Integrand size = 17, antiderivative size = 123 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=-\frac {e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {\sqrt {c} \left (c d^2-a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )^2}+\frac {2 c d e \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {c d e \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^2} \]
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Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {724, 815, 649, 211, 266} \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d^2-a e^2\right )}{\sqrt {a} \left (a e^2+c d^2\right )^2}-\frac {c d e \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^2}-\frac {e}{(d+e x) \left (a e^2+c d^2\right )}+\frac {2 c d e \log (d+e x)}{\left (a e^2+c d^2\right )^2} \]
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Rule 211
Rule 266
Rule 649
Rule 724
Rule 815
Rubi steps \begin{align*} \text {integral}& = -\frac {e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {c \int \frac {d-e x}{(d+e x) \left (a+c x^2\right )} \, dx}{c d^2+a e^2} \\ & = -\frac {e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {c \int \left (\frac {2 d e^2}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {c d^2-a e^2-2 c d e x}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{c d^2+a e^2} \\ & = -\frac {e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {2 c d e \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {c \int \frac {c d^2-a e^2-2 c d e x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2} \\ & = -\frac {e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {2 c d e \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {\left (2 c^2 d e\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2}+\frac {\left (c \left (c d^2-a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^2} \\ & = -\frac {e}{\left (c d^2+a e^2\right ) (d+e x)}+\frac {\sqrt {c} \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^2+a e^2\right )^2}+\frac {2 c d e \log (d+e x)}{\left (c d^2+a e^2\right )^2}-\frac {c d e \log \left (a+c x^2\right )}{\left (c d^2+a e^2\right )^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\sqrt {c} \left (c d^2-a e^2\right ) (d+e x) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )-\sqrt {a} e \left (c d^2+a e^2-2 c d (d+e x) \log (d+e x)+c d (d+e x) \log \left (a+c x^2\right )\right )}{\sqrt {a} \left (c d^2+a e^2\right )^2 (d+e x)} \]
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Time = 2.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {c \left (d e \ln \left (c \,x^{2}+a \right )+\frac {\left (e^{2} a -c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}-\frac {e}{\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )}+\frac {2 c d e \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}\) | \(104\) |
risch | \(-\frac {e}{\left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )}+\frac {2 c d e \ln \left (e x +d \right )}{a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (e^{4} a^{3}+2 d^{2} e^{2} a^{2} c +d^{4} c^{2} a \right ) \textit {\_Z}^{2}+4 a c d e \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 e^{6} a^{3}+5 d^{2} e^{4} a^{2} c +d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \textit {\_R}^{2}+\left (4 d \,e^{3} a c +4 d^{3} e \,c^{2}\right ) \textit {\_R} +2 e^{2} c \right ) x +\left (4 d \,e^{5} a^{3}+8 a^{2} c \,d^{3} e^{3}+4 a \,c^{2} d^{5} e \right ) \textit {\_R}^{2}+\left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \textit {\_R} -2 c d e \right )\right )}{2}\) | \(255\) |
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Time = 0.39 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.85 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\left [-\frac {2 \, c d^{2} e + 2 \, a e^{3} + {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 2 \, {\left (c d e^{2} x + c d^{2} e\right )} \log \left (c x^{2} + a\right ) - 4 \, {\left (c d e^{2} x + c d^{2} e\right )} \log \left (e x + d\right )}{2 \, {\left (c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )}}, -\frac {c d^{2} e + a e^{3} - {\left (c d^{3} - a d e^{2} + {\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + {\left (c d e^{2} x + c d^{2} e\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (c d e^{2} x + c d^{2} e\right )} \log \left (e x + d\right )}{c^{2} d^{5} + 2 \, a c d^{3} e^{2} + a^{2} d e^{4} + {\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}\right ] \]
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Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=-\frac {c d e \log \left (c x^{2} + a\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {2 \, c d e \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (c^{2} d^{2} - a c e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} - \frac {e}{c d^{3} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x} \]
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Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.57 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=-\frac {c d e \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac {e^{3}}{{\left (c d^{2} e^{2} + a e^{4}\right )} {\left (e x + d\right )}} + \frac {{\left (c^{2} d^{2} e^{2} - a c e^{4}\right )} \arctan \left (\frac {c d - \frac {c d^{2}}{e x + d} - \frac {a e^{2}}{e x + d}}{\sqrt {a c} e}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c} e^{2}} \]
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Time = 10.05 (sec) , antiderivative size = 452, normalized size of antiderivative = 3.67 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\ln \left (a^5\,e^8\,\sqrt {-a\,c}-c^3\,d^8\,{\left (-a\,c\right )}^{3/2}-36\,a^3\,d^2\,e^6\,{\left (-a\,c\right )}^{3/2}+a\,c^5\,d^8\,x+a^5\,c\,e^8\,x+70\,a\,d^4\,e^4\,{\left (-a\,c\right )}^{5/2}+36\,c\,d^6\,e^2\,{\left (-a\,c\right )}^{5/2}+36\,a^2\,c^4\,d^6\,e^2\,x+70\,a^3\,c^3\,d^4\,e^4\,x+36\,a^4\,c^2\,d^2\,e^6\,x\right )\,\left (c\,\left (\frac {d^2\,\sqrt {-a\,c}}{2}-a\,d\,e\right )-\frac {a\,e^2\,\sqrt {-a\,c}}{2}\right )}{a^3\,e^4+2\,a^2\,c\,d^2\,e^2+a\,c^2\,d^4}+\frac {\ln \left (c^3\,d^8\,{\left (-a\,c\right )}^{3/2}-a^5\,e^8\,\sqrt {-a\,c}+36\,a^3\,d^2\,e^6\,{\left (-a\,c\right )}^{3/2}+a\,c^5\,d^8\,x+a^5\,c\,e^8\,x-70\,a\,d^4\,e^4\,{\left (-a\,c\right )}^{5/2}-36\,c\,d^6\,e^2\,{\left (-a\,c\right )}^{5/2}+36\,a^2\,c^4\,d^6\,e^2\,x+70\,a^3\,c^3\,d^4\,e^4\,x+36\,a^4\,c^2\,d^2\,e^6\,x\right )\,\left (a\,\left (\frac {e^2\,\sqrt {-a\,c}}{2}-c\,d\,e\right )-\frac {c\,d^2\,\sqrt {-a\,c}}{2}\right )}{a^3\,e^4+2\,a^2\,c\,d^2\,e^2+a\,c^2\,d^4}-\frac {e}{\left (c\,d^2+a\,e^2\right )\,\left (d+e\,x\right )}+\frac {2\,c\,d\,e\,\ln \left (d+e\,x\right )}{{\left (c\,d^2+a\,e^2\right )}^2} \]
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