Optimal. Leaf size=98 \[ \frac{7 (d x)^{7/2} e^{-\frac{7 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{7}{2}\right /n} \text{Ei}\left (\frac{7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0913091, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2306, 2310, 2178} \[ \frac{7 (d x)^{7/2} e^{-\frac{7 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{7}{2}\right /n} \text{Ei}\left (\frac{7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 2306
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int \frac{(d x)^{5/2}}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx &=-\frac{(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{7 \int \frac{(d x)^{5/2}}{a+b \log \left (c x^n\right )} \, dx}{2 b n}\\ &=-\frac{(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}+\frac{\left (7 (d x)^{7/2} \left (c x^n\right )^{\left .-\frac{7}{2}\right /n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{7 x}{2 n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 b d n^2}\\ &=\frac{7 e^{-\frac{7 a}{2 b n}} (d x)^{7/2} \left (c x^n\right )^{\left .-\frac{7}{2}\right /n} \text{Ei}\left (\frac{7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )}{2 b^2 d n^2}-\frac{(d x)^{7/2}}{b d n \left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.153984, size = 84, normalized size = 0.86 \[ \frac{x (d x)^{5/2} \left (7 e^{-\frac{7 a}{2 b n}} \left (c x^n\right )^{\left .-\frac{7}{2}\right /n} \text{Ei}\left (\frac{7 \left (a+b \log \left (c x^n\right )\right )}{2 b n}\right )-\frac{2 b n}{a+b \log \left (c x^n\right )}\right )}{2 b^2 n^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.648, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b d^{\frac{5}{2}} n \int \frac{x^{\frac{5}{2}}}{7 \,{\left (b^{3} \log \left (c\right )^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + a^{3} + 3 \,{\left (b^{3} \log \left (c\right ) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \,{\left (b^{3} \log \left (c\right )^{2} + 2 \, a b^{2} \log \left (c\right ) + a^{2} b\right )} \log \left (x^{n}\right )\right )}}\,{d x} + \frac{2 \, d^{\frac{5}{2}} x^{\frac{7}{2}}}{7 \,{\left (b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x} d^{2} x^{2}}{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{\frac{5}{2}}}{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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