Integrand size = 20, antiderivative size = 25 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\frac {c^2 \sqrt {c x^2} \log (a+b x)}{b x} \]
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Time = 0.00 (sec) , antiderivative size = 25, 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} \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\frac {c^2 \sqrt {c x^2} \log (a+b x)}{b x} \]
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Rule 15
Rule 31
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{x} \\ & = \frac {c^2 \sqrt {c x^2} \log (a+b x)}{b x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\frac {\left (c x^2\right )^{5/2} \log (a+b x)}{b x^5} \]
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Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \ln \left (b x +a \right )}{x^{5} b}\) | \(21\) |
risch | \(\frac {c^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b x}\) | \(24\) |
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Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\frac {\sqrt {c x^{2}} c^{2} \log \left (b x + a\right )}{b x} \]
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\[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{5} \left (a + b x\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\frac {c^{\frac {5}{2}} \log \left (b x + a\right )}{b} \]
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Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx={\left (\frac {c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b} - \frac {c^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{5/2}}{x^5\,\left (a+b\,x\right )} \,d x \]
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