Integrand size = 17, antiderivative size = 295 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \]
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Time = 0.25 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {755, 837, 815, 649, 211, 266} \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {6 a^2 e^3+c d x \left (11 a e^2+5 c d^2\right )}{24 a^2 \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {8 a^3 e^5+c d x \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )}{16 a^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (35 a^3 e^6+35 a^2 c d^2 e^4+21 a c^2 d^4 e^2+5 c^3 d^6\right )}{16 a^{7/2} \left (a e^2+c d^2\right )^4}+\frac {a e+c d x}{6 a \left (a+c x^2\right )^3 \left (a e^2+c d^2\right )}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (a e^2+c d^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (a e^2+c d^2\right )^4} \]
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Rule 211
Rule 266
Rule 649
Rule 755
Rule 815
Rule 837
Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}-\frac {\int \frac {-5 c d^2-6 a e^2-5 c d e x}{(d+e x) \left (a+c x^2\right )^3} \, dx}{6 a \left (c d^2+a e^2\right )} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {\int \frac {3 c \left (5 c^2 d^4+11 a c d^2 e^2+8 a^2 e^4\right )+3 c^2 d e \left (5 c d^2+11 a e^2\right ) x}{(d+e x) \left (a+c x^2\right )^2} \, dx}{24 a^2 c \left (c d^2+a e^2\right )^2} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \frac {-3 c^2 \left (5 c^3 d^6+16 a c^2 d^4 e^2+19 a^2 c d^2 e^4+16 a^3 e^6\right )-3 c^3 d e \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{(d+e x) \left (a+c x^2\right )} \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}-\frac {\int \left (-\frac {48 a^3 c^2 e^8}{\left (c d^2+a e^2\right ) (d+e x)}-\frac {3 c^3 \left (5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx}{48 a^3 c^2 \left (c d^2+a e^2\right )^3} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {c \int \frac {5 c^3 d^7+21 a c^2 d^5 e^2+35 a^2 c d^3 e^4+35 a^3 d e^6-16 a^3 e^7 x}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {\left (c e^7\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (c d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right )\right ) \int \frac {1}{a+c x^2} \, dx}{16 a^3 \left (c d^2+a e^2\right )^4} \\ & = \frac {a e+c d x}{6 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^3}+\frac {6 a^2 e^3+c d \left (5 c d^2+11 a e^2\right ) x}{24 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {8 a^3 e^5+c d \left (5 c^2 d^4+16 a c d^2 e^2+19 a^2 e^4\right ) x}{16 a^3 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{16 a^{7/2} \left (c d^2+a e^2\right )^4}+\frac {e^7 \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e^7 \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {\frac {8 \left (c d^2+a e^2\right )^3 (a e+c d x)}{a \left (a+c x^2\right )^3}+\frac {2 \left (c d^2+a e^2\right )^2 \left (6 a^2 e^3+5 c^2 d^3 x+11 a c d e^2 x\right )}{a^2 \left (a+c x^2\right )^2}+\frac {3 \left (c d^2+a e^2\right ) \left (8 a^3 e^5+5 c^3 d^5 x+16 a c^2 d^3 e^2 x+19 a^2 c d e^4 x\right )}{a^3 \left (a+c x^2\right )}+\frac {3 \sqrt {c} d \left (5 c^3 d^6+21 a c^2 d^4 e^2+35 a^2 c d^2 e^4+35 a^3 e^6\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{7/2}}+48 e^7 \log (d+e x)-24 e^7 \log \left (a+c x^2\right )}{48 \left (c d^2+a e^2\right )^4} \]
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Time = 2.28 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.32
method | result | size |
default | \(\frac {c \left (\frac {\frac {c^{2} d \left (19 e^{6} a^{3}+35 d^{2} e^{4} a^{2} c +21 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}\right ) x^{5}}{16 a^{3}}+\left (\frac {1}{2} a \,e^{7} c +\frac {1}{2} d^{2} e^{5} c^{2}\right ) x^{4}+\frac {c d \left (17 e^{6} a^{3}+33 d^{2} e^{4} a^{2} c +21 d^{4} e^{2} c^{2} a +5 c^{3} d^{6}\right ) x^{3}}{6 a^{2}}+\left (\frac {5}{4} a^{2} e^{7}+\frac {3}{2} a \,d^{2} e^{5} c +\frac {1}{4} c^{2} d^{4} e^{3}\right ) x^{2}+\frac {d \left (29 e^{6} a^{3}+61 d^{2} e^{4} a^{2} c +43 d^{4} e^{2} c^{2} a +11 c^{3} d^{6}\right ) x}{16 a}+\frac {e \left (11 e^{6} a^{3}+18 d^{2} e^{4} a^{2} c +9 d^{4} e^{2} c^{2} a +2 c^{3} d^{6}\right )}{12 c}}{\left (c \,x^{2}+a \right )^{3}}+\frac {-\frac {8 a^{3} e^{7} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (35 a^{3} d \,e^{6}+35 a^{2} c \,d^{3} e^{4}+21 a \,c^{2} d^{5} e^{2}+5 c^{3} d^{7}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{16 a^{3}}\right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}+\frac {e^{7} \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}\) | \(390\) |
risch | \(\text {Expression too large to display}\) | \(1139\) |
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Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (277) = 554\).
