Integrand size = 29, antiderivative size = 49 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {d (b c-a d) x}{b^2}+\frac {(c+d x)^2}{2 b}+\frac {(b c-a d)^2 \log (a+b x)}{b^3} \]
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Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {(b c-a d)^2 \log (a+b x)}{b^3}+\frac {d x (b c-a d)}{b^2}+\frac {(c+d x)^2}{2 b} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(c+d x)^2}{a+b x} \, dx \\ & = \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx \\ & = \frac {d (b c-a d) x}{b^2}+\frac {(c+d x)^2}{2 b}+\frac {(b c-a d)^2 \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {b d x (4 b c-2 a d+b d x)+2 (b c-a d)^2 \log (a+b x)}{2 b^3} \]
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Time = 2.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.14
method | result | size |
default | \(-\frac {d \left (-\frac {1}{2} b d \,x^{2}+a d x -2 b c x \right )}{b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{3}}\) | \(56\) |
parallelrisch | \(\frac {d^{2} x^{2} b^{2}+2 \ln \left (b x +a \right ) a^{2} d^{2}-4 \ln \left (b x +a \right ) a b c d +2 \ln \left (b x +a \right ) b^{2} c^{2}-2 x a b \,d^{2}+4 x \,b^{2} c d}{2 b^{3}}\) | \(73\) |
risch | \(\frac {d^{2} x^{2}}{2 b}-\frac {d^{2} a x}{b^{2}}+\frac {2 d c x}{b}+\frac {\ln \left (b x +a \right ) a^{2} d^{2}}{b^{3}}-\frac {2 \ln \left (b x +a \right ) a c d}{b^{2}}+\frac {\ln \left (b x +a \right ) c^{2}}{b}\) | \(74\) |
norman | \(\frac {\frac {a^{2} \left (3 a^{2} d^{2}-8 a b c d \right )}{2 b^{3}}+\frac {b \,d^{2} x^{4}}{2}+\frac {2 a \left (a^{2} d^{2}-3 a b c d \right ) x}{b^{2}}+2 b c d \,x^{3}}{\left (b x +a \right )^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{3}}\) | \(103\) |
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Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {b^{2} d^{2} x^{2} + 2 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x + 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
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Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=x \left (- \frac {a d^{2}}{b^{2}} + \frac {2 c d}{b}\right ) + \frac {d^{2} x^{2}}{2 b} + \frac {\left (a d - b c\right )^{2} \log {\left (a + b x \right )}}{b^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {b d^{2} x^{2} + 2 \, {\left (2 \, b c d - a d^{2}\right )} x}{2 \, b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {b d^{2} x^{2} + 4 \, b c d x - 2 \, a d^{2} x}{2 \, b^{2}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \]
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Time = 0.07 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^3} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{b^3}-x\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )+\frac {d^2\,x^2}{2\,b} \]
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