Integrand size = 17, antiderivative size = 100 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac {2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5} \]
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Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {4 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)}-\frac {\left (a e^2+c d^2\right )^2}{2 e^5 (d+e x)^2}+\frac {2 c \left (a e^2+3 c d^2\right ) \log (d+e x)}{e^5}-\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 c^2 d}{e^4}+\frac {c^2 x}{e^3}+\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^3}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^2}+\frac {2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx \\ & = -\frac {3 c^2 d x}{e^4}+\frac {c^2 x^2}{2 e^3}-\frac {\left (c d^2+a e^2\right )^2}{2 e^5 (d+e x)^2}+\frac {4 c d \left (c d^2+a e^2\right )}{e^5 (d+e x)}+\frac {2 c \left (3 c d^2+a e^2\right ) \log (d+e x)}{e^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {-a^2 e^4+2 a c d e^2 (3 d+4 e x)+c^2 \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+4 c \left (3 c d^2+a e^2\right ) (d+e x)^2 \log (d+e x)}{2 e^5 (d+e x)^2} \]
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Time = 2.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {c^{2} \left (-\frac {1}{2} e \,x^{2}+3 d x \right )}{e^{4}}+\frac {4 c d \left (e^{2} a +c \,d^{2}\right )}{e^{5} \left (e x +d \right )}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(107\) |
norman | \(\frac {-\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}-18 c^{2} d^{4}}{2 e^{5}}+\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{e^{2}}+\frac {2 d \left (2 a c \,e^{2}+6 c^{2} d^{2}\right ) x}{e^{4}}}{\left (e x +d \right )^{2}}+\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(112\) |
risch | \(\frac {c^{2} x^{2}}{2 e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\left (4 d \,e^{2} a c +4 c^{2} d^{3}\right ) x -\frac {a^{2} e^{4}-6 a c \,d^{2} e^{2}-7 c^{2} d^{4}}{2 e}}{e^{4} \left (e x +d \right )^{2}}+\frac {2 c \ln \left (e x +d \right ) a}{e^{3}}+\frac {6 c^{2} \ln \left (e x +d \right ) d^{2}}{e^{5}}\) | \(115\) |
parallelrisch | \(\frac {c^{2} x^{4} e^{4}+4 \ln \left (e x +d \right ) x^{2} a c \,e^{4}+12 \ln \left (e x +d \right ) x^{2} c^{2} d^{2} e^{2}-4 x^{3} c^{2} d \,e^{3}+8 \ln \left (e x +d \right ) x a c d \,e^{3}+24 \ln \left (e x +d \right ) x \,c^{2} d^{3} e +4 \ln \left (e x +d \right ) a c \,d^{2} e^{2}+12 \ln \left (e x +d \right ) c^{2} d^{4}+8 x a c d \,e^{3}+24 x \,c^{2} d^{3} e -a^{2} e^{4}+6 a c \,d^{2} e^{2}+18 c^{2} d^{4}}{2 e^{5} \left (e x +d \right )^{2}}\) | \(178\) |
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Time = 0.40 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {c^{2} e^{4} x^{4} - 4 \, c^{2} d e^{3} x^{3} - 11 \, c^{2} d^{2} e^{2} x^{2} + 7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 2 \, {\left (c^{2} d^{3} e + 4 \, a c d e^{3}\right )} x + 4 \, {\left (3 \, c^{2} d^{4} + a c d^{2} e^{2} + {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
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Time = 0.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=- \frac {3 c^{2} d x}{e^{4}} + \frac {c^{2} x^{2}}{2 e^{3}} + \frac {2 c \left (a e^{2} + 3 c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4} + x \left (8 a c d e^{3} + 8 c^{2} d^{3} e\right )}{2 d^{2} e^{5} + 4 d e^{6} x + 2 e^{7} x^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} + \frac {c^{2} e x^{2} - 6 \, c^{2} d x}{2 \, e^{4}} + \frac {2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {2 \, {\left (3 \, c^{2} d^{2} + a c e^{2}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{5}} + \frac {c^{2} e^{3} x^{2} - 6 \, c^{2} d e^{2} x}{2 \, e^{6}} + \frac {7 \, c^{2} d^{4} + 6 \, a c d^{2} e^{2} - a^{2} e^{4} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{2 \, {\left (e x + d\right )}^{2} e^{5}} \]
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Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^3} \, dx=\frac {x\,\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )+\frac {-a^2\,e^4+6\,a\,c\,d^2\,e^2+7\,c^2\,d^4}{2\,e}}{d^2\,e^4+2\,d\,e^5\,x+e^6\,x^2}+\frac {\ln \left (d+e\,x\right )\,\left (6\,c^2\,d^2+2\,a\,c\,e^2\right )}{e^5}+\frac {c^2\,x^2}{2\,e^3}-\frac {3\,c^2\,d\,x}{e^4} \]
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