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 = 0.0409941, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, 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 2305
Rule 2304
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*}
Mathematica [A] time = 0.016092, 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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Maple [C] time = 0.126, size = 710, normalized size = 9.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10184, size = 138, normalized size = 1.89 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.895862, size = 243, normalized size = 3.33 \begin{align*} \frac{2}{27} \,{\left (9 \, b^{2} n^{2} x \log \left (x\right )^{2} + 9 \, b^{2} x \log \left (c\right )^{2} - 6 \,{\left (2 \, b^{2} n - 3 \, a b\right )} x \log \left (c\right ) +{\left (8 \, b^{2} n^{2} - 12 \, a b n + 9 \, a^{2}\right )} x + 6 \,{\left (3 \, b^{2} n x \log \left (c\right ) -{\left (2 \, b^{2} n^{2} - 3 \, a b n\right )} x\right )} \log \left (x\right )\right )} \sqrt{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.88766, size = 216, normalized size = 2.96 \begin{align*} \frac{2 a^{2} \sqrt{d} x^{\frac{3}{2}}}{3} + \frac{4 a b \sqrt{d} n x^{\frac{3}{2}} \log{\left (x \right )}}{3} - \frac{8 a b \sqrt{d} n x^{\frac{3}{2}}}{9} + \frac{4 a b \sqrt{d} x^{\frac{3}{2}} \log{\left (c \right )}}{3} + \frac{2 b^{2} \sqrt{d} n^{2} x^{\frac{3}{2}} \log{\left (x \right )}^{2}}{3} - \frac{8 b^{2} \sqrt{d} n^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{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{\left (c \right )} \log{\left (x \right )}}{3} - \frac{8 b^{2} \sqrt{d} n x^{\frac{3}{2}} \log{\left (c \right )}}{9} + \frac{2 b^{2} \sqrt{d} x^{\frac{3}{2}} \log{\left (c \right )}^{2}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.97107, size = 517, normalized size = 7.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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