Integrand size = 25, antiderivative size = 200 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=-\frac {2 e f^2 p}{15 d x^3}+\frac {2 e^2 f^2 p}{5 d^2 x}-\frac {4 e f g p}{3 d x}+\frac {2 e^{5/2} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {4 e^{3/2} f g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \]
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Time = 0.11 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2526, 2505, 331, 211} \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2 e^{5/2} f^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {4 e^{3/2} f g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac {2 e^2 f^2 p}{5 d^2 x}-\frac {2 e f^2 p}{15 d x^3}-\frac {4 e f g p}{3 d x} \]
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Rule 211
Rule 331
Rule 2505
Rule 2526
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6}+\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{x^4}+\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^2}\right ) \, dx \\ & = f^2 \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx+(2 f g) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^4} \, dx+g^2 \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x^2} \, dx \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac {1}{5} \left (2 e f^2 p\right ) \int \frac {1}{x^4 \left (d+e x^2\right )} \, dx+\frac {1}{3} (4 e f g p) \int \frac {1}{x^2 \left (d+e x^2\right )} \, dx+\left (2 e g^2 p\right ) \int \frac {1}{d+e x^2} \, dx \\ & = -\frac {2 e f^2 p}{15 d x^3}-\frac {4 e f g p}{3 d x}+\frac {2 \sqrt {e} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}-\frac {\left (2 e^2 f^2 p\right ) \int \frac {1}{x^2 \left (d+e x^2\right )} \, dx}{5 d}-\frac {\left (4 e^2 f g p\right ) \int \frac {1}{d+e x^2} \, dx}{3 d} \\ & = -\frac {2 e f^2 p}{15 d x^3}+\frac {2 e^2 f^2 p}{5 d^2 x}-\frac {4 e f g p}{3 d x}-\frac {4 e^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x}+\frac {\left (2 e^3 f^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{5 d^2} \\ & = -\frac {2 e f^2 p}{15 d x^3}+\frac {2 e^2 f^2 p}{5 d^2 x}-\frac {4 e f g p}{3 d x}+\frac {2 e^{5/2} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 d^{5/2}}-\frac {4 e^{3/2} f g p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 d^{3/2}}+\frac {2 \sqrt {e} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2 \sqrt {e} g^2 p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 e f^2 p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {e x^2}{d}\right )}{15 d x^3}-\frac {4 e f g p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^2}{d}\right )}{3 d x}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 x^5}-\frac {2 f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^3}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{x} \]
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Time = 1.90 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.67
method | result | size |
parts | \(-\frac {g^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{x}-\frac {2 f g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3 x^{3}}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{5 x^{5}}-\frac {2 p e \left (\frac {f^{2}}{d \,x^{3}}+\frac {f \left (10 d g -3 e f \right )}{d^{2} x}+\frac {\left (-15 g^{2} d^{2}+10 d e f g -3 e^{2} f^{2}\right ) \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}\right )}{15}\) | \(134\) |
