Integrand size = 20, antiderivative size = 92 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=\frac {a^2 c^2 \sqrt {c x^2}}{b^3}-\frac {a c^2 x \sqrt {c x^2}}{2 b^2}+\frac {c^2 x^2 \sqrt {c x^2}}{3 b}-\frac {a^3 c^2 \sqrt {c x^2} \log (a+b x)}{b^4 x} \]
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Time = 0.02 (sec) , antiderivative size = 92, 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} \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=-\frac {a^3 c^2 \sqrt {c x^2} \log (a+b x)}{b^4 x}+\frac {a^2 c^2 \sqrt {c x^2}}{b^3}-\frac {a c^2 x \sqrt {c x^2}}{2 b^2}+\frac {c^2 x^2 \sqrt {c x^2}}{3 b} \]
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Rule 15
Rule 45
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {x^3}{a+b x} \, dx}{x} \\ & = \frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx}{x} \\ & = \frac {a^2 c^2 \sqrt {c x^2}}{b^3}-\frac {a c^2 x \sqrt {c x^2}}{2 b^2}+\frac {c^2 x^2 \sqrt {c x^2}}{3 b}-\frac {a^3 c^2 \sqrt {c x^2} \log (a+b x)}{b^4 x} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.59 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=\frac {c \left (c x^2\right )^{3/2} \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 x^3} \]
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Time = 0.14 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.57
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 b^{4} x^{5}}\) | \(52\) |
risch | \(\frac {c^{2} \sqrt {c \,x^{2}}\, \left (\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x \right )}{x \,b^{3}}-\frac {a^{3} c^{2} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{4} x}\) | \(67\) |
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Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=\frac {{\left (2 \, b^{3} c^{2} x^{3} - 3 \, a b^{2} c^{2} x^{2} + 6 \, a^{2} b c^{2} x - 6 \, a^{3} c^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} x} \]
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\[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{2} \left (a + b x\right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=-\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{3} c^{\frac {5}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{4}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} c^{\frac {5}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4}} - \frac {\sqrt {c x^{2}} a c^{2} x}{2 \, b^{2}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} c}{3 \, b} + \frac {\sqrt {c x^{2}} a^{2} c^{2}}{b^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.91 \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=-\frac {1}{6} \, {\left (\frac {6 \, a^{3} c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4}} - \frac {6 \, a^{3} c^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4}} - \frac {2 \, b^{2} c^{2} x^{3} \mathrm {sgn}\left (x\right ) - 3 \, a b c^{2} x^{2} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} c^{2} x \mathrm {sgn}\left (x\right )}{b^{3}}\right )} \sqrt {c} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{5/2}}{x^2 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{5/2}}{x^2\,\left (a+b\,x\right )} \,d x \]
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