Optimal. Leaf size=73 \[ \frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac {8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac {16 b^2 n^2 (d x)^{5/2}}{125 d} \]
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Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2305, 2304} \[ \frac {2 (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{5 d}-\frac {8 b n (d x)^{5/2} \left (a+b \log \left (c x^n\right )\right )}{25 d}+\frac {16 b^2 n^2 (d x)^{5/2}}{125 d} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \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 {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*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.84 \[ \frac {2}{125} x (d x)^{3/2} \left (25 a^2+10 b (5 a-2 b n) \log \left (c x^n\right )-20 a b n+25 b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 121, normalized size = 1.66 \[ \frac {2}{125} \, {\left (25 \, b^{2} d n^{2} x^{2} \log \relax (x)^{2} + 25 \, b^{2} d x^{2} \log \relax (c)^{2} - 10 \, {\left (2 \, b^{2} d n - 5 \, a b d\right )} x^{2} \log \relax (c) + {\left (8 \, b^{2} d n^{2} - 20 \, a b d n + 25 \, a^{2} d\right )} x^{2} + 10 \, {\left (5 \, b^{2} d n x^{2} \log \relax (c) - {\left (2 \, b^{2} d n^{2} - 5 \, a b d n\right )} x^{2}\right )} \log \relax (x)\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.42, size = 386, normalized size = 5.29 \[ -\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}\relax (d)\right ) \log \relax (x)^{2} + \left (25 i - 25\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \relax (x)^{2} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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}\relax (d)\right ) \log \relax (x) - \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}\relax (d)\right ) \log \relax (c) \log \relax (x) - \left (20 i - 20\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (50 i - 50\right ) \, \sqrt {2} b^{2} n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \relax (c) \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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}\relax (d)\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}\relax (d)\right ) \log \relax (c) - \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}\relax (d)\right ) \log \relax (x) + \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}\relax (d)\right ) - \left (20 i - 20\right ) \, \sqrt {2} b^{2} n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \relax (c) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (50 i - 50\right ) \, \sqrt {2} a b n x^{\frac {5}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\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}\relax (d)\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}\relax (d)\right ) - 50 \, b^{2} \sqrt {d} x^{\frac {5}{2}} \log \relax (c)^{2} - 100 \, a b \sqrt {d} x^{\frac {5}{2}} \log \relax (c) - 50 \, a^{2} \sqrt {d} x^{\frac {5}{2}}\right )} d \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 716, normalized size = 9.81 \[ \frac {2 b^{2} d^{2} x^{3} \ln \left (x^{n}\right )^{2}}{5 \sqrt {d x}}+\frac {2 \left (-5 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+5 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+5 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-5 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 b n +10 b \ln \relax (c )+10 a \right ) b \,d^{2} x^{3} \ln \left (x^{n}\right )}{25 \sqrt {d x}}+\frac {\left (-25 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+50 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+50 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-100 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+40 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-100 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-100 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+100 a^{2}-40 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-40 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+100 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+100 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+100 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+100 i \pi a b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+32 b^{2} n^{2}-25 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+50 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-25 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+50 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+200 a b \ln \relax (c )-80 b^{2} n \ln \relax (c )+100 b^{2} \ln \relax (c )^{2}-80 a b n -25 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-100 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )-100 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+40 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )\right ) d^{2} x^{3}}{250 \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 102, normalized size = 1.40 \[ \frac {2 \, \left (d x\right )^{\frac {5}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{5 \, d} - \frac {8 \, \left (d x\right )^{\frac {5}{2}} a b n}{25 \, d} + \frac {4 \, \left (d x\right )^{\frac {5}{2}} a b \log \left (c x^{n}\right )}{5 \, d} + \frac {2 \, \left (d x\right )^{\frac {5}{2}} a^{2}}{5 \, d} + \frac {8}{125} \, {\left (\frac {2 \, \left (d x\right )^{\frac {5}{2}} n^{2}}{d} - \frac {5 \, \left (d x\right )^{\frac {5}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x\right )}^{3/2}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 50.36, size = 216, normalized size = 2.96 \[ \frac {2 a^{2} d^{\frac {3}{2}} x^{\frac {5}{2}}}{5} + \frac {4 a b d^{\frac {3}{2}} n x^{\frac {5}{2}} \log {\relax (x )}}{5} - \frac {8 a b d^{\frac {3}{2}} n x^{\frac {5}{2}}}{25} + \frac {4 a b d^{\frac {3}{2}} x^{\frac {5}{2}} \log {\relax (c )}}{5} + \frac {2 b^{2} d^{\frac {3}{2}} n^{2} x^{\frac {5}{2}} \log {\relax (x )}^{2}}{5} - \frac {8 b^{2} d^{\frac {3}{2}} n^{2} x^{\frac {5}{2}} \log {\relax (x )}}{25} + \frac {16 b^{2} d^{\frac {3}{2}} n^{2} x^{\frac {5}{2}}}{125} + \frac {4 b^{2} d^{\frac {3}{2}} n x^{\frac {5}{2}} \log {\relax (c )} \log {\relax (x )}}{5} - \frac {8 b^{2} d^{\frac {3}{2}} n x^{\frac {5}{2}} \log {\relax (c )}}{25} + \frac {2 b^{2} d^{\frac {3}{2}} x^{\frac {5}{2}} \log {\relax (c )}^{2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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