Optimal. Leaf size=73 \[ -\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac {16 b^2 n^2}{27 d (d x)^{3/2}} \]
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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} \[ -\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac {16 b^2 n^2}{27 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d x)^{5/2}} \, dx &=-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}+\frac {1}{3} (4 b n) \int \frac {a+b \log \left (c x^n\right )}{(d x)^{5/2}} \, dx\\ &=-\frac {16 b^2 n^2}{27 d (d x)^{3/2}}-\frac {8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac {2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.84 \[ -\frac {2 x \left (9 a^2+6 b (3 a+2 b n) \log \left (c x^n\right )+12 a b n+9 b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{27 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 94, normalized size = 1.29 \[ -\frac {2 \, {\left (9 \, b^{2} n^{2} \log \relax (x)^{2} + 8 \, b^{2} n^{2} + 9 \, b^{2} \log \relax (c)^{2} + 12 \, a b n + 9 \, a^{2} + 6 \, {\left (2 \, b^{2} n + 3 \, a b\right )} \log \relax (c) + 6 \, {\left (2 \, b^{2} n^{2} + 3 \, b^{2} n \log \relax (c) + 3 \, a b n\right )} \log \relax (x)\right )} \sqrt {d x}}{27 \, d^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 213, normalized size = 2.92 \[ -\frac {2 \, {\left (\frac {9 \, b^{2} d n^{2} \log \left (d x\right )^{2}}{\sqrt {d x} x} - \frac {6 \, {\left (3 \, b^{2} d^{2} n^{2} \log \relax (d) - 2 \, b^{2} d^{2} n^{2} - 3 \, b^{2} d^{2} n \log \relax (c) - 3 \, a b d^{2} n\right )} \log \left (d x\right )}{\sqrt {d x} d x} + \frac {9 \, b^{2} d^{2} n^{2} \log \relax (d)^{2} - 12 \, b^{2} d^{2} n^{2} \log \relax (d) - 18 \, b^{2} d^{2} n \log \relax (c) \log \relax (d) + 8 \, b^{2} d^{2} n^{2} + 12 \, b^{2} d^{2} n \log \relax (c) + 9 \, b^{2} d^{2} \log \relax (c)^{2} - 18 \, a b d^{2} n \log \relax (d) + 12 \, a b d^{2} n + 18 \, a b d^{2} \log \relax (c) + 9 \, a^{2} d^{2}}{\sqrt {d x} d x}\right )}}{27 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 716, normalized size = 9.81 \[ -\frac {2 b^{2} \ln \left (x^{n}\right )^{2}}{3 \sqrt {d x}\, d^{2} x}-\frac {2 \left (-3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+3 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+3 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-3 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 b n +6 b \ln \relax (c )+6 a \right ) b \ln \left (x^{n}\right )}{9 \sqrt {d x}\, d^{2} x}-\frac {-9 \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}+18 \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}+18 \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}-36 \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}-24 i \pi \,b^{2} n \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-36 i \pi \,b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )-36 i \pi a b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+36 a^{2}+24 i \pi \,b^{2} n \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+24 i \pi \,b^{2} n \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 i \pi \,b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+36 i \pi \,b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+36 i \pi a b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+36 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}-9 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-9 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+18 \pi ^{2} b^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+72 a b \ln \relax (c )+48 b^{2} n \ln \relax (c )+36 b^{2} \ln \relax (c )^{2}+48 a b n -9 \pi ^{2} b^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-36 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 )-36 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 )-24 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 )}{54 \sqrt {d x}\, d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 102, normalized size = 1.40 \[ -\frac {8}{27} \, b^{2} {\left (\frac {2 \, n^{2}}{\left (d x\right )^{\frac {3}{2}} d} + \frac {3 \, n \log \left (c x^{n}\right )}{\left (d x\right )^{\frac {3}{2}} d}\right )} - \frac {2 \, b^{2} \log \left (c x^{n}\right )^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {8 \, a b n}{9 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {4 \, a b \log \left (c x^{n}\right )}{3 \, \left (d x\right )^{\frac {3}{2}} d} - \frac {2 \, a^{2}}{3 \, \left (d x\right )^{\frac {3}{2}} d} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 26.71, size = 218, normalized size = 2.99 \[ - \frac {2 a^{2}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 a b n \log {\relax (x )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {8 a b n}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 a b \log {\relax (c )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b^{2} n^{2} \log {\relax (x )}^{2}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {8 b^{2} n^{2} \log {\relax (x )}}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {16 b^{2} n^{2}}{27 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {4 b^{2} n \log {\relax (c )} \log {\relax (x )}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {8 b^{2} n \log {\relax (c )}}{9 d^{\frac {5}{2}} x^{\frac {3}{2}}} - \frac {2 b^{2} \log {\relax (c )}^{2}}{3 d^{\frac {5}{2}} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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