Integrand size = 16, antiderivative size = 87 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=\frac {a^2 (A b-a B) x}{b^4}-\frac {a (A b-a B) x^2}{2 b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^4}{4 b}-\frac {a^3 (A b-a B) \log (a+b x)}{b^5} \]
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Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=-\frac {a^3 (A b-a B) \log (a+b x)}{b^5}+\frac {a^2 x (A b-a B)}{b^4}-\frac {a x^2 (A b-a B)}{2 b^3}+\frac {x^3 (A b-a B)}{3 b^2}+\frac {B x^4}{4 b} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^2 (-A b+a B)}{b^4}+\frac {a (-A b+a B) x}{b^3}+\frac {(A b-a B) x^2}{b^2}+\frac {B x^3}{b}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {a^2 (A b-a B) x}{b^4}-\frac {a (A b-a B) x^2}{2 b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^4}{4 b}-\frac {a^3 (A b-a B) \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=\frac {b x \left (-12 a^3 B+6 a^2 b (2 A+B x)-2 a b^2 x (3 A+2 B x)+b^3 x^2 (4 A+3 B x)\right )+12 a^3 (-A b+a B) \log (a+b x)}{12 b^5} \]
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Time = 1.16 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94
method | result | size |
norman | \(\frac {a^{2} \left (A b -B a \right ) x}{b^{4}}-\frac {a \left (A b -B a \right ) x^{2}}{2 b^{3}}+\frac {\left (A b -B a \right ) x^{3}}{3 b^{2}}+\frac {B \,x^{4}}{4 b}-\frac {a^{3} \left (A b -B a \right ) \ln \left (b x +a \right )}{b^{5}}\) | \(82\) |
default | \(\frac {\frac {1}{4} b^{3} B \,x^{4}+\frac {1}{3} A \,b^{3} x^{3}-\frac {1}{3} B a \,b^{2} x^{3}-\frac {1}{2} a A \,b^{2} x^{2}+\frac {1}{2} B \,a^{2} b \,x^{2}+a^{2} A b x -a^{3} B x}{b^{4}}-\frac {a^{3} \left (A b -B a \right ) \ln \left (b x +a \right )}{b^{5}}\) | \(91\) |
risch | \(\frac {B \,x^{4}}{4 b}+\frac {A \,x^{3}}{3 b}-\frac {B a \,x^{3}}{3 b^{2}}-\frac {a A \,x^{2}}{2 b^{2}}+\frac {B \,a^{2} x^{2}}{2 b^{3}}+\frac {a^{2} A x}{b^{3}}-\frac {a^{3} B x}{b^{4}}-\frac {a^{3} \ln \left (b x +a \right ) A}{b^{4}}+\frac {a^{4} \ln \left (b x +a \right ) B}{b^{5}}\) | \(100\) |
parallelrisch | \(-\frac {-3 B \,x^{4} b^{4}-4 A \,x^{3} b^{4}+4 B \,x^{3} a \,b^{3}+6 A \,x^{2} a \,b^{3}-6 B \,x^{2} a^{2} b^{2}+12 A \ln \left (b x +a \right ) a^{3} b -12 A x \,a^{2} b^{2}-12 B \ln \left (b x +a \right ) a^{4}+12 B x \,a^{3} b}{12 b^{5}}\) | \(100\) |
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none
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=\frac {3 \, B b^{4} x^{4} - 4 \, {\left (B a b^{3} - A b^{4}\right )} x^{3} + 6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 12 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x + 12 \, {\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{12 \, b^{5}} \]
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Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=\frac {B x^{4}}{4 b} + \frac {a^{3} \left (- A b + B a\right ) \log {\left (a + b x \right )}}{b^{5}} + x^{3} \left (\frac {A}{3 b} - \frac {B a}{3 b^{2}}\right ) + x^{2} \left (- \frac {A a}{2 b^{2}} + \frac {B a^{2}}{2 b^{3}}\right ) + x \left (\frac {A a^{2}}{b^{3}} - \frac {B a^{3}}{b^{4}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=\frac {3 \, B b^{3} x^{4} - 4 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} + 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} - 12 \, {\left (B a^{3} - A a^{2} b\right )} x}{12 \, b^{4}} + \frac {{\left (B a^{4} - A a^{3} b\right )} \log \left (b x + a\right )}{b^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=\frac {3 \, B b^{3} x^{4} - 4 \, B a b^{2} x^{3} + 4 \, A b^{3} x^{3} + 6 \, B a^{2} b x^{2} - 6 \, A a b^{2} x^{2} - 12 \, B a^{3} x + 12 \, A a^{2} b x}{12 \, b^{4}} + \frac {{\left (B a^{4} - A a^{3} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} \]
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Time = 0.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 (A+B x)}{a+b x} \, dx=x^3\,\left (\frac {A}{3\,b}-\frac {B\,a}{3\,b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4-A\,a^3\,b\right )}{b^5}+\frac {B\,x^4}{4\,b}-\frac {a\,x^2\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{2\,b}+\frac {a^2\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b^2} \]
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