Integrand size = 17, antiderivative size = 94 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {\left (c d^2+a e^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (c d^2+a e^2\right ) \log (d+e x)}{e^5} \]
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Time = 0.05 (sec) , antiderivative size = 94, 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} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}+\frac {c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c \left (3 c d^2+2 a e^2\right )}{e^4}-\frac {2 c^2 d x}{e^3}+\frac {c^2 x^2}{e^2}+\frac {\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^2}-\frac {4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)}\right ) \, dx \\ & = \frac {c \left (3 c d^2+2 a e^2\right ) x}{e^4}-\frac {c^2 d x^2}{e^3}+\frac {c^2 x^3}{3 e^2}-\frac {\left (c d^2+a e^2\right )^2}{e^5 (d+e x)}-\frac {4 c d \left (c d^2+a e^2\right ) \log (d+e x)}{e^5} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {3 c e \left (3 c d^2+2 a e^2\right ) x-3 c^2 d e^2 x^2+c^2 e^3 x^3-\frac {3 \left (c d^2+a e^2\right )^2}{d+e x}-12 c d \left (c d^2+a e^2\right ) \log (d+e x)}{3 e^5} \]
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Time = 2.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {c \left (\frac {1}{3} c \,e^{2} x^{3}-c d e \,x^{2}+2 a \,e^{2} x +3 c \,d^{2} x \right )}{e^{4}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 c d \left (e^{2} a +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(100\) |
norman | \(\frac {-\frac {a^{2} e^{4}+4 a c \,d^{2} e^{2}+4 c^{2} d^{4}}{e^{5}}+\frac {c^{2} x^{4}}{3 e}+\frac {2 c \left (e^{2} a +c \,d^{2}\right ) x^{2}}{e^{3}}-\frac {2 c^{2} d \,x^{3}}{3 e^{2}}}{e x +d}-\frac {4 c d \left (e^{2} a +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{5}}\) | \(109\) |
risch | \(\frac {c^{2} x^{3}}{3 e^{2}}-\frac {c^{2} d \,x^{2}}{e^{3}}+\frac {2 c a x}{e^{2}}+\frac {3 c^{2} d^{2} x}{e^{4}}-\frac {a^{2}}{e \left (e x +d \right )}-\frac {2 a c \,d^{2}}{e^{3} \left (e x +d \right )}-\frac {c^{2} d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 c d \ln \left (e x +d \right ) a}{e^{3}}-\frac {4 c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}\) | \(126\) |
parallelrisch | \(-\frac {-c^{2} x^{4} e^{4}+2 x^{3} c^{2} d \,e^{3}+12 \ln \left (e x +d \right ) x a c d \,e^{3}+12 \ln \left (e x +d \right ) x \,c^{2} d^{3} e -6 x^{2} a c \,e^{4}-6 x^{2} c^{2} d^{2} e^{2}+12 \ln \left (e x +d \right ) a c \,d^{2} e^{2}+12 \ln \left (e x +d \right ) c^{2} d^{4}+3 a^{2} e^{4}+12 a c \,d^{2} e^{2}+12 c^{2} d^{4}}{3 e^{5} \left (e x +d \right )}\) | \(148\) |
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Time = 0.47 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {c^{2} e^{4} x^{4} - 2 \, c^{2} d e^{3} x^{3} - 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 6 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 3 \, {\left (3 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \, {\left (c^{2} d^{4} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{6} x + d e^{5}\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=- \frac {c^{2} d x^{2}}{e^{3}} + \frac {c^{2} x^{3}}{3 e^{2}} - \frac {4 c d \left (a e^{2} + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{5}} + x \left (\frac {2 a c}{e^{2}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {- a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4}}{d e^{5} + e^{6} x} \]
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Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=-\frac {c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{6} x + d e^{5}} + \frac {c^{2} e^{2} x^{3} - 3 \, c^{2} d e x^{2} + 3 \, {\left (3 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} x}{3 \, e^{4}} - \frac {4 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=\frac {{\left (c^{2} - \frac {6 \, c^{2} d}{e x + d} + \frac {6 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}}\right )} {\left (e x + d\right )}^{3}}{3 \, e^{5}} + \frac {4 \, {\left (c^{2} d^{3} + a c d e^{2}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{5}} - \frac {\frac {c^{2} d^{4} e^{3}}{e x + d} + \frac {2 \, a c d^{2} e^{5}}{e x + d} + \frac {a^{2} e^{7}}{e x + d}}{e^{8}} \]
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Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^2} \, dx=x\,\left (\frac {3\,c^2\,d^2}{e^4}+\frac {2\,a\,c}{e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (4\,c^2\,d^3+4\,a\,c\,d\,e^2\right )}{e^5}+\frac {c^2\,x^3}{3\,e^2}-\frac {a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}{e\,\left (x\,e^5+d\,e^4\right )}-\frac {c^2\,d\,x^2}{e^3} \]
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