Integrand size = 20, antiderivative size = 73 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {16 b^2 n^2 (d x)^{5/2}}{125 d}-\frac {8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d} \]
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Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2342, 2341} \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac {8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac {16 b^2 n^2 (d x)^{5/2}}{125 d} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac {1}{5} (4 b n) \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \frac {16 b^2 n^2 (d x)^{5/2}}{125 d}-\frac {8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{125} x (d x)^{3/2} \left (25 a^2-20 a b n+8 b^2 n^2+10 b (5 a-2 b n) \log \left (c x^n\right )+25 b^2 \log ^2\left (c x^n\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 716, normalized size of antiderivative = 9.81
method | result | size |
risch | \(\frac {2 d^{2} b^{2} x^{3} \ln \left (x^{n}\right )^{2}}{5 \sqrt {d x}}+\frac {2 d^{2} b \,x^{3} \left (-5 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+5 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+5 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-5 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+10 b \ln \left (c \right )-4 b n +10 a \right ) \ln \left (x^{n}\right )}{25 \sqrt {d x}}+\frac {d^{2} \left (100 a^{2}+40 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+50 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-100 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-100 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-25 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+40 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-100 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-100 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+50 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-100 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+50 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-25 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+50 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-25 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+32 b^{2} n^{2}+200 \ln \left (c \right ) a b +100 \ln \left (c \right )^{2} b^{2}-80 b^{2} \ln \left (c \right ) n -80 a b n -25 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+100 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+100 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-40 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+100 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-40 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+100 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}\right ) x^{3}}{250 \sqrt {d x}}\) | \(716\) |
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Time = 0.30 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.66 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{125} \, {\left (25 \, b^{2} d n^{2} x^{2} \log \left (x\right )^{2} + 25 \, b^{2} d x^{2} \log \left (c\right )^{2} - 10 \, {\left (2 \, b^{2} d n - 5 \, a b d\right )} x^{2} \log \left (c\right ) + {\left (8 \, b^{2} d n^{2} - 20 \, a b d n + 25 \, a^{2} d\right )} x^{2} + 10 \, {\left (5 \, b^{2} d n x^{2} \log \left (c\right ) - {\left (2 \, b^{2} d n^{2} - 5 \, a b d n\right )} x^{2}\right )} \log \left (x\right )\right )} \sqrt {d x} \]
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Time = 3.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 a^{2} x \left (d x\right )^{\frac {3}{2}}}{5} - \frac {8 a b n x \left (d x\right )^{\frac {3}{2}}}{25} + \frac {4 a b x \left (d x\right )^{\frac {3}{2}} \log {\left (c x^{n} \right )}}{5} + \frac {16 b^{2} n^{2} x \left (d x\right )^{\frac {3}{2}}}{125} - \frac {8 b^{2} n x \left (d x\right )^{\frac {3}{2}} \log {\left (c x^{n} \right )}}{25} + \frac {2 b^{2} x \left (d x\right )^{\frac {3}{2}} \log {\left (c x^{n} \right )}^{2}}{5} \]
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Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 \, \left (d x\right )^{\frac {5}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{5 \, d} - \frac {8 \, \left (d x\right )^{\frac {5}{2}} a b n}{25 \, d} + \frac {4 \, \left (d x\right )^{\frac {5}{2}} a b \log \left (c x^{n}\right )}{5 \, d} + \frac {2 \, \left (d x\right )^{\frac {5}{2}} a^{2}}{5 \, d} + \frac {8}{125} \, {\left (\frac {2 \, \left (d x\right )^{\frac {5}{2}} n^{2}}{d} - \frac {5 \, \left (d x\right )^{\frac {5}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} \]
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Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 386, normalized size of antiderivative = 5.29 \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {1}{125} \, {\left (-\left (25 i + 25\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right )^{2} + \left (25 i - 25\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \left (x\right )^{2} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (20 i + 20\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) - \left (50 i + 50\right ) \, \sqrt {2} b^{2} n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (c\right ) \log \left (x\right ) - \left (20 i - 20\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (50 i - 50\right ) \, \sqrt {2} b^{2} n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \left (c\right ) \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (8 i + 8\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (20 i + 20\right ) \, \sqrt {2} b^{2} n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (c\right ) - \left (50 i + 50\right ) \, \sqrt {2} a b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) + \left (8 i - 8\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (20 i - 20\right ) \, \sqrt {2} b^{2} n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \left (c\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (50 i - 50\right ) \, \sqrt {2} a b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (20 i + 20\right ) \, \sqrt {2} a b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (20 i - 20\right ) \, \sqrt {2} a b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - 50 \, b^{2} \sqrt {d} x^{\frac {5}{2}} \log \left (c\right )^{2} - 100 \, a b \sqrt {d} x^{\frac {5}{2}} \log \left (c\right ) - 50 \, a^{2} \sqrt {d} x^{\frac {5}{2}}\right )} d \]
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Timed out. \[ \int (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int {\left (d\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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