Optimal. Leaf size=67 \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt{d x}}-\frac{16 b^2 n^2}{d \sqrt{d x}} \]
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Rubi [A] time = 0.0468623, antiderivative size = 67, 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} \[ -\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt{d x}}-\frac{16 b^2 n^2}{d \sqrt{d x}} \]
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)^{3/2}} \, dx &=-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt{d x}}+(4 b n) \int \frac{a+b \log \left (c x^n\right )}{(d x)^{3/2}} \, dx\\ &=-\frac{16 b^2 n^2}{d \sqrt{d x}}-\frac{8 b n \left (a+b \log \left (c x^n\right )\right )}{d \sqrt{d x}}-\frac{2 \left (a+b \log \left (c x^n\right )\right )^2}{d \sqrt{d x}}\\ \end{align*}
Mathematica [A] time = 0.0127942, size = 54, normalized size = 0.81 \[ -\frac{2 x \left (a^2+2 b (a+2 b n) \log \left (c x^n\right )+4 a b n+b^2 \log ^2\left (c x^n\right )+8 b^2 n^2\right )}{(d x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.131, size = 707, normalized size = 10.6 \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 = 1.08424, size = 136, normalized size = 2.03 \begin{align*} -8 \, b^{2}{\left (\frac{2 \, n^{2}}{\sqrt{d x} d} + \frac{n \log \left (c x^{n}\right )}{\sqrt{d x} d}\right )} - \frac{2 \, b^{2} \log \left (c x^{n}\right )^{2}}{\sqrt{d x} d} - \frac{8 \, a b n}{\sqrt{d x} d} - \frac{4 \, a b \log \left (c x^{n}\right )}{\sqrt{d x} d} - \frac{2 \, a^{2}}{\sqrt{d x} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.938875, size = 212, normalized size = 3.16 \begin{align*} -\frac{2 \,{\left (b^{2} n^{2} \log \left (x\right )^{2} + 8 \, b^{2} n^{2} + b^{2} \log \left (c\right )^{2} + 4 \, a b n + a^{2} + 2 \,{\left (2 \, b^{2} n + a b\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} n^{2} + b^{2} n \log \left (c\right ) + a b n\right )} \log \left (x\right )\right )} \sqrt{d x}}{d^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.55021, size = 201, normalized size = 3. \begin{align*} - \frac{2 a^{2}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 a b n \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{8 a b n}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 a b \log{\left (c \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b^{2} n^{2} \log{\left (x \right )}^{2}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{8 b^{2} n^{2} \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{16 b^{2} n^{2}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{4 b^{2} n \log{\left (c \right )} \log{\left (x \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{8 b^{2} n \log{\left (c \right )}}{d^{\frac{3}{2}} \sqrt{x}} - \frac{2 b^{2} \log{\left (c \right )}^{2}}{d^{\frac{3}{2}} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26972, size = 201, normalized size = 3. \begin{align*} -\frac{2 \,{\left (\frac{b^{2} n^{2} \log \left (d x\right )^{2}}{\sqrt{d x}} - \frac{2 \,{\left (b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n^{2} - b^{2} n \log \left (c\right ) - a b n\right )} \log \left (d x\right )}{\sqrt{d x}} + \frac{b^{2} n^{2} \log \left (d\right )^{2} - 4 \, b^{2} n^{2} \log \left (d\right ) - 2 \, b^{2} n \log \left (c\right ) \log \left (d\right ) + 8 \, b^{2} n^{2} + 4 \, b^{2} n \log \left (c\right ) + b^{2} \log \left (c\right )^{2} - 2 \, a b n \log \left (d\right ) + 4 \, a b n + 2 \, a b \log \left (c\right ) + a^{2}}{\sqrt{d x}}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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