risch | \(-\frac {\left (15 g^{2} x^{4}+10 f g \,x^{2}+3 f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{15 x^{5}}-\frac {-10 i \pi \,d^{3} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-15 i \pi \,d^{3} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-3 i \pi \,d^{3} f^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{3} f^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-30 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) g^{2} d^{2} x^{5}+20 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) e f g d \,x^{5}-6 \sqrt {-d e}\, p \ln \left (-e x -\sqrt {-d e}\right ) e^{2} f^{2} x^{5}+30 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) g^{2} d^{2} x^{5}-20 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) e f g d \,x^{5}+6 \sqrt {-d e}\, p \ln \left (-e x +\sqrt {-d e}\right ) e^{2} f^{2} x^{5}-10 i \pi \,d^{3} f g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+10 i \pi \,d^{3} f g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-15 i \pi \,d^{3} g^{2} x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+3 i \pi \,d^{3} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+15 i \pi \,d^{3} g^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+10 i \pi \,d^{3} f g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-3 i \pi \,d^{3} f^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}+15 i \pi \,d^{3} g^{2} x^{4} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+30 \ln \left (c \right ) d^{3} g^{2} x^{4}+40 d^{2} e f g p \,x^{4}-12 d \,e^{2} f^{2} p \,x^{4}+20 \ln \left (c \right ) d^{3} f g \,x^{2}+4 d^{2} e \,f^{2} p \,x^{2}+6 \ln \left (c \right ) d^{3} f^{2}}{30 d^{3} x^{5}}\) | \(736\) |
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Time = 0.34 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.76 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\left [\frac {{\left (3 \, e^{2} f^{2} - 10 \, d e f g + 15 \, d^{2} g^{2}\right )} p x^{5} \sqrt {-\frac {e}{d}} \log \left (\frac {e x^{2} + 2 \, d x \sqrt {-\frac {e}{d}} - d}{e x^{2} + d}\right ) - 2 \, d e f^{2} p x^{2} + 2 \, {\left (3 \, e^{2} f^{2} - 10 \, d e f g\right )} p x^{4} - {\left (15 \, d^{2} g^{2} p x^{4} + 10 \, d^{2} f g p x^{2} + 3 \, d^{2} f^{2} p\right )} \log \left (e x^{2} + d\right ) - {\left (15 \, d^{2} g^{2} x^{4} + 10 \, d^{2} f g x^{2} + 3 \, d^{2} f^{2}\right )} \log \left (c\right )}{15 \, d^{2} x^{5}}, \frac {2 \, {\left (3 \, e^{2} f^{2} - 10 \, d e f g + 15 \, d^{2} g^{2}\right )} p x^{5} \sqrt {\frac {e}{d}} \arctan \left (x \sqrt {\frac {e}{d}}\right ) - 2 \, d e f^{2} p x^{2} + 2 \, {\left (3 \, e^{2} f^{2} - 10 \, d e f g\right )} p x^{4} - {\left (15 \, d^{2} g^{2} p x^{4} + 10 \, d^{2} f g p x^{2} + 3 \, d^{2} f^{2} p\right )} \log \left (e x^{2} + d\right ) - {\left (15 \, d^{2} g^{2} x^{4} + 10 \, d^{2} f g x^{2} + 3 \, d^{2} f^{2}\right )} \log \left (c\right )}{15 \, d^{2} x^{5}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 1603 vs. \(2 (199) = 398\).
Time = 156.42 (sec) , antiderivative size = 1603, normalized size of antiderivative = 8.02 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.82 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2 \, {\left (3 \, e^{3} f^{2} p - 10 \, d e^{2} f g p + 15 \, d^{2} e g^{2} p\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{15 \, \sqrt {d e} d^{2}} - \frac {{\left (15 \, g^{2} p x^{4} + 10 \, f g p x^{2} + 3 \, f^{2} p\right )} \log \left (e x^{2} + d\right )}{15 \, x^{5}} + \frac {6 \, e^{2} f^{2} p x^{4} - 20 \, d e f g p x^{4} - 15 \, d^{2} g^{2} x^{4} \log \left (c\right ) - 2 \, d e f^{2} p x^{2} - 10 \, d^{2} f g x^{2} \log \left (c\right ) - 3 \, d^{2} f^{2} \log \left (c\right )}{15 \, d^{2} x^{5}} \]
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Time = 1.61 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.58 \[ \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^6} \, dx=\frac {2\,\sqrt {e}\,p\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (15\,d^2\,g^2-10\,d\,e\,f\,g+3\,e^2\,f^2\right )}{15\,d^{5/2}}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2}{5}+\frac {2\,f\,g\,x^2}{3}+g^2\,x^4\right )}{x^5}-\frac {\frac {2\,e\,f^2\,p}{d}+\frac {2\,e\,f\,p\,x^2\,\left (10\,d\,g-3\,e\,f\right )}{d^2}}{15\,x^3} \]
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