Integrand size = 18, antiderivative size = 33 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a}{c^2 \sqrt {c x^2}}+\frac {b x \log (x)}{c^2 \sqrt {c x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {b x \log (x)}{c^2 \sqrt {c x^2}}-\frac {a}{c^2 \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {a+b x}{x^2} \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{c^2 \sqrt {c x^2}} \\ & = -\frac {a}{c^2 \sqrt {c x^2}}+\frac {b x \log (x)}{c^2 \sqrt {c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {-a x^4+b x^5 \log (x)}{\left (c x^2\right )^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {x^{4} \left (b \ln \left (x \right ) x -a \right )}{\left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(21\) |
risch | \(-\frac {a}{c^{2} \sqrt {c \,x^{2}}}+\frac {b x \ln \left (x \right )}{c^{2} \sqrt {c \,x^{2}}}\) | \(30\) |
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none
Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c x^{2}} {\left (b x \log \left (x\right ) - a\right )}}{c^{3} x^{2}} \]
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Time = 1.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=- \frac {a x^{4}}{\left (c x^{2}\right )^{\frac {5}{2}}} + \frac {b x^{5} \log {\left (x \right )}}{\left (c x^{2}\right )^{\frac {5}{2}}} \]
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none
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=-\frac {a x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} + \frac {b \log \left (x\right )}{c^{\frac {5}{2}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\frac {b \log \left ({\left | x \right |}\right )}{c^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} - \frac {a}{c^{\frac {5}{2}} x \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx=\int \frac {x^3\,\left (a+b\,x\right )}{{\left (c\,x^2\right )}^{5/2}} \,d x \]
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