Integrand size = 23, antiderivative size = 68 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b n}{8 e^2 \left (d+e x^2\right )}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d e^2} \]
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Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2373, 272, 45} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n}{8 e^2 \left (d+e x^2\right )}-\frac {b n \log \left (d+e x^2\right )}{8 d e^2} \]
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Rule 45
Rule 272
Rule 2373
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b n) \int \frac {x^3}{\left (d+e x^2\right )^2} \, dx}{4 d} \\ & = \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b n) \text {Subst}\left (\int \frac {x}{(d+e x)^2} \, dx,x,x^2\right )}{8 d} \\ & = \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b n) \text {Subst}\left (\int \left (-\frac {d}{e (d+e x)^2}+\frac {1}{e (d+e x)}\right ) \, dx,x,x^2\right )}{8 d} \\ & = -\frac {b n}{8 e^2 \left (d+e x^2\right )}+\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac {b n \log \left (d+e x^2\right )}{8 d e^2} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.90 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {2 a d^2+b d^2 n+4 a d e x^2+b d e n x^2-2 b n \left (d+e x^2\right )^2 \log (x)+2 b d \left (d+2 e x^2\right ) \log \left (c x^n\right )+b d^2 n \log \left (d+e x^2\right )+2 b d e n x^2 \log \left (d+e x^2\right )+b e^2 n x^4 \log \left (d+e x^2\right )}{8 d e^2 \left (d+e x^2\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(62)=124\).
Time = 0.62 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.91
method | result | size |
parallelrisch | \(\frac {-\ln \left (e \,x^{2}+d \right ) x^{4} b \,e^{2} n^{2}+2 x^{4} \ln \left (c \,x^{n}\right ) b \,e^{2} n -2 \ln \left (e \,x^{2}+d \right ) x^{2} b d e \,n^{2}-x^{2} b d e \,n^{2}-\ln \left (e \,x^{2}+d \right ) b \,d^{2} n^{2}-4 x^{2} a d e n -b \,d^{2} n^{2}-2 a \,d^{2} n}{8 n d \,e^{2} \left (e \,x^{2}+d \right )^{2}}\) | \(130\) |
risch | \(-\frac {b \left (2 e \,x^{2}+d \right ) \ln \left (x^{n}\right )}{4 \left (e \,x^{2}+d \right )^{2} e^{2}}-\frac {-2 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i \pi b \,d^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+\ln \left (e \,x^{2}+d \right ) b \,e^{2} n \,x^{4}-2 \ln \left (x \right ) b \,e^{2} n \,x^{4}+2 i \pi b d e \,x^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+2 i \pi b d e \,x^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b \,d^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \left (e \,x^{2}+d \right ) b d e n \,x^{2}-4 \ln \left (x \right ) b d e n \,x^{2}+4 e \ln \left (c \right ) b d \,x^{2}+b d e n \,x^{2}+\ln \left (e \,x^{2}+d \right ) b \,d^{2} n -2 \ln \left (x \right ) b \,d^{2} n +4 a d e \,x^{2}+2 d^{2} b \ln \left (c \right )+b \,d^{2} n +2 a \,d^{2}}{8 e^{2} d \left (e \,x^{2}+d \right )^{2}}\) | \(369\) |
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (62) = 124\).
Time = 0.33 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.85 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {2 \, b e^{2} n x^{4} \log \left (x\right ) - b d^{2} n - 2 \, a d^{2} - {\left (b d e n + 4 \, a d e\right )} x^{2} - {\left (b e^{2} n x^{4} + 2 \, b d e n x^{2} + b d^{2} n\right )} \log \left (e x^{2} + d\right ) - 2 \, {\left (2 \, b d e x^{2} + b d^{2}\right )} \log \left (c\right )}{8 \, {\left (d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{2} + d^{3} e^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (58) = 116\).
Time = 167.21 (sec) , antiderivative size = 612, normalized size of antiderivative = 9.00 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {a}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 x^{2}}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {\frac {a x^{4}}{4} - \frac {b n x^{4}}{16} + \frac {b x^{4} \log {\left (c x^{n} \right )}}{4}}{d^{3}} & \text {for}\: e = 0 \\\frac {- \frac {a}{2 x^{2}} - \frac {b n}{4 x^{2}} - \frac {b \log {\left (c x^{n} \right )}}{2 x^{2}}}{e^{3}} & \text {for}\: d = 0 \\- \frac {2 a d^{2}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {4 a d e x^{2}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {b d^{2} n \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {b d^{2} n \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {b d^{2} n}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {2 b d e n x^{2} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {2 b d e n x^{2} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {b d e n x^{2}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {b e^{2} n x^{4} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} - \frac {b e^{2} n x^{4} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} + \frac {2 b e^{2} x^{4} \log {\left (c x^{n} \right )}}{8 d^{3} e^{2} + 16 d^{2} e^{3} x^{2} + 8 d e^{4} x^{4}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (62) = 124\).
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.88 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {1}{8} \, b n {\left (\frac {1}{e^{3} x^{2} + d e^{2}} + \frac {\log \left (e x^{2} + d\right )}{d e^{2}} - \frac {\log \left (x^{2}\right )}{d e^{2}}\right )} - \frac {{\left (2 \, e x^{2} + d\right )} b \log \left (c x^{n}\right )}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} - \frac {{\left (2 \, e x^{2} + d\right )} a}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (62) = 124\).
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.06 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {{\left (2 \, b e n x^{2} + b d n\right )} \log \left (x\right )}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} - \frac {b e n x^{2} + 4 \, b e x^{2} \log \left (c\right ) + 4 \, a e x^{2} + b d n + 2 \, b d \log \left (c\right ) + 2 \, a d}{8 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} - \frac {b n \log \left (e x^{2} + d\right )}{8 \, d e^{2}} + \frac {b n \log \left (x\right )}{4 \, d e^{2}} \]
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Time = 0.44 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.90 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b\,n\,\ln \left (x\right )}{4\,d\,e^2}-\frac {\ln \left (c\,x^n\right )\,\left (\frac {b\,x^2}{2\,e}+\frac {b\,d}{4\,e^2}\right )}{d^2+2\,d\,e\,x^2+e^2\,x^4}-\frac {b\,n\,\ln \left (e\,x^2+d\right )}{8\,d\,e^2}-\frac {\left (2\,a\,e+\frac {b\,e\,n}{2}\right )\,x^2+a\,d+\frac {b\,d\,n}{2}}{4\,d^2\,e^2+8\,d\,e^3\,x^2+4\,e^4\,x^4} \]
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