Time = 12.85 (sec) , antiderivative size = 1784, normalized size of antiderivative = 6.05 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (277) = 554\).
Time = 0.30 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=-\frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {e^{7} \log \left (e x + d\right )}{c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {24 \, a^{3} c^{2} e^{5} x^{4} + 8 \, a^{3} c^{2} d^{4} e + 28 \, a^{4} c d^{2} e^{3} + 44 \, a^{5} e^{5} + 3 \, {\left (5 \, c^{5} d^{5} + 16 \, a c^{4} d^{3} e^{2} + 19 \, a^{2} c^{3} d e^{4}\right )} x^{5} + 8 \, {\left (5 \, a c^{4} d^{5} + 16 \, a^{2} c^{3} d^{3} e^{2} + 17 \, a^{3} c^{2} d e^{4}\right )} x^{3} + 12 \, {\left (a^{3} c^{2} d^{2} e^{3} + 5 \, a^{4} c e^{5}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{3} d^{5} + 32 \, a^{3} c^{2} d^{3} e^{2} + 29 \, a^{4} c d e^{4}\right )} x}{48 \, {\left (a^{6} c^{3} d^{6} + 3 \, a^{7} c^{2} d^{4} e^{2} + 3 \, a^{8} c d^{2} e^{4} + a^{9} e^{6} + {\left (a^{3} c^{6} d^{6} + 3 \, a^{4} c^{5} d^{4} e^{2} + 3 \, a^{5} c^{4} d^{2} e^{4} + a^{6} c^{3} e^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{5} d^{6} + 3 \, a^{5} c^{4} d^{4} e^{2} + 3 \, a^{6} c^{3} d^{2} e^{4} + a^{7} c^{2} e^{6}\right )} x^{4} + 3 \, {\left (a^{5} c^{4} d^{6} + 3 \, a^{6} c^{3} d^{4} e^{2} + 3 \, a^{7} c^{2} d^{2} e^{4} + a^{8} c e^{6}\right )} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 562 vs. \(2 (277) = 554\).
Time = 0.28 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.91 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\frac {e^{8} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e + 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} + 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}} - \frac {e^{7} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} + \frac {{\left (5 \, c^{4} d^{7} + 21 \, a c^{3} d^{5} e^{2} + 35 \, a^{2} c^{2} d^{3} e^{4} + 35 \, a^{3} c d e^{6}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{16 \, {\left (a^{3} c^{4} d^{8} + 4 \, a^{4} c^{3} d^{6} e^{2} + 6 \, a^{5} c^{2} d^{4} e^{4} + 4 \, a^{6} c d^{2} e^{6} + a^{7} e^{8}\right )} \sqrt {a c}} + \frac {8 \, a^{3} c^{3} d^{6} e + 36 \, a^{4} c^{2} d^{4} e^{3} + 72 \, a^{5} c d^{2} e^{5} + 44 \, a^{6} e^{7} + 3 \, {\left (5 \, c^{6} d^{7} + 21 \, a c^{5} d^{5} e^{2} + 35 \, a^{2} c^{4} d^{3} e^{4} + 19 \, a^{3} c^{3} d e^{6}\right )} x^{5} + 24 \, {\left (a^{3} c^{3} d^{2} e^{5} + a^{4} c^{2} e^{7}\right )} x^{4} + 8 \, {\left (5 \, a c^{5} d^{7} + 21 \, a^{2} c^{4} d^{5} e^{2} + 33 \, a^{3} c^{3} d^{3} e^{4} + 17 \, a^{4} c^{2} d e^{6}\right )} x^{3} + 12 \, {\left (a^{3} c^{3} d^{4} e^{3} + 6 \, a^{4} c^{2} d^{2} e^{5} + 5 \, a^{5} c e^{7}\right )} x^{2} + 3 \, {\left (11 \, a^{2} c^{4} d^{7} + 43 \, a^{3} c^{3} d^{5} e^{2} + 61 \, a^{4} c^{2} d^{3} e^{4} + 29 \, a^{5} c d e^{6}\right )} x}{48 \, {\left (c d^{2} + a e^{2}\right )}^{4} {\left (c x^{2} + a\right )}^{3} a^{3}} \]
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Time = 10.85 (sec) , antiderivative size = 1470, normalized size of antiderivative = 4.98 \[ \int \frac {1}{(d+e x) \left (a+c x^2\right )^4} \, dx=\text {Too large to display} \]
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