Integrand size = 15, antiderivative size = 59 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=-\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}-\frac {2 d (b c-a d)}{b^3 (a+b x)}+\frac {d^2 \log (a+b x)}{b^3} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=-\frac {2 d (b c-a d)}{b^3 (a+b x)}-\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac {d^2 \log (a+b x)}{b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^3}+\frac {2 d (b c-a d)}{b^2 (a+b x)^2}+\frac {d^2}{b^2 (a+b x)}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{2 b^3 (a+b x)^2}-\frac {2 d (b c-a d)}{b^3 (a+b x)}+\frac {d^2 \log (a+b x)}{b^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=\frac {-\frac {(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}+2 d^2 \log (a+b x)}{2 b^3} \]
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Time = 0.20 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {\frac {2 d \left (a d -b c \right ) x}{b^{2}}+\frac {3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}}{2 b^{3}}}{\left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(67\) |
default | \(\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 b^{3} \left (b x +a \right )^{2}}+\frac {2 d \left (a d -b c \right )}{b^{3} \left (b x +a \right )}\) | \(69\) |
norman | \(\frac {\frac {3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}}{2 b^{3}}+\frac {2 \left (a \,d^{2}-b c d \right ) x}{b^{2}}}{\left (b x +a \right )^{2}}+\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(69\) |
parallelrisch | \(\frac {2 \ln \left (b x +a \right ) x^{2} b^{2} d^{2}+4 \ln \left (b x +a \right ) x a b \,d^{2}+2 \ln \left (b x +a \right ) a^{2} d^{2}+4 x a b \,d^{2}-4 x \,b^{2} c d +3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}}{2 b^{3} \left (b x +a \right )^{2}}\) | \(97\) |
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Time = 0.22 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.68 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=-\frac {b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \, {\left (b^{2} c d - a b d^{2}\right )} x - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=\frac {3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {d^{2} \log {\left (a + b x \right )}}{b^{3}} \]
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Time = 0.21 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.34 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=-\frac {b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \, {\left (b^{2} c d - a b d^{2}\right )} x}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {d^{2} \log \left (b x + a\right )}{b^{3}} \]
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Time = 0.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.15 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=\frac {d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} - \frac {4 \, {\left (b c d - a d^{2}\right )} x + \frac {b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2}}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.31 \[ \int \frac {(c+d x)^2}{(a+b x)^3} \, dx=\frac {d^2\,\ln \left (a+b\,x\right )}{b^3}-\frac {\frac {-3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2}{2\,b^3}-\frac {2\,d\,x\,\left (a\,d-b\,c\right )}{b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \]
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