Integrand size = 20, antiderivative size = 44 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\frac {c \sqrt {c x^2} \log (x)}{a x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a x} \]
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Time = 0.00 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\frac {c \sqrt {c x^2} \log (x)}{a x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a x} \]
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x (a+b x)} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x} \, dx}{a x}-\frac {\left (b c \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{a x} \\ & = \frac {c \sqrt {c x^2} \log (x)}{a x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a x} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\frac {\left (c x^2\right )^{3/2} (\log (x)-\log (a (a+b x)))}{a x^3} \]
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Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (\ln \left (x \right )-\ln \left (b x +a \right )\right )}{a \,x^{3}}\) | \(26\) |
risch | \(\frac {c \sqrt {c \,x^{2}}\, \ln \left (-x \right )}{x a}-\frac {c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a x}\) | \(43\) |
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Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.50 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\left [\frac {\sqrt {c x^{2}} c \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} c \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\right ] \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{4} \left (a + b x\right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.55 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=-\frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{a} + \frac {c^{\frac {3}{2}} \log \left (x\right )}{a} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^4\,\left (a+b\,x\right )} \,d x \]
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