Integrand size = 20, antiderivative size = 73 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {16 b^2 n^2 (d x)^{7/2}}{343 d}-\frac {8 b n (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d} \]
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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} \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {8 b n (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{49 d}+\frac {16 b^2 n^2 (d x)^{7/2}}{343 d} \]
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Rule 2341
Rule 2342
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d}-\frac {1}{7} (4 b n) \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right ) \, dx \\ & = \frac {16 b^2 n^2 (d x)^{7/2}}{343 d}-\frac {8 b n (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )}{49 d}+\frac {2 (d x)^{7/2} \left (a+b \log \left (c x^n\right )\right )^2}{7 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{343} x (d x)^{5/2} \left (49 a^2-28 a b n+8 b^2 n^2+14 b (7 a-2 b n) \log \left (c x^n\right )+49 b^2 \log ^2\left (c x^n\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 716, normalized size of antiderivative = 9.81
method | result | size |
risch | \(\frac {2 d^{3} x^{4} b^{2} \ln \left (x^{n}\right )^{2}}{7 \sqrt {d x}}+\frac {2 d^{3} b \,x^{4} \left (-7 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+7 i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+7 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-7 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+14 b \ln \left (c \right )-4 b n +14 a \right ) \ln \left (x^{n}\right )}{49 \sqrt {d x}}+\frac {d^{3} \left (196 a^{2}+56 i \pi \,b^{2} n \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+98 \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}-196 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-196 i \pi a b \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-49 \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}+56 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 )-196 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 )-196 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 )+98 \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}-196 \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}+98 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-49 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right )^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{4}+98 \pi ^{2} b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{5}-49 \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}+392 \ln \left (c \right ) a b +196 \ln \left (c \right )^{2} b^{2}-112 b^{2} \ln \left (c \right ) n -112 a b n -49 \pi ^{2} b^{2} \operatorname {csgn}\left (i c \,x^{n}\right )^{6}+196 i \pi a b \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+196 i \pi a b \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-56 i \pi \,b^{2} n \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+196 i \ln \left (c \right ) \pi \,b^{2} \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-56 i \pi \,b^{2} n \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+196 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^{4}}{686 \sqrt {d x}}\) | \(716\) |
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Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (61) = 122\).
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2}{343} \, {\left (49 \, b^{2} d^{2} n^{2} x^{3} \log \left (x\right )^{2} + 49 \, b^{2} d^{2} x^{3} \log \left (c\right )^{2} - 14 \, {\left (2 \, b^{2} d^{2} n - 7 \, a b d^{2}\right )} x^{3} \log \left (c\right ) + {\left (8 \, b^{2} d^{2} n^{2} - 28 \, a b d^{2} n + 49 \, a^{2} d^{2}\right )} x^{3} + 14 \, {\left (7 \, b^{2} d^{2} n x^{3} \log \left (c\right ) - {\left (2 \, b^{2} d^{2} n^{2} - 7 \, a b d^{2} n\right )} x^{3}\right )} \log \left (x\right )\right )} \sqrt {d x} \]
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Time = 22.49 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.63 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 a^{2} x \left (d x\right )^{\frac {5}{2}}}{7} - \frac {8 a b n x \left (d x\right )^{\frac {5}{2}}}{49} + \frac {4 a b x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{7} + \frac {16 b^{2} n^{2} x \left (d x\right )^{\frac {5}{2}}}{343} - \frac {8 b^{2} n x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}}{49} + \frac {2 b^{2} x \left (d x\right )^{\frac {5}{2}} \log {\left (c x^{n} \right )}^{2}}{7} \]
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none
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {2 \, \left (d x\right )^{\frac {7}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{7 \, d} - \frac {8 \, \left (d x\right )^{\frac {7}{2}} a b n}{49 \, d} + \frac {4 \, \left (d x\right )^{\frac {7}{2}} a b \log \left (c x^{n}\right )}{7 \, d} + \frac {2 \, \left (d x\right )^{\frac {7}{2}} a^{2}}{7 \, d} + \frac {8}{343} \, {\left (\frac {2 \, \left (d x\right )^{\frac {7}{2}} n^{2}}{d} - \frac {7 \, \left (d x\right )^{\frac {7}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} \]
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Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 425, normalized size of antiderivative = 5.82 \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\left (\frac {1}{7} i + \frac {1}{7}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right )^{2} - \left (\frac {1}{7} i - \frac {1}{7}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (x\right )^{2} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{49} i + \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) + \left (\frac {2}{7} i + \frac {2}{7}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{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 (\frac {4}{49} i - \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{7} i - \frac {2}{7}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{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 (\frac {8}{343} i + \frac {8}{343}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{49} i + \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (c\right ) + \left (\frac {2}{7} i + \frac {2}{7}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) \log \left (x\right ) - \left (\frac {8}{343} i - \frac {8}{343}\right ) \, \sqrt {2} b^{2} d^{2} n^{2} x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {4}{49} i - \frac {4}{49}\right ) \, \sqrt {2} b^{2} d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (c\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {2}{7} i - \frac {2}{7}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \log \left (x\right ) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) - \left (\frac {4}{49} i + \frac {4}{49}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \left (\frac {4}{49} i - \frac {4}{49}\right ) \, \sqrt {2} a b d^{2} n x^{\frac {7}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\left (d\right )\right ) + \frac {2}{7} \, b^{2} d^{\frac {5}{2}} x^{\frac {7}{2}} \log \left (c\right )^{2} + \frac {4}{7} \, a b d^{\frac {5}{2}} x^{\frac {7}{2}} \log \left (c\right ) + \frac {2}{7} \, a^{2} d^{\frac {5}{2}} x^{\frac {7}{2}} \]
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Timed out. \[ \int (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int {\left (d\,x\right )}^{5/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
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