Integrand size = 20, antiderivative size = 40 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=\frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=\frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {x}{a+b x} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{x} \\ & = \frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=c \sqrt {c x^2} \left (\frac {1}{b}-\frac {a \log (a+b x)}{b^2 x}\right ) \]
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Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (a \ln \left (b x +a \right )-b x \right )}{b^{2} x^{3}}\) | \(29\) |
risch | \(\frac {c \sqrt {c \,x^{2}}}{b}-\frac {a c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{2} x}\) | \(37\) |
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Time = 0.22 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=\frac {{\left (b c x - a c \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{2} x} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (36) = 72\).
Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.88 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=-\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} + \frac {\sqrt {c x^{2}} c}{b} \]
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Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=c^{\frac {3}{2}} {\left (\frac {x \mathrm {sgn}\left (x\right )}{b} - \frac {a \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2}} + \frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2}}\right )} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^2\,\left (a+b\,x\right )} \,d x \]
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