Optimal. Leaf size=82 \[ \frac {b n}{8 d e \left (d+e x^2\right )}+\frac {b n \log (x)}{4 d^2 e}-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e} \]
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Rubi [A]
time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2376, 272, 46}
\begin {gather*} -\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e}+\frac {b n \log (x)}{4 d^2 e}+\frac {b n}{8 d e \left (d+e x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rule 2376
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}+\frac {(b n) \int \frac {1}{x \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}+\frac {(b n) \text {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^2\right )}{8 e}\\ &=-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}+\frac {(b n) \text {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^2\right )}{8 e}\\ &=\frac {b n}{8 d e \left (d+e x^2\right )}+\frac {b n \log (x)}{4 d^2 e}-\frac {a+b \log \left (c x^n\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 111, normalized size = 1.35 \begin {gather*} \frac {b n}{8 d e \left (d+e x^2\right )}+\frac {b n \log (x)}{4 d^2 e}-\frac {b n \log (x)}{4 e \left (d+e x^2\right )^2}+\frac {-a-b \left (-n \log (x)+\log \left (c x^n\right )\right )}{4 e \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d^2 e} \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 = 0.13, size = 243, normalized size = 2.96
method | result | size |
risch | \(-\frac {b \ln \left (x^{n}\right )}{4 e \left (e \,x^{2}+d \right )^{2}}-\frac {-2 \ln \left (x \right ) b \,e^{2} n \,x^{4}+\ln \left (e \,x^{2}+d \right ) b \,e^{2} n \,x^{4}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \ln \left (x \right ) b d e n \,x^{2}+2 \ln \left (e \,x^{2}+d \right ) b d e n \,x^{2}-b d e n \,x^{2}-2 \ln \left (x \right ) b \,d^{2} n +\ln \left (e \,x^{2}+d \right ) b \,d^{2} n +2 d^{2} b \ln \left (c \right )-b \,d^{2} n +2 a \,d^{2}}{8 d^{2} e \left (e \,x^{2}+d \right )^{2}}\) | \(243\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 108, normalized size = 1.32 \begin {gather*} -\frac {1}{8} \, b n {\left (\frac {e^{\left (-1\right )} \log \left (x^{2} e + d\right )}{d^{2}} - \frac {e^{\left (-1\right )} \log \left (x^{2}\right )}{d^{2}} - \frac {1}{d x^{2} e^{2} + d^{2} e}\right )} - \frac {b \log \left (c x^{n}\right )}{4 \, {\left (x^{4} e^{3} + 2 \, d x^{2} e^{2} + d^{2} e\right )}} - \frac {a}{4 \, {\left (x^{4} e^{3} + 2 \, d x^{2} e^{2} + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 119, normalized size = 1.45 \begin {gather*} \frac {b d n x^{2} e + b d^{2} n - 2 \, b d^{2} \log \left (c\right ) - 2 \, a d^{2} - {\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e + b d^{2} n\right )} \log \left (x^{2} e + d\right ) + 2 \, {\left (b n x^{4} e^{2} + 2 \, b d n x^{2} e\right )} \log \left (x\right )}{8 \, {\left (d^{2} x^{4} e^{3} + 2 \, d^{3} x^{2} e^{2} + d^{4} e\right )}} \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. 619 vs.
\(2 (71) = 142\).
time = 201.82, size = 619, normalized size = 7.55 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {a}{4 x^{4}} - \frac {b n}{16 x^{4}} - \frac {b \log {\left (c x^{n} \right )}}{4 x^{4}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {- \frac {a}{4 x^{4}} - \frac {b n}{16 x^{4}} - \frac {b \log {\left (c x^{n} \right )}}{4 x^{4}}}{e^{3}} & \text {for}\: d = 0 \\\frac {\frac {a x^{2}}{2} - \frac {b n x^{2}}{4} + \frac {b x^{2} \log {\left (c x^{n} \right )}}{2}}{d^{3}} & \text {for}\: e = 0 \\- \frac {2 a d^{2}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b d^{2} n \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b d^{2} n \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {b d^{2} n}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {2 b d e n x^{2} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {2 b d e n x^{2} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {b d e n x^{2}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {4 b d e x^{2} \log {\left (c x^{n} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b e^{2} n x^{4} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} - \frac {b e^{2} n x^{4} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} + \frac {2 b e^{2} x^{4} \log {\left (c x^{n} \right )}}{8 d^{4} e + 16 d^{3} e^{2} x^{2} + 8 d^{2} e^{3} x^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.57, size = 136, normalized size = 1.66 \begin {gather*} -\frac {b n x^{4} e^{2} \log \left (x^{2} e + d\right ) - 2 \, b n x^{4} e^{2} \log \left (x\right ) + 2 \, b d n x^{2} e \log \left (x^{2} e + d\right ) - 4 \, b d n x^{2} e \log \left (x\right ) - b d n x^{2} e + b d^{2} n \log \left (x^{2} e + d\right ) - b d^{2} n + 2 \, b d^{2} \log \left (c\right ) + 2 \, a d^{2}}{8 \, {\left (d^{2} x^{4} e^{3} + 2 \, d^{3} x^{2} e^{2} + d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 109, normalized size = 1.33 \begin {gather*} \frac {\frac {b\,n}{2}-a+\frac {b\,e\,n\,x^2}{2\,d}}{4\,d^2\,e+8\,d\,e^2\,x^2+4\,e^3\,x^4}-\frac {b\,\ln \left (c\,x^n\right )}{4\,e\,\left (d^2+2\,d\,e\,x^2+e^2\,x^4\right )}-\frac {b\,n\,\ln \left (e\,x^2+d\right )}{8\,d^2\,e}+\frac {b\,n\,\ln \left (x\right )}{4\,d^2\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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