Integrand size = 23, antiderivative size = 125 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {e f p}{12 d x^4}+\frac {e (2 e f-3 d g) p}{12 d^2 x^2}+\frac {e^2 (2 e f-3 d g) p \log (x)}{6 d^3}-\frac {e^2 (2 e f-3 d g) p \log \left (d+e x^2\right )}{12 d^3}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \]
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Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2525, 45, 2461, 12, 78} \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {e^2 p (2 e f-3 d g) \log \left (d+e x^2\right )}{12 d^3}+\frac {e^2 p \log (x) (2 e f-3 d g)}{6 d^3}+\frac {e p (2 e f-3 d g)}{12 d^2 x^2}-\frac {e f p}{12 d x^4} \]
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Rule 12
Rule 45
Rule 78
Rule 2461
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x^4} \, dx,x,x^2\right ) \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {-2 f-3 g x}{6 x^3 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{12} (e p) \text {Subst}\left (\int \frac {-2 f-3 g x}{x^3 (d+e x)} \, dx,x,x^2\right ) \\ & = -\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{12} (e p) \text {Subst}\left (\int \left (-\frac {2 f}{d x^3}+\frac {2 e f-3 d g}{d^2 x^2}+\frac {e (-2 e f+3 d g)}{d^3 x}-\frac {e^2 (-2 e f+3 d g)}{d^3 (d+e x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {e f p}{12 d x^4}+\frac {e (2 e f-3 d g) p}{12 d^2 x^2}+\frac {e^2 (2 e f-3 d g) p \log (x)}{6 d^3}-\frac {e^2 (2 e f-3 d g) p \log \left (d+e x^2\right )}{12 d^3}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {1}{4} e g p \left (-\frac {1}{d x^2}-\frac {2 e \log (x)}{d^2}+\frac {e \log \left (d+e x^2\right )}{d^2}\right )+\frac {1}{3} e f p \left (-\frac {1}{4 d x^4}+\frac {e}{2 d^2 x^2}+\frac {e^2 \log (x)}{d^3}-\frac {e^2 \log \left (d+e x^2\right )}{2 d^3}\right )-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \]
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Time = 0.89 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.86
method | result | size |
parts | \(-\frac {g \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}-\frac {f \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{6 x^{6}}-\frac {p e \left (-\frac {-3 d g +2 e f}{2 d^{2} x^{2}}+\frac {\left (3 d g -2 e f \right ) e \ln \left (x \right )}{d^{3}}+\frac {f}{2 d \,x^{4}}-\frac {e \left (3 d g -2 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{3}}\right )}{6}\) | \(108\) |
parallelrisch | \(-\frac {6 \ln \left (x \right ) x^{6} d \,e^{2} g \,p^{2}-4 \ln \left (x \right ) x^{6} e^{3} f \,p^{2}-3 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d \,e^{2} g p +2 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{3} f p -3 x^{6} d \,e^{2} g \,p^{2}+2 x^{6} e^{3} f \,p^{2}+3 x^{4} d^{2} e g \,p^{2}-2 x^{4} d \,e^{2} f \,p^{2}+3 x^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} g p +x^{2} d^{2} e f \,p^{2}+2 \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d^{3} f p}{12 x^{6} p \,d^{3}}\) | \(191\) |
