Integrand size = 19, antiderivative size = 76 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=-\frac {2 a \sqrt {-a+b (c x)^n}}{n}+\frac {2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{n} \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {374, 12, 272, 52, 65, 211} \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\frac {2 a^{3/2} \arctan \left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )}{n}-\frac {2 a \sqrt {b (c x)^n-a}}{n}+\frac {2 \left (b (c x)^n-a\right )^{3/2}}{3 n} \]
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Rule 12
Rule 52
Rule 65
Rule 211
Rule 272
Rule 374
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {c \left (-a+b x^n\right )^{3/2}}{x} \, dx,x,c x\right )}{c} \\ & = \text {Subst}\left (\int \frac {\left (-a+b x^n\right )^{3/2}}{x} \, dx,x,c x\right ) \\ & = \frac {\text {Subst}\left (\int \frac {(-a+b x)^{3/2}}{x} \, dx,x,(c x)^n\right )}{n} \\ & = \frac {2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}-\frac {a \text {Subst}\left (\int \frac {\sqrt {-a+b x}}{x} \, dx,x,(c x)^n\right )}{n} \\ & = -\frac {2 a \sqrt {-a+b (c x)^n}}{n}+\frac {2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {-a+b x}} \, dx,x,(c x)^n\right )}{n} \\ & = -\frac {2 a \sqrt {-a+b (c x)^n}}{n}+\frac {2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac {\left (2 a^2\right ) \text {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 \sqrt {-a+b (c x)^n}}{n}+\frac {2 \left (-a+b (c x)^n\right )^{3/2}}{3 n}+\frac {2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{n} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\frac {-2 \left (4 a-b (c x)^n\right ) \sqrt {-a+b (c x)^n}+6 a^{3/2} \arctan \left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{3 n} \]
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Time = 1.41 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (-a +b \left (c x \right )^{n}\right )^{\frac {3}{2}}}{3}-2 a \sqrt {-a +b \left (c x \right )^{n}}+2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(60\) |
default | \(\frac {\frac {2 \left (-a +b \left (c x \right )^{n}\right )^{\frac {3}{2}}}{3}-2 a \sqrt {-a +b \left (c x \right )^{n}}+2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n}\) | \(60\) |
risch | \(\frac {2 \left (-b \,{\mathrm e}^{n \ln \left (c x \right )}+4 a \right ) \left (a -b \,{\mathrm e}^{n \ln \left (c x \right )}\right )}{3 n \sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}+\frac {2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a +b \,{\mathrm e}^{n \ln \left (c x \right )}}}{\sqrt {a}}\right )}{n}\) | \(76\) |
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Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.78 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\left [\frac {3 \, \sqrt {-a} a \log \left (\frac {\left (c x\right )^{n} b + 2 \, \sqrt {\left (c x\right )^{n} b - a} \sqrt {-a} - 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt {\left (c x\right )^{n} b - a} {\left (\left (c x\right )^{n} b - 4 \, a\right )}}{3 \, n}, \frac {2 \, {\left (3 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b - a}}{\sqrt {a}}\right ) + \sqrt {\left (c x\right )^{n} b - a} {\left (\left (c x\right )^{n} b - 4 \, a\right )}\right )}}{3 \, n}\right ] \]
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Time = 11.99 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.24 \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\begin {cases} \frac {\begin {cases} 2 a^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {- a + b \left (c x\right )^{n}}}{\sqrt {a}} \right )} - 2 a \sqrt {- a + b \left (c x\right )^{n}} + \frac {2 \left (- a + b \left (c x\right )^{n}\right )^{\frac {3}{2}}}{3} & \text {for}\: b \neq 0 \\- a \sqrt {- a} \log {\left (\left (c x\right )^{n} \right )} & \text {otherwise} \end {cases}}{n} & \text {for}\: n \neq 0 \\\left (- a \sqrt {- a + b} + b \sqrt {- a + b}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]
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\[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b - a\right )}^{\frac {3}{2}}}{x} \,d x } \]
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\[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\int { \frac {{\left (\left (c x\right )^{n} b - a\right )}^{\frac {3}{2}}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (-a+b (c x)^n\right )^{3/2}}{x} \, dx=\int \frac {{\left (b\,{\left (c\,x\right )}^n-a\right )}^{3/2}}{x} \,d x \]
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