Integrand size = 20, antiderivative size = 45 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 45, 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} \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {x}{a+b x} \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{c \sqrt {c x^2}} \\ & = \frac {x^2}{b c \sqrt {c x^2}}-\frac {a x \log (a+b x)}{b^2 c \sqrt {c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^3 (b x-a \log (a+b x))}{b^2 \left (c x^2\right )^{3/2}} \]
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Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64
method | result | size |
default | \(-\frac {x^{3} \left (a \ln \left (b x +a \right )-b x \right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} b^{2}}\) | \(29\) |
risch | \(\frac {x^{2}}{b c \sqrt {c \,x^{2}}}-\frac {a x \ln \left (b x +a \right )}{b^{2} c \sqrt {c \,x^{2}}}\) | \(42\) |
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none
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.67 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c^{2} x} \]
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\[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\int \frac {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (41) = 82\).
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.58 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {x^{2}}{\sqrt {c x^{2}} b c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} c^{\frac {3}{2}}} + \frac {2 \, a x}{\sqrt {c x^{2}} b^{2} c} - \frac {a \log \left (b x\right )}{b^{2} c^{\frac {3}{2}}} - \frac {2 \, a^{2}}{\sqrt {c x^{2}} b^{3} c} + \frac {2 \, a^{2}}{b^{3} c^{\frac {3}{2}} x} \]
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Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {\frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{2} \sqrt {c}} + \frac {x}{b \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a \log \left ({\left | b x + a \right |}\right )}{b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \]
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Timed out. \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\int \frac {x^4}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \]
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