Optimal. Leaf size=73 \[ -\frac {8 b n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d}+\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{3 d}+\frac {16 b^2 n^2 (d x)^{3/2}}{27 d} \]
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Rubi [A] time = 0.04, 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} \[ -\frac {8 b n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d}+\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{3 d}+\frac {16 b^2 n^2 (d x)^{3/2}}{27 d} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2305
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx &=\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{3 d}-\frac {1}{3} (4 b n) \int \sqrt {d x} \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac {16 b^2 n^2 (d x)^{3/2}}{27 d}-\frac {8 b n (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d}+\frac {2 (d x)^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.84 \[ \frac {2}{27} x \sqrt {d x} \left (9 a^2+6 b (3 a-2 b n) \log \left (c x^n\right )-12 a b n+9 b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 99, normalized size = 1.36 \[ \frac {2}{27} \, {\left (9 \, b^{2} n^{2} x \log \relax (x)^{2} + 9 \, b^{2} x \log \relax (c)^{2} - 6 \, {\left (2 \, b^{2} n - 3 \, a b\right )} x \log \relax (c) + {\left (8 \, b^{2} n^{2} - 12 \, a b n + 9 \, a^{2}\right )} x + 6 \, {\left (3 \, b^{2} n x \log \relax (c) - {\left (2 \, b^{2} n^{2} - 3 \, a b n\right )} x\right )} \log \relax (x)\right )} \sqrt {d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.28, size = 383, normalized size = 5.25 \[ \left (\frac {1}{3} i + \frac {1}{3}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (x)^{2} - \left (\frac {1}{3} i - \frac {1}{3}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \relax (x)^{2} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {4}{9} i + \frac {4}{9}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (x) + \left (\frac {2}{3} i + \frac {2}{3}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (c) \log \relax (x) + \left (\frac {4}{9} i - \frac {4}{9}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {2}{3} i - \frac {2}{3}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \relax (c) \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (\frac {8}{27} i + \frac {8}{27}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {4}{9} i + \frac {4}{9}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (c) + \left (\frac {2}{3} i + \frac {2}{3}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) \log \relax (x) - \left (\frac {8}{27} i - \frac {8}{27}\right ) \, \sqrt {2} b^{2} n^{2} x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (\frac {4}{9} i - \frac {4}{9}\right ) \, \sqrt {2} b^{2} n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \relax (c) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {2}{3} i - \frac {2}{3}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \log \relax (x) \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) - \left (\frac {4}{9} i + \frac {4}{9}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \cos \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \left (\frac {4}{9} i - \frac {4}{9}\right ) \, \sqrt {2} a b n x^{\frac {3}{2}} \sqrt {{\left | d \right |}} \sin \left (\frac {1}{4} \, \pi \mathrm {sgn}\relax (d)\right ) + \frac {2}{3} \, b^{2} \sqrt {d} x^{\frac {3}{2}} \log \relax (c)^{2} + \frac {4}{3} \, a b \sqrt {d} x^{\frac {3}{2}} \log \relax (c) + \frac {2}{3} \, a^{2} \sqrt {d} x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.18, size = 710, normalized size = 9.73 \[ \frac {2 b^{2} d \,x^{2} \ln \left (x^{n}\right )^{2}}{3 \sqrt {d 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 d \,x^{2} \ln \left (x^{n}\right )}{9 \sqrt {d x}}+\frac {\left (-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 )\right ) d \,x^{2}}{54 \sqrt {d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 102, normalized size = 1.40 \[ \frac {2 \, \left (d x\right )^{\frac {3}{2}} b^{2} \log \left (c x^{n}\right )^{2}}{3 \, d} - \frac {8 \, \left (d x\right )^{\frac {3}{2}} a b n}{9 \, d} + \frac {4 \, \left (d x\right )^{\frac {3}{2}} a b \log \left (c x^{n}\right )}{3 \, d} + \frac {8}{27} \, {\left (\frac {2 \, \left (d x\right )^{\frac {3}{2}} n^{2}}{d} - \frac {3 \, \left (d x\right )^{\frac {3}{2}} n \log \left (c x^{n}\right )}{d}\right )} b^{2} + \frac {2 \, \left (d x\right )^{\frac {3}{2}} a^{2}}{3 \, d} \]
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 \[ \int \sqrt {d\,x}\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.46, size = 216, normalized size = 2.96 \[ \frac {2 a^{2} \sqrt {d} x^{\frac {3}{2}}}{3} + \frac {4 a b \sqrt {d} n x^{\frac {3}{2}} \log {\relax (x )}}{3} - \frac {8 a b \sqrt {d} n x^{\frac {3}{2}}}{9} + \frac {4 a b \sqrt {d} x^{\frac {3}{2}} \log {\relax (c )}}{3} + \frac {2 b^{2} \sqrt {d} n^{2} x^{\frac {3}{2}} \log {\relax (x )}^{2}}{3} - \frac {8 b^{2} \sqrt {d} n^{2} x^{\frac {3}{2}} \log {\relax (x )}}{9} + \frac {16 b^{2} \sqrt {d} n^{2} x^{\frac {3}{2}}}{27} + \frac {4 b^{2} \sqrt {d} n x^{\frac {3}{2}} \log {\relax (c )} \log {\relax (x )}}{3} - \frac {8 b^{2} \sqrt {d} n x^{\frac {3}{2}} \log {\relax (c )}}{9} + \frac {2 b^{2} \sqrt {d} x^{\frac {3}{2}} \log {\relax (c )}^{2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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