Integrand size = 20, antiderivative size = 29 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {(a+b x)^3}{3 a c x^2 \sqrt {c x^2}} \]
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Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37} \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {(a+b x)^3}{3 a c x^2 \sqrt {c x^2}} \]
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Rule 15
Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(a+b x)^2}{x^4} \, dx}{c \sqrt {c x^2}} \\ & = -\frac {(a+b x)^3}{3 a c x^2 \sqrt {c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {c x^2 \left (a^2+3 a b x+3 b^2 x^2\right )}{3 \left (c x^2\right )^{5/2}} \]
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Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
gosper | \(-\frac {3 b^{2} x^{2}+3 a b x +a^{2}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(27\) |
default | \(-\frac {3 b^{2} x^{2}+3 a b x +a^{2}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(27\) |
risch | \(\frac {-b^{2} x^{2}-a b x -\frac {1}{3} a^{2}}{c \,x^{2} \sqrt {c \,x^{2}}}\) | \(34\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{2} x^{2}+3 a b \,x^{2}+3 b^{2} x^{2}+a^{2} x +3 a b x +a^{2}\right ) \sqrt {c \,x^{2}}}{3 c^{2} x^{4}}\) | \(55\) |
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Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt {c x^{2}}}{3 \, c^{2} x^{4}} \]
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Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=- \frac {a^{2}}{3 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {a b x}{\left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b^{2} x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}} \]
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Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {b^{2}}{\sqrt {c x^{2}} c} - \frac {a b}{c^{\frac {3}{2}} x^{2}} - \frac {a^{2}}{3 \, c^{\frac {3}{2}} x^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, c^{\frac {3}{2}} x^{3} \mathrm {sgn}\left (x\right )} \]
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Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {a^2\,x^2+3\,a\,b\,x^3+3\,b^2\,x^4}{3\,c^{3/2}\,{\left (x^2\right )}^{5/2}} \]
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