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 = 0.0464623, 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} \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{9 d (d x)^{3/2}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d x)^{3/2}}-\frac{16 b^2 n^2}{27 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
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*}
Mathematica [A] time = 0.0146199, 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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Maple [C] time = 0.137, size = 716, normalized size = 9.8 \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 = 1.13115, size = 138, normalized size = 1.89 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.848769, size = 236, normalized size = 3.23 \begin{align*} -\frac{2 \,{\left (9 \, b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + 9 \, b^{2} \log \left (c\right )^{2} + 12 \, a b n + 9 \, a^{2} + 6 \,{\left (2 \, b^{2} n + 3 \, a b\right )} \log \left (c\right ) + 6 \,{\left (2 \, b^{2} n^{2} + 3 \, b^{2} n \log \left (c\right ) + 3 \, a b n\right )} \log \left (x\right )\right )} \sqrt{d x}}{27 \, d^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 42.5333, size = 218, normalized size = 2.99 \begin{align*} - \frac{2 a^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{4 a b n \log{\left (x \right )}}{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{\left (c \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b^{2} n^{2} \log{\left (x \right )}^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{8 b^{2} n^{2} \log{\left (x \right )}}{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{\left (c \right )} \log{\left (x \right )}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{8 b^{2} n \log{\left (c \right )}}{9 d^{\frac{5}{2}} x^{\frac{3}{2}}} - \frac{2 b^{2} \log{\left (c \right )}^{2}}{3 d^{\frac{5}{2}} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30257, size = 288, normalized size = 3.95 \begin{align*} -\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 \left (d\right ) - 2 \, b^{2} d^{2} n^{2} - 3 \, b^{2} d^{2} n \log \left (c\right ) - 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 \left (d\right )^{2} - 12 \, b^{2} d^{2} n^{2} \log \left (d\right ) - 18 \, b^{2} d^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} d^{2} n^{2} + 12 \, b^{2} d^{2} n \log \left (c\right ) + 9 \, b^{2} d^{2} \log \left (c\right )^{2} - 18 \, a b d^{2} n \log \left (d\right ) + 12 \, a b d^{2} n + 18 \, a b d^{2} \log \left (c\right ) + 9 \, a^{2} d^{2}}{\sqrt{d x} d x}\right )}}{27 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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