Optimal. Leaf size=93 \[ \frac{2 a^2 \sqrt{a+b (c x)^n}}{n}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n} \]
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Rubi [A] time = 0.0752527, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {367, 12, 266, 50, 63, 208} \[ \frac{2 a^2 \sqrt{a+b (c x)^n}}{n}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n} \]
Antiderivative was successfully verified.
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Rule 367
Rule 12
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b (c x)^n\right )^{5/2}}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{c \left (a+b x^n\right )^{5/2}}{x} \, dx,x,c x\right )}{c}\\ &=\operatorname{Subst}\left (\int \frac{\left (a+b x^n\right )^{5/2}}{x} \, dx,x,c x\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac{a \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 a^2 \sqrt{a+b (c x)^n}}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac{2 a^2 \sqrt{a+b (c x)^n}}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b (c x)^n}\right )}{b n}\\ &=\frac{2 a^2 \sqrt{a+b (c x)^n}}{n}+\frac{2 a \left (a+b (c x)^n\right )^{3/2}}{3 n}+\frac{2 \left (a+b (c x)^n\right )^{5/2}}{5 n}-\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0780542, size = 77, normalized size = 0.83 \[ \frac{2 \sqrt{a+b (c x)^n} \left (23 a^2+11 a b (c x)^n+3 b^2 (c x)^{2 n}\right )-30 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b (c x)^n}}{\sqrt{a}}\right )}{15 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 70, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ({\frac{2}{5} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{5}{2}}}}+{\frac{2\,a}{3} \left ( a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{3}{2}}}}+2\,\sqrt{a+b \left ( cx \right ) ^{n}}{a}^{2}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) \right ) } \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*} \int \frac{{\left (\left (c x\right )^{n} b + a\right )}^{\frac{5}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84996, size = 393, normalized size = 4.23 \begin{align*} \left [\frac{15 \, a^{\frac{5}{2}} \log \left (\frac{\left (c x\right )^{n} b - 2 \, \sqrt{\left (c x\right )^{n} b + a} \sqrt{a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \,{\left (11 \, \left (c x\right )^{n} a b + 3 \, \left (c x\right )^{2 \, n} b^{2} + 23 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b + a}}{15 \, n}, \frac{2 \,{\left (15 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b + a} \sqrt{-a}}{a}\right ) +{\left (11 \, \left (c x\right )^{n} a b + 3 \, \left (c x\right )^{2 \, n} b^{2} + 23 \, a^{2}\right )} \sqrt{\left (c x\right )^{n} b + a}\right )}}{15 \, n}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 73.9411, size = 124, normalized size = 1.33 \begin{align*} - \begin{cases} - \left (a^{2} \sqrt{a + b} + 2 a b \sqrt{a + b} + b^{2} \sqrt{a + b}\right ) \log{\left (c x \right )} & \text{for}\: n = 0 \\- \frac{\frac{2 a^{3} \operatorname{atan}{\left (\frac{\sqrt{a + b \left (c x\right )^{n}}}{\sqrt{- a}} \right )}}{\sqrt{- a}} + 2 a^{2} \sqrt{a + b \left (c x\right )^{n}} + \frac{2 a \left (a + b \left (c x\right )^{n}\right )^{\frac{3}{2}}}{3} + \frac{2 \left (a + b \left (c x\right )^{n}\right )^{\frac{5}{2}}}{5}}{n} & \text{otherwise} \end{cases} \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 (\left (c x\right )^{n} b + a\right )}^{\frac{5}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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