 A a^{2} c - 18 A a b^{2} - 18 B a^{2} b\right ) + x \left (- 9 A a^{2} b - 3 B a^{3}\right )}{6 x^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=\frac {1}{4} \, B c^{3} x^{4} + \frac {1}{3} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + \frac {3}{2} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{2} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} \log \left (x\right ) - \frac {2 \, A a^{3} + 18 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.09 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=\frac {1}{4} \, B c^{3} x^{4} + B b c^{2} x^{3} + \frac {1}{3} \, A c^{3} x^{3} + \frac {3}{2} \, B b^{2} c x^{2} + \frac {3}{2} \, B a c^{2} x^{2} + \frac {3}{2} \, A b c^{2} x^{2} + B b^{3} x + 6 \, B a b c x + 3 \, A b^{2} c x + 3 \, A a c^{2} x + {\left (3 \, B a b^{2} + A b^{3} + 3 \, B a^{2} c + 6 \, A a b c\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, A a^{3} + 18 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{6 \, x^{3}} \]
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Time = 0.06 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^4} \, dx=\ln \left (x\right )\,\left (3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3\right )-\frac {x\,\left (\frac {B\,a^3}{2}+\frac {3\,A\,b\,a^2}{2}\right )+\frac {A\,a^3}{3}+x^2\,\left (3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2\right )}{x^3}+x^3\,\left (\frac {A\,c^3}{3}+B\,b\,c^2\right )+x^2\,\left (\frac {3\,B\,b^2\,c}{2}+\frac {3\,A\,b\,c^2}{2}+\frac {3\,B\,a\,c^2}{2}\right )+x\,\left (B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2\right )+\frac {B\,c^3\,x^4}{4} \]
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