Integrand size = 20, antiderivative size = 23 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\frac {c \sqrt {c x^2} \log (a+b x)}{b x} \]
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Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 31} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\frac {c \sqrt {c x^2} \log (a+b x)}{b x} \]
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Rule 15
Rule 31
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{x} \\ & = \frac {c \sqrt {c x^2} \log (a+b x)}{b x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\frac {\left (c x^2\right )^{3/2} \log (a+b x)}{b x^3} \]
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \ln \left (b x +a \right )}{x^{3} b}\) | \(21\) |
risch | \(\frac {c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b x}\) | \(22\) |
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Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\frac {\sqrt {c x^{2}} c \log \left (b x + a\right )}{b x} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{3} \left (a + b x\right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{b} \]
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Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=c^{\frac {3}{2}} {\left (\frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b} - \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b}\right )} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^3\,\left (a+b\,x\right )} \,d x \]
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