Integrand size = 20, antiderivative size = 49 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {a c \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Time = 0.01 (sec) , antiderivative size = 49, 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} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {a c \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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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)^2} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx}{x} \\ & = \frac {a c \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {c \sqrt {c x^2} \log (a+b x)}{b^2 x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {c^2 x (a+(a+b x) \log (a+b x))}{b^2 \sqrt {c x^2} (a+b x)} \]
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Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b \ln \left (b x +a \right ) x +a \ln \left (b x +a \right )+a \right )}{x^{3} b^{2} \left (b x +a \right )}\) | \(41\) |
risch | \(\frac {a c \sqrt {c \,x^{2}}}{b^{2} x \left (b x +a \right )}+\frac {c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{2} x}\) | \(46\) |
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Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} {\left (a c + {\left (b c x + a c\right )} \log \left (b x + a\right )\right )}}{b^{3} x^{2} + a b^{2} x} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.63 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\frac {\left (-1\right )^{\frac {2 \, c x}{b}} c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} - \frac {\sqrt {c x^{2}} c}{b^{2} x + a b} \]
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Time = 0.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=-c^{\frac {3}{2}} {\left (\frac {{\left (\log \left ({\left | a \right |}\right ) + 1\right )} \mathrm {sgn}\left (x\right )}{b^{2}} - \frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2}} - \frac {a \mathrm {sgn}\left (x\right )}{{\left (b x + a\right )} b^{2}}\right )} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^2\,{\left (a+b\,x\right )}^2} \,d x \]
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