Optimal. Leaf size=67 \[ -\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {16 b^2 n^2 \sqrt {d x}}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 67, 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} \[ -\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac {16 b^2 n^2 \sqrt {d x}}{d} \]
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}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}-(4 b n) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d x}} \, dx\\ &=\frac {16 b^2 n^2 \sqrt {d x}}{d}-\frac {8 b n \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )}{d}+\frac {2 \sqrt {d x} \left (a+b \log \left (c x^n\right )\right )^2}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 54, normalized size = 0.81 \[ \frac {2 x \left (a^2+2 b (a-2 b n) \log \left (c x^n\right )-4 a b n+b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{\sqrt {d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 87, normalized size = 1.30 \[ \frac {2 \, {\left (b^{2} n^{2} \log \relax (x)^{2} + 8 \, b^{2} n^{2} + b^{2} \log \relax (c)^{2} - 4 \, a b n + a^{2} - 2 \, {\left (2 \, b^{2} n - a b\right )} \log \relax (c) - 2 \, {\left (2 \, b^{2} n^{2} - b^{2} n \log \relax (c) - a b n\right )} \log \relax (x)\right )} \sqrt {d x}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 118, normalized size = 1.76 \[ \frac {2 \, {\left ({\left (\sqrt {d x} \log \relax (x)^{2} - 4 \, \sqrt {d x} \log \relax (x) + 8 \, \sqrt {d x}\right )} b^{2} n^{2} + 2 \, {\left (\sqrt {d x} \log \relax (x) - 2 \, \sqrt {d x}\right )} b^{2} n \log \relax (c) + \sqrt {d x} b^{2} \log \relax (c)^{2} + 2 \, {\left (\sqrt {d x} \log \relax (x) - 2 \, \sqrt {d x}\right )} a b n + 2 \, \sqrt {d x} a b \log \relax (c) + \sqrt {d x} a^{2}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 107, normalized size = 1.60 \[ \frac {16 \sqrt {d x}\, b^{2} n^{2}}{d}-\frac {8 \sqrt {d x}\, b^{2} n \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )}{d}+\frac {2 \sqrt {d x}\, b^{2} \ln \left (c \,{\mathrm e}^{n \ln \relax (x )}\right )^{2}}{d}-\frac {8 \sqrt {d x}\, a b n}{d}+\frac {4 \sqrt {d x}\, a b \ln \left (c \,x^{n}\right )}{d}+\frac {2 \sqrt {d x}\, a^{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 102, normalized size = 1.52 \[ \frac {2 \, \sqrt {d x} b^{2} \log \left (c x^{n}\right )^{2}}{d} + 8 \, {\left (\frac {2 \, \sqrt {d x} n^{2}}{d} - \frac {\sqrt {d x} n \log \left (c x^{n}\right )}{d}\right )} b^{2} - \frac {8 \, \sqrt {d x} a b n}{d} + \frac {4 \, \sqrt {d x} a b \log \left (c x^{n}\right )}{d} + \frac {2 \, \sqrt {d x} a^{2}}{d} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{\sqrt {d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.46, size = 199, normalized size = 2.97 \[ \frac {2 a^{2} \sqrt {x}}{\sqrt {d}} + \frac {4 a b n \sqrt {x} \log {\relax (x )}}{\sqrt {d}} - \frac {8 a b n \sqrt {x}}{\sqrt {d}} + \frac {4 a b \sqrt {x} \log {\relax (c )}}{\sqrt {d}} + \frac {2 b^{2} n^{2} \sqrt {x} \log {\relax (x )}^{2}}{\sqrt {d}} - \frac {8 b^{2} n^{2} \sqrt {x} \log {\relax (x )}}{\sqrt {d}} + \frac {16 b^{2} n^{2} \sqrt {x}}{\sqrt {d}} + \frac {4 b^{2} n \sqrt {x} \log {\relax (c )} \log {\relax (x )}}{\sqrt {d}} - \frac {8 b^{2} n \sqrt {x} \log {\relax (c )}}{\sqrt {d}} + \frac {2 b^{2} \sqrt {x} \log {\relax (c )}^{2}}{\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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