Optimal. Leaf size=125 \[ -\frac {e f p}{12 d x^4}+\frac {e (2 e f-3 d g) p}{12 d^2 x^2}+\frac {e^2 (2 e f-3 d g) p \log (x)}{6 d^3}-\frac {e^2 (2 e f-3 d g) p \log \left (d+e x^2\right )}{12 d^3}-\frac {f \log \left (c \left (d+e x^2\right )^p\right )}{6 x^6}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4} \]
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Rubi [A]
time = 0.11, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2525, 45, 2461,
12, 78} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 78
Rule 2461
Rule 2525
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{x^7} \, dx &=\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*}
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Mathematica [A]
time = 0.05, size = 130, normalized size = 1.04 \begin {gather*} \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}{6} e f p \left (-\frac {1}{2 d x^4}+\frac {e}{d^2 x^2}+\frac {2 e^2 \log (x)}{d^3}-\frac {e^2 \log \left (d+e x^2\right )}{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} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.12, size = 428, normalized size = 3.42
method | result | size |
risch | \(-\frac {\left (3 g \,x^{2}+2 f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{12 x^{6}}-\frac {12 \ln \left (x \right ) d \,e^{2} g p \,x^{6}-8 \ln \left (x \right ) e^{3} f p \,x^{6}-6 \ln \left (-e \,x^{2}-d \right ) d \,e^{2} g p \,x^{6}+4 \ln \left (-e \,x^{2}-d \right ) e^{3} f p \,x^{6}-2 i \pi \,d^{3} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+2 i \pi \,d^{3} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-2 i \pi \,d^{3} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )+3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+2 i \pi \,d^{3} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-3 i \pi \,d^{3} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+6 d^{2} e g p \,x^{4}-4 d \,e^{2} f p \,x^{4}+6 \ln \left (c \right ) d^{3} g \,x^{2}+2 d^{2} e f p \,x^{2}+4 \ln \left (c \right ) d^{3} f}{24 d^{3} x^{6}}\) | \(428\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 107, normalized size = 0.86 \begin {gather*} \frac {1}{12} \, p {\left (\frac {{\left (3 \, d g e - 2 \, f e^{2}\right )} \log \left (x^{2} e + d\right )}{d^{3}} - \frac {{\left (3 \, d g e - 2 \, f e^{2}\right )} \log \left (x^{2}\right )}{d^{3}} - \frac {{\left (3 \, d g - 2 \, f e\right )} x^{2} + d f}{d^{2} x^{4}}\right )} e - \frac {{\left (3 \, g x^{2} + 2 \, f\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 134, normalized size = 1.07 \begin {gather*} \frac {2 \, d f p x^{4} e^{2} - {\left (3 \, d^{2} g p x^{4} + d^{2} f p x^{2}\right )} e + {\left (3 \, d g p x^{6} e^{2} - 2 \, f p x^{6} e^{3} - 3 \, d^{3} g p x^{2} - 2 \, d^{3} f p\right )} \log \left (x^{2} e + d\right ) - {\left (3 \, d^{3} g x^{2} + 2 \, d^{3} f\right )} \log \left (c\right ) - 2 \, {\left (3 \, d g p x^{6} e^{2} - 2 \, f p x^{6} e^{3}\right )} \log \left (x\right )}{12 \, d^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 515 vs.
\(2 (119) = 238\).
time = 4.84, size = 515, normalized size = 4.12 \begin {gather*} \frac {{\left (3 \, {\left (x^{2} e + d\right )}^{3} d g p e^{3} \log \left (x^{2} e + d\right ) - 9 \, {\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )} d^{3} g p e^{3} \log \left (x^{2} e + d\right ) - 3 \, {\left (x^{2} e + d\right )}^{3} d g p e^{3} \log \left (x^{2} e\right ) + 9 \, {\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} \log \left (x^{2} e\right ) - 9 \, {\left (x^{2} e + d\right )} d^{3} g p e^{3} \log \left (x^{2} e\right ) + 3 \, d^{4} g p e^{3} \log \left (x^{2} e\right ) - 3 \, {\left (x^{2} e + d\right )}^{2} d^{2} g p e^{3} + 6 \, {\left (x^{2} e + d\right )} d^{3} g p e^{3} - 3 \, d^{4} g p e^{3} - 2 \, {\left (x^{2} e + d\right )}^{3} f p e^{4} \log \left (x^{2} e + d\right ) + 6 \, {\left (x^{2} e + d\right )}^{2} d f p e^{4} \log \left (x^{2} e + d\right ) - 6 \, {\left (x^{2} e + d\right )} d^{2} f p e^{4} \log \left (x^{2} e + d\right ) + 2 \, {\left (x^{2} e + d\right )}^{3} f p e^{4} \log \left (x^{2} e\right ) - 6 \, {\left (x^{2} e + d\right )}^{2} d f p e^{4} \log \left (x^{2} e\right ) + 6 \, {\left (x^{2} e + d\right )} d^{2} f p e^{4} \log \left (x^{2} e\right ) - 2 \, d^{3} f p e^{4} \log \left (x^{2} e\right ) - 3 \, {\left (x^{2} e + d\right )} d^{3} g e^{3} \log \left (c\right ) + 3 \, d^{4} g e^{3} \log \left (c\right ) + 2 \, {\left (x^{2} e + d\right )}^{2} d f p e^{4} - 5 \, {\left (x^{2} e + d\right )} d^{2} f p e^{4} + 3 \, d^{3} f p e^{4} - 2 \, d^{3} f e^{4} \log \left (c\right )\right )} e^{\left (-1\right )}}{12 \, {\left ({\left (x^{2} e + d\right )}^{3} d^{3} - 3 \, {\left (x^{2} e + d\right )}^{2} d^{4} + 3 \, {\left (x^{2} e + d\right )} d^{5} - d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 113, normalized size = 0.90 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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