Integrand size = 20, antiderivative size = 64 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=-\frac {c \sqrt {c x^2}}{a x^2}-\frac {b c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c \sqrt {c x^2} \log (a+b x)}{a^2 x} \]
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Time = 0.01 (sec) , antiderivative size = 64, 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, 46} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=-\frac {b c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {c \sqrt {c x^2}}{a x^2} \]
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Rule 15
Rule 46
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^2 (a+b x)} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{x} \\ & = -\frac {c \sqrt {c x^2}}{a x^2}-\frac {b c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b c \sqrt {c x^2} \log (a+b x)}{a^2 x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.53 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=-\frac {c^2 (a+b x \log (x)-b x \log (a+b x))}{a^2 \sqrt {c x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b \ln \left (x \right ) x -b \ln \left (b x +a \right ) x +a \right )}{a^{2} x^{4}}\) | \(33\) |
risch | \(-\frac {c \sqrt {c \,x^{2}}}{a \,x^{2}}+\frac {c \sqrt {c \,x^{2}}\, b \ln \left (-b x -a \right )}{x \,a^{2}}-\frac {b c \ln \left (x \right ) \sqrt {c \,x^{2}}}{a^{2} x}\) | \(62\) |
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Time = 0.23 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=\frac {{\left (b c x \log \left (\frac {b x + a}{x}\right ) - a c\right )} \sqrt {c x^{2}}}{a^{2} x^{2}} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{5} \left (a + b x\right )}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.58 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=\frac {b c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{2}} - \frac {b c^{\frac {3}{2}} \log \left (x\right )}{a^{2}} - \frac {c^{\frac {3}{2}}}{a x} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^5\,\left (a+b\,x\right )} \,d x \]
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