risch | \(-\frac {\left (3 g \,x^{2}+2 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{12 x^{6}}-\frac {12 \ln \left (x \right ) d \,e^{2} g p \,x^{6}-8 \ln \left (x \right ) e^{3} f p \,x^{6}-6 \ln \left (-e \,x^{2}-d \right ) d \,e^{2} g p \,x^{6}+4 \ln \left (-e \,x^{2}-d \right ) e^{3} f p \,x^{6}+3 i \pi \,d^{3} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+3 i \pi \,d^{3} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-2 i \pi \,d^{3} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-3 i \pi \,d^{3} g \,x^{2} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-2 i \pi \,d^{3} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+2 i \pi \,d^{3} f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-3 i \pi \,d^{3} g \,x^{2} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+2 i \pi \,d^{3} f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+6 d^{2} e g p \,x^{4}-4 d \,e^{2} f p \,x^{4}+6 \ln \left (c \right ) d^{3} g \,x^{2}+2 d^{2} e f p \,x^{2}+4 \ln \left (c \right ) d^{3} f}{24 d^{3} x^{6}}\) | \(428\) |
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Time = 0.32 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.03 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {2 \, {\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} \log \left (x\right ) - d^{2} e f p x^{2} + {\left (2 \, d e^{2} f - 3 \, d^{2} e g\right )} p x^{4} - {\left ({\left (2 \, e^{3} f - 3 \, d e^{2} g\right )} p x^{6} + 3 \, d^{3} g p x^{2} + 2 \, d^{3} f p\right )} \log \left (e x^{2} + d\right ) - {\left (3 \, d^{3} g x^{2} + 2 \, d^{3} f\right )} \log \left (c\right )}{12 \, d^{3} x^{6}} \]
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Timed out. \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.83 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {1}{12} \, e p {\left (\frac {{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {{\left (2 \, e^{2} f - 3 \, d e g\right )} \log \left (x^{2}\right )}{d^{3}} - \frac {{\left (2 \, e f - 3 \, d g\right )} x^{2} - d f}{d^{2} x^{4}}\right )} - \frac {{\left (3 \, g x^{2} + 2 \, f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{12 \, x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (113) = 226\).
Time = 0.33 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.53 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=-\frac {\frac {{\left (2 \, e^{4} f p + 3 \, {\left (e x^{2} + d\right )} e^{3} g p - 3 \, d e^{3} g p\right )} \log \left (e x^{2} + d\right )}{{\left (e x^{2} + d\right )}^{3} - 3 \, {\left (e x^{2} + d\right )}^{2} d + 3 \, {\left (e x^{2} + d\right )} d^{2} - d^{3}} - \frac {2 \, {\left (e x^{2} + d\right )}^{2} e^{4} f p - 5 \, {\left (e x^{2} + d\right )} d e^{4} f p + 3 \, d^{2} e^{4} f p - 3 \, {\left (e x^{2} + d\right )}^{2} d e^{3} g p + 6 \, {\left (e x^{2} + d\right )} d^{2} e^{3} g p - 3 \, d^{3} e^{3} g p - 2 \, d^{2} e^{4} f \log \left (c\right ) - 3 \, {\left (e x^{2} + d\right )} d^{2} e^{3} g \log \left (c\right ) + 3 \, d^{3} e^{3} g \log \left (c\right )}{{\left (e x^{2} + d\right )}^{3} d^{2} - 3 \, {\left (e x^{2} + d\right )}^{2} d^{3} + 3 \, {\left (e x^{2} + d\right )} d^{4} - d^{5}} + \frac {{\left (2 \, e^{4} f p - 3 \, d e^{3} g p\right )} \log \left (e x^{2} + d\right )}{d^{3}} - \frac {{\left (2 \, e^{4} f p - 3 \, d e^{3} g p\right )} \log \left (e x^{2}\right )}{d^{3}}}{12 \, e} \]
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Time = 1.72 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx=\frac {\ln \left (x\right )\,\left (2\,e^3\,f\,p-3\,d\,e^2\,g\,p\right )}{6\,d^3}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{4}+\frac {f}{6}\right )}{x^6}-\frac {\ln \left (e\,x^2+d\right )\,\left (2\,e^3\,f\,p-3\,d\,e^2\,g\,p\right )}{12\,d^3}-\frac {\frac {e\,f\,p}{2\,d}+\frac {e\,p\,x^2\,\left (3\,d\,g-2\,e\,f\right )}{2\,d^2}}{6\,x^4} \]
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