Optimal. Leaf size=93 \[ -\frac {e f p}{4 d x^2}-\frac {e (e f-2 d g) p \log (x)}{2 d^2}+\frac {(e f-d g)^2 p \log \left (d+e x^2\right )}{4 d^2 f}-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 93, 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, 37, 2461,
12, 90} \begin {gather*} -\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4}+\frac {p (e f-d g)^2 \log \left (d+e x^2\right )}{4 d^2 f}-\frac {e p \log (x) (e f-2 d g)}{2 d^2}-\frac {e f p}{4 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 90
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^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(f+g x) \log \left (c (d+e x)^p\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4}-\frac {1}{2} (e p) \text {Subst}\left (\int -\frac {(f+g x)^2}{2 f x^2 (d+e x)} \, dx,x,x^2\right )\\ &=-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4}+\frac {(e p) \text {Subst}\left (\int \frac {(f+g x)^2}{x^2 (d+e x)} \, dx,x,x^2\right )}{4 f}\\ &=-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4}+\frac {(e p) \text {Subst}\left (\int \left (\frac {f^2}{d x^2}+\frac {f (-e f+2 d g)}{d^2 x}+\frac {(-e f+d g)^2}{d^2 (d+e x)}\right ) \, dx,x,x^2\right )}{4 f}\\ &=-\frac {e f p}{4 d x^2}-\frac {e (e f-2 d g) p \log (x)}{2 d^2}+\frac {(e f-d g)^2 p \log \left (d+e x^2\right )}{4 d^2 f}-\frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 f x^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 105, normalized size = 1.13 \begin {gather*} \frac {e g p \log (x)}{d}-\frac {e g p \log \left (d+e x^2\right )}{2 d}+\frac {1}{4} e f 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 {f \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {g \log \left (c \left (d+e x^2\right )^p\right )}{2 x^2} \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 = 0.15, size = 392, normalized size = 4.22
method | result | size |
risch | \(-\frac {\left (2 g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{4 x^{4}}-\frac {2 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-2 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-2 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+2 i \pi \,d^{2} g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+4 \ln \left (e \,x^{2}+d \right ) d e g p \,x^{4}-2 \ln \left (e \,x^{2}+d \right ) e^{2} f p \,x^{4}-8 \ln \left (x \right ) d e g p \,x^{4}+4 \ln \left (x \right ) e^{2} f p \,x^{4}+i \pi \,d^{2} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}-i \pi \,d^{2} f \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \,d^{2} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}+i \pi \,d^{2} f \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+4 \ln \left (c \right ) d^{2} g \,x^{2}+2 d e f p \,x^{2}+2 \ln \left (c \right ) d^{2} f}{8 d^{2} x^{4}}\) | \(392\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 83, normalized size = 0.89 \begin {gather*} -\frac {1}{4} \, p {\left (\frac {{\left (2 \, d g - f e\right )} \log \left (x^{2} e + d\right )}{d^{2}} - \frac {{\left (2 \, d g - f e\right )} \log \left (x^{2}\right )}{d^{2}} + \frac {f}{d x^{2}}\right )} e - \frac {{\left (2 \, g x^{2} + f\right )} \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 106, normalized size = 1.14 \begin {gather*} -\frac {d f p x^{2} e + {\left (2 \, d g p x^{4} e - f p x^{4} e^{2} + 2 \, d^{2} g p x^{2} + d^{2} f p\right )} \log \left (x^{2} e + d\right ) + {\left (2 \, d^{2} g x^{2} + d^{2} f\right )} \log \left (c\right ) - 2 \, {\left (2 \, d g p x^{4} e - f p x^{4} e^{2}\right )} \log \left (x\right )}{4 \, d^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs.
\(2 (83) = 166\).
time = 79.75, size = 167, normalized size = 1.80 \begin {gather*} \begin {cases} - \frac {f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{4 x^{4}} - \frac {g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 x^{2}} - \frac {e f p}{4 d x^{2}} + \frac {e g p \log {\left (x \right )}}{d} - \frac {e g \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{2 d} - \frac {e^{2} f p \log {\left (x \right )}}{2 d^{2}} + \frac {e^{2} f \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{4 d^{2}} & \text {for}\: d \neq 0 \\- \frac {f p}{8 x^{4}} - \frac {f \log {\left (c \left (e x^{2}\right )^{p} \right )}}{4 x^{4}} - \frac {g p}{2 x^{2}} - \frac {g \log {\left (c \left (e x^{2}\right )^{p} \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \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. 322 vs.
\(2 (92) = 184\).
time = 3.79, size = 322, normalized size = 3.46 \begin {gather*} -\frac {{\left (2 \, {\left (x^{2} e + d\right )}^{2} d g p e^{2} \log \left (x^{2} e + d\right ) - 2 \, {\left (x^{2} e + d\right )} d^{2} g p e^{2} \log \left (x^{2} e + d\right ) - 2 \, {\left (x^{2} e + d\right )}^{2} d g p e^{2} \log \left (x^{2} e\right ) + 4 \, {\left (x^{2} e + d\right )} d^{2} g p e^{2} \log \left (x^{2} e\right ) - 2 \, d^{3} g p e^{2} \log \left (x^{2} e\right ) - {\left (x^{2} e + d\right )}^{2} f p e^{3} \log \left (x^{2} e + d\right ) + 2 \, {\left (x^{2} e + d\right )} d f p e^{3} \log \left (x^{2} e + d\right ) + {\left (x^{2} e + d\right )}^{2} f p e^{3} \log \left (x^{2} e\right ) - 2 \, {\left (x^{2} e + d\right )} d f p e^{3} \log \left (x^{2} e\right ) + d^{2} f p e^{3} \log \left (x^{2} e\right ) + 2 \, {\left (x^{2} e + d\right )} d^{2} g e^{2} \log \left (c\right ) - 2 \, d^{3} g e^{2} \log \left (c\right ) + {\left (x^{2} e + d\right )} d f p e^{3} - d^{2} f p e^{3} + d^{2} f e^{3} \log \left (c\right )\right )} e^{\left (-1\right )}}{4 \, {\left ({\left (x^{2} e + d\right )}^{2} d^{2} - 2 \, {\left (x^{2} e + d\right )} d^{3} + d^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.37, size = 85, normalized size = 0.91 \begin {gather*} \frac {\ln \left (e\,x^2+d\right )\,\left (e^2\,f\,p-2\,d\,e\,g\,p\right )}{4\,d^2}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {g\,x^2}{2}+\frac {f}{4}\right )}{x^4}-\frac {\ln \left (x\right )\,\left (e^2\,f\,p-2\,d\,e\,g\,p\right )}{2\,d^2}-\frac {e\,f\,p}{4\,d\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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