Optimal. Leaf size=101 \[ \frac{2 a^2 \sqrt{b (c x)^n-a}}{n}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}-\frac{2 a \left (b (c x)^n-a\right )^{3/2}}{3 n}+\frac{2 \left (b (c x)^n-a\right )^{5/2}}{5 n} \]
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Rubi [A] time = 0.0730843, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {367, 12, 266, 50, 63, 205} \[ \frac{2 a^2 \sqrt{b (c x)^n-a}}{n}-\frac{2 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{n}-\frac{2 a \left (b (c x)^n-a\right )^{3/2}}{3 n}+\frac{2 \left (b (c x)^n-a\right )^{5/2}}{5 n} \]
Antiderivative was successfully verified.
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Rule 367
Rule 12
Rule 266
Rule 50
Rule 63
Rule 205
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} \tan ^{-1}\left (\frac{\sqrt{-a+b (c x)^n}}{\sqrt{a}}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0802183, size = 81, normalized size = 0.8 \[ \frac{2 \sqrt{b (c x)^n-a} \left (23 a^2-11 a b (c x)^n+3 b^2 (c x)^{2 n}\right )-30 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b (c x)^n-a}}{\sqrt{a}}\right )}{15 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 86, normalized size = 0.9 \begin{align*} -{\frac{2\,a}{3\,n} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{3}{2}}}}+{\frac{2}{5\,n} \left ( -a+b \left ( cx \right ) ^{n} \right ) ^{{\frac{5}{2}}}}-2\,{\frac{{a}^{5/2}}{n}\arctan \left ({\frac{\sqrt{-a+b \left ( cx \right ) ^{n}}}{\sqrt{a}}} \right ) }+2\,{\frac{{a}^{2}\sqrt{-a+b \left ( cx \right ) ^{n}}}{n}} \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 = 1.98703, size = 392, normalized size = 3.88 \begin{align*} \left [\frac{15 \, \sqrt{-a} a^{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 \, a^{\frac{5}{2}} \arctan \left (\frac{\sqrt{\left (c x\right )^{n} b - a}}{\sqrt{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 = 69.661, size = 117, normalized size = 1.16 \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{- 2 a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{- a + b \left (c x\right )^{n}}}{\sqrt{a}} \right )} + 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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