Integrand size = 16, antiderivative size = 71 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=\frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^2}{2 b^2}+\frac {d^2 x^3}{3 b}-\frac {a (b c-a d)^2 \log (a+b x)}{b^4} \]
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Time = 0.04 (sec) , antiderivative size = 71, 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} \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=-\frac {a (b c-a d)^2 \log (a+b x)}{b^4}+\frac {x (b c-a d)^2}{b^3}+\frac {d x^2 (2 b c-a d)}{2 b^2}+\frac {d^2 x^3}{3 b} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^2}{b^3}+\frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^2}{b}-\frac {a (-b c+a d)^2}{b^3 (a+b x)}\right ) \, dx \\ & = \frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^2}{2 b^2}+\frac {d^2 x^3}{3 b}-\frac {a (b c-a d)^2 \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.04 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=\frac {b x \left (6 a^2 d^2-3 a b d (4 c+d x)+2 b^2 \left (3 c^2+3 c d x+d^2 x^2\right )\right )-6 a (b c-a d)^2 \log (a+b x)}{6 b^4} \]
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Time = 0.42 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.25
method | result | size |
norman | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x}{b^{3}}+\frac {d^{2} x^{3}}{3 b}-\frac {d \left (a d -2 b c \right ) x^{2}}{2 b^{2}}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\) | \(89\) |
default | \(\frac {\frac {1}{3} d^{2} x^{3} b^{2}-\frac {1}{2} x^{2} a b \,d^{2}+x^{2} b^{2} c d +a^{2} d^{2} x -2 a b c d x +b^{2} c^{2} x}{b^{3}}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\) | \(93\) |
risch | \(\frac {d^{2} x^{3}}{3 b}-\frac {x^{2} a \,d^{2}}{2 b^{2}}+\frac {x^{2} c d}{b}+\frac {a^{2} d^{2} x}{b^{3}}-\frac {2 a c d x}{b^{2}}+\frac {c^{2} x}{b}-\frac {a^{3} \ln \left (b x +a \right ) d^{2}}{b^{4}}+\frac {2 a^{2} \ln \left (b x +a \right ) c d}{b^{3}}-\frac {a \ln \left (b x +a \right ) c^{2}}{b^{2}}\) | \(110\) |
parallelrisch | \(-\frac {-2 d^{2} x^{3} b^{3}+3 x^{2} a \,b^{2} d^{2}-6 x^{2} b^{3} c d +6 \ln \left (b x +a \right ) a^{3} d^{2}-12 \ln \left (b x +a \right ) a^{2} b c d +6 \ln \left (b x +a \right ) a \,b^{2} c^{2}-6 x \,a^{2} b \,d^{2}+12 x a \,b^{2} c d -6 x \,b^{3} c^{2}}{6 b^{4}}\) | \(111\) |
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Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.38 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=\frac {2 \, b^{3} d^{2} x^{3} + 3 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x - 6 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
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Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.06 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=- \frac {a \left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{4}} + x^{2} \left (- \frac {a d^{2}}{2 b^{2}} + \frac {c d}{b}\right ) + x \left (\frac {a^{2} d^{2}}{b^{3}} - \frac {2 a c d}{b^{2}} + \frac {c^{2}}{b}\right ) + \frac {d^{2} x^{3}}{3 b} \]
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Time = 0.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.37 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=\frac {2 \, b^{2} d^{2} x^{3} + 3 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{3}} - \frac {{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left (b x + a\right )}{b^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.39 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=\frac {2 \, b^{2} d^{2} x^{3} + 6 \, b^{2} c d x^{2} - 3 \, a b d^{2} x^{2} + 6 \, b^{2} c^{2} x - 12 \, a b c d x + 6 \, a^{2} d^{2} x}{6 \, b^{3}} - \frac {{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \]
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Time = 0.05 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {x (c+d x)^2}{a+b x} \, dx=x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )-x^2\,\left (\frac {a\,d^2}{2\,b^2}-\frac {c\,d}{b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2\right )}{b^4}+\frac {d^2\,x^3}{3\,b} \]
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