Integrand size = 20, antiderivative size = 100 \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=-\frac {b^2}{a^3 \sqrt {c x^2}}-\frac {1}{3 a x^2 \sqrt {c x^2}}+\frac {b}{2 a^2 x \sqrt {c x^2}}-\frac {b^3 x \log (x)}{a^4 \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 100, 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, 46} \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=-\frac {b^3 x \log (x)}{a^4 \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 \sqrt {c x^2}}-\frac {b^2}{a^3 \sqrt {c x^2}}+\frac {b}{2 a^2 x \sqrt {c x^2}}-\frac {1}{3 a x^2 \sqrt {c x^2}} \]
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Rule 15
Rule 46
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x^4 (a+b x)} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx}{\sqrt {c x^2}} \\ & = -\frac {b^2}{a^3 \sqrt {c x^2}}-\frac {1}{3 a x^2 \sqrt {c x^2}}+\frac {b}{2 a^2 x \sqrt {c x^2}}-\frac {b^3 x \log (x)}{a^4 \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 \sqrt {c x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=\frac {c \left (a \left (-2 a^2+3 a b x-6 b^2 x^2\right )-6 b^3 x^3 \log (x)+6 b^3 x^3 \log (a+b x)\right )}{6 a^4 \left (c x^2\right )^{3/2}} \]
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Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.62
method | result | size |
default | \(-\frac {6 b^{3} \ln \left (x \right ) x^{3}-6 b^{3} \ln \left (b x +a \right ) x^{3}+6 a \,b^{2} x^{2}-3 a^{2} b x +2 a^{3}}{6 x^{2} \sqrt {c \,x^{2}}\, a^{4}}\) | \(62\) |
risch | \(\frac {-\frac {1}{3 a}+\frac {b x}{2 a^{2}}-\frac {b^{2} x^{2}}{a^{3}}}{\sqrt {c \,x^{2}}\, x^{2}}-\frac {b^{3} x \ln \left (x \right )}{a^{4} \sqrt {c \,x^{2}}}+\frac {x \,b^{3} \ln \left (-b x -a \right )}{\sqrt {c \,x^{2}}\, a^{4}}\) | \(79\) |
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Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=\frac {{\left (6 \, b^{3} x^{3} \log \left (\frac {b x + a}{x}\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {c x^{2}}}{6 \, a^{4} c x^{4}} \]
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\[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=\int \frac {1}{x^{3} \sqrt {c x^{2}} \left (a + b x\right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=\frac {b^{3} \log \left (b x + a\right )}{a^{4} \sqrt {c}} - \frac {b^{3} \log \left (x\right )}{a^{4} \sqrt {c}} - \frac {6 \, b^{2} \sqrt {c} x^{2} - 3 \, a b \sqrt {c} x + 2 \, a^{2} \sqrt {c}}{6 \, a^{3} c x^{3}} \]
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Exception generated. \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{x^3 \sqrt {c x^2} (a+b x)} \, dx=\int \frac {1}{x^3\,\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \]
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