Integrand size = 17, antiderivative size = 145 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=-\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac {\left (5 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {753, 653, 205, 211} \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+5 c d^2\right )}{16 a^{7/2} c^{3/2}}+\frac {x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac {4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac {(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]
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Rule 205
Rule 211
Rule 653
Rule 753
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}+\frac {\int \frac {5 c d^2+a e^2+4 c d e x}{\left (a+c x^2\right )^3} \, dx}{6 a c} \\ & = -\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) \int \frac {1}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c} \\ & = -\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac {\left (5 c d^2+a e^2\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 c} \\ & = -\frac {(a e-c d x) (d+e x)}{6 a c \left (a+c x^2\right )^3}-\frac {4 a d e-\left (5 c d^2+a e^2\right ) x}{24 a^2 c \left (a+c x^2\right )^2}+\frac {\left (5 c d^2+a e^2\right ) x}{16 a^3 c \left (a+c x^2\right )}+\frac {\left (5 c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=\frac {15 c^3 d^2 x^5-a^3 e (16 d+3 e x)+a c^2 x^3 \left (40 d^2+3 e^2 x^2\right )+a^2 c x \left (33 d^2+8 e^2 x^2\right )}{48 a^3 c \left (a+c x^2\right )^3}+\frac {\left (5 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} c^{3/2}} \]
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Time = 2.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\frac {\left (e^{2} a +5 c \,d^{2}\right ) c \,x^{5}}{16 a^{3}}+\frac {\left (e^{2} a +5 c \,d^{2}\right ) x^{3}}{6 a^{2}}-\frac {\left (e^{2} a -11 c \,d^{2}\right ) x}{16 a c}-\frac {d e}{3 c}}{\left (c \,x^{2}+a \right )^{3}}+\frac {\left (e^{2} a +5 c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 c \,a^{3} \sqrt {a c}}\) | \(116\) |
risch | \(\frac {\frac {\left (e^{2} a +5 c \,d^{2}\right ) c \,x^{5}}{16 a^{3}}+\frac {\left (e^{2} a +5 c \,d^{2}\right ) x^{3}}{6 a^{2}}-\frac {\left (e^{2} a -11 c \,d^{2}\right ) x}{16 a c}-\frac {d e}{3 c}}{\left (c \,x^{2}+a \right )^{3}}-\frac {\ln \left (c x +\sqrt {-a c}\right ) e^{2}}{32 \sqrt {-a c}\, c \,a^{2}}-\frac {5 \ln \left (c x +\sqrt {-a c}\right ) d^{2}}{32 \sqrt {-a c}\, a^{3}}+\frac {\ln \left (-c x +\sqrt {-a c}\right ) e^{2}}{32 \sqrt {-a c}\, c \,a^{2}}+\frac {5 \ln \left (-c x +\sqrt {-a c}\right ) d^{2}}{32 \sqrt {-a c}\, a^{3}}\) | \(190\) |
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Time = 0.28 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.38 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=\left [-\frac {32 \, a^{4} c d e - 6 \, {\left (5 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )} x^{5} - 16 \, {\left (5 \, a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} x^{3} + 3 \, {\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \, {\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 6 \, {\left (11 \, a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} x}{96 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}, -\frac {16 \, a^{4} c d e - 3 \, {\left (5 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )} x^{5} - 8 \, {\left (5 \, a^{2} c^{3} d^{2} + a^{3} c^{2} e^{2}\right )} x^{3} - 3 \, {\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \, {\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \, {\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - 3 \, {\left (11 \, a^{3} c^{2} d^{2} - a^{4} c e^{2}\right )} x}{48 \, {\left (a^{4} c^{5} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{6} c^{3} x^{2} + a^{7} c^{2}\right )}}\right ] \]
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Time = 0.55 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.48 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=- \frac {\sqrt {- \frac {1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log {\left (- a^{4} c \sqrt {- \frac {1}{a^{7} c^{3}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log {\left (a^{4} c \sqrt {- \frac {1}{a^{7} c^{3}}} + x \right )}}{32} + \frac {- 16 a^{3} d e + x^{5} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right ) + x^{3} \cdot \left (8 a^{2} c e^{2} + 40 a c^{2} d^{2}\right ) + x \left (- 3 a^{3} e^{2} + 33 a^{2} c d^{2}\right )}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=\frac {3 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{5} - 16 \, a^{3} d e + 8 \, {\left (5 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{3} + 3 \, {\left (11 \, a^{2} c d^{2} - a^{3} e^{2}\right )} x}{48 \, {\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )}} + \frac {{\left (5 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} \]
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Time = 0.30 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=\frac {{\left (5 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, \sqrt {a c} a^{3} c} + \frac {15 \, c^{3} d^{2} x^{5} + 3 \, a c^{2} e^{2} x^{5} + 40 \, a c^{2} d^{2} x^{3} + 8 \, a^{2} c e^{2} x^{3} + 33 \, a^{2} c d^{2} x - 3 \, a^{3} e^{2} x - 16 \, a^{3} d e}{48 \, {\left (c x^{2} + a\right )}^{3} a^{3} c} \]
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Time = 9.46 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\left (a+c x^2\right )^4} \, dx=\frac {\frac {x^3\,\left (5\,c\,d^2+a\,e^2\right )}{6\,a^2}-\frac {d\,e}{3\,c}-\frac {x\,\left (a\,e^2-11\,c\,d^2\right )}{16\,a\,c}+\frac {c\,x^5\,\left (5\,c\,d^2+a\,e^2\right )}{16\,a^3}}{a^3+3\,a^2\,c\,x^2+3\,a\,c^2\,x^4+c^3\,x^6}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (5\,c\,d^2+a\,e^2\right )}{16\,a^{7/2}\,c^{3/2}} \]
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