Integrand size = 18, antiderivative size = 90 \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {1}{a^2 c \sqrt {c x^2}}-\frac {b x}{a^2 c \sqrt {c x^2} (a+b x)}-\frac {2 b x \log (x)}{a^3 c \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 c \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 90, 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, 46} \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {2 b x \log (x)}{a^3 c \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 c \sqrt {c x^2}}-\frac {b x}{a^2 c \sqrt {c x^2} (a+b x)}-\frac {1}{a^2 c \sqrt {c x^2}} \]
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Rule 15
Rule 46
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x^2 (a+b x)^2} \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}} \\ & = -\frac {1}{a^2 c \sqrt {c x^2}}-\frac {b x}{a^2 c \sqrt {c x^2} (a+b x)}-\frac {2 b x \log (x)}{a^3 c \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 c \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.54 \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {x^2 \left (-\frac {a (a+2 b x)}{a+b x}-2 b x \log (x)+2 b x \log (a+b x)\right )}{a^3 \left (c x^2\right )^{3/2}} \]
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Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.82
method | result | size |
default | \(-\frac {x^{2} \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b \ln \left (x \right ) x -2 \ln \left (b x +a \right ) x a b +2 a b x +a^{2}\right )}{\left (c \,x^{2}\right )^{\frac {3}{2}} a^{3} \left (b x +a \right )}\) | \(74\) |
risch | \(\frac {-\frac {2 b x}{a^{2}}-\frac {1}{a}}{c \sqrt {c \,x^{2}}\, \left (b x +a \right )}-\frac {2 b x \ln \left (x \right )}{a^{3} c \sqrt {c \,x^{2}}}+\frac {2 x b \ln \left (-b x -a \right )}{c \sqrt {c \,x^{2}}\, a^{3}}\) | \(78\) |
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Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=-\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b c^{2} x^{3} + a^{4} c^{2} x^{2}} \]
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\[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\int \frac {x}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\frac {1}{\sqrt {c x^{2}} a b c x + \sqrt {c x^{2}} a^{2} c} + \frac {2 \, \left (-1\right )^{\frac {2 \, a c x}{b}} b \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{3} c^{\frac {3}{2}}} - \frac {2}{\sqrt {c x^{2}} a^{2} c} \]
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Exception generated. \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx=\int \frac {x}{{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2} \,d x \]
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