Integrand size = 20, antiderivative size = 49 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]
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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} \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {x}{(a+b x)^2} \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx}{c \sqrt {c x^2}} \\ & = \frac {a x}{b^2 c \sqrt {c x^2} (a+b x)}+\frac {x \log (a+b x)}{b^2 c \sqrt {c x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {\frac {a x^3}{b^2 (a+b x)}+\frac {x^3 \log (a+b x)}{b^2}}{\left (c x^2\right )^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84
method | result | size |
default | \(\frac {x^{3} \left (b \ln \left (b x +a \right ) x +a \ln \left (b x +a \right )+a \right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} b^{2} \left (b x +a \right )}\) | \(41\) |
risch | \(\frac {a x}{b^{2} c \left (b x +a \right ) \sqrt {c \,x^{2}}}+\frac {x \ln \left (b x +a \right )}{b^{2} c \sqrt {c \,x^{2}}}\) | \(46\) |
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none
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c^{2} x^{2} + a b^{2} c^{2} x} \]
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\[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\int \frac {x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (45) = 90\).
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.20 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {a^{2}}{\sqrt {c x^{2}} b^{4} c x + \sqrt {c x^{2}} a b^{3} c} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2} c^{\frac {3}{2}}} + \frac {\log \left (b x\right )}{b^{2} c^{\frac {3}{2}}} + \frac {3 \, a}{\sqrt {c x^{2}} b^{3} c} - \frac {2 \, a}{b^{3} c^{\frac {3}{2}} x} \]
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none
Time = 0.32 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.20 \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {\frac {{\left (\log \left ({\left | a \right |}\right ) + 1\right )} \mathrm {sgn}\left (x\right )}{b^{2} \sqrt {c}} - \frac {\log \left ({\left | b x + a \right |}\right )}{b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )} - \frac {a}{{\left (b x + a\right )} b^{2} \sqrt {c} \mathrm {sgn}\left (x\right )}}{c} \]
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Timed out. \[ \int \frac {x^4}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\int \frac {x^4}{{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2} \,d x \]
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