Integrand size = 15, antiderivative size = 53 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {-c d^2-a e^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \]
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Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=-\frac {a e^2+c d^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c d^2+a e^2}{e^2 (d+e x)^3}-\frac {2 c d}{e^2 (d+e x)^2}+\frac {c}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {c d^2+a e^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {-a e^2+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \]
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Time = 2.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.89
method | result | size |
norman | \(\frac {-\frac {e^{2} a -3 c \,d^{2}}{2 e^{3}}+\frac {2 c d x}{e^{2}}}{\left (e x +d \right )^{2}}+\frac {c \ln \left (e x +d \right )}{e^{3}}\) | \(47\) |
risch | \(\frac {-\frac {e^{2} a -3 c \,d^{2}}{2 e^{3}}+\frac {2 c d x}{e^{2}}}{\left (e x +d \right )^{2}}+\frac {c \ln \left (e x +d \right )}{e^{3}}\) | \(47\) |
default | \(\frac {2 c d}{e^{3} \left (e x +d \right )}-\frac {e^{2} a +c \,d^{2}}{2 e^{3} \left (e x +d \right )^{2}}+\frac {c \ln \left (e x +d \right )}{e^{3}}\) | \(50\) |
parallelrisch | \(\frac {2 \ln \left (e x +d \right ) x^{2} c \,e^{2}+4 \ln \left (e x +d \right ) x c d e +2 \ln \left (e x +d \right ) c \,d^{2}+4 x c d e -e^{2} a +3 c \,d^{2}}{2 e^{3} \left (e x +d \right )^{2}}\) | \(71\) |
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Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.38 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {4 \, c d e x + 3 \, c d^{2} - a e^{2} + 2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
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Time = 0.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {c \log {\left (d + e x \right )}}{e^{3}} + \frac {- a e^{2} + 3 c d^{2} + 4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.08 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {4 \, c d e x + 3 \, c d^{2} - a e^{2}}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac {c \log \left (e x + d\right )}{e^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.91 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {c \log \left ({\left | e x + d \right |}\right )}{e^{3}} + \frac {4 \, c d x + \frac {3 \, c d^{2} - a e^{2}}{e}}{2 \, {\left (e x + d\right )}^{2} e^{2}} \]
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Time = 9.53 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.09 \[ \int \frac {a+c x^2}{(d+e x)^3} \, dx=\frac {c\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {a\,e^2-3\,c\,d^2}{2\,e^3}-\frac {2\,c\,d\,x}{e^2}}{d^2+2\,d\,e\,x+e^2\,x^2} \]
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