Integrand size = 17, antiderivative size = 89 \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {b}{a^2 c \sqrt {c x^2}}-\frac {1}{2 a c x \sqrt {c x^2}}+\frac {b^2 x \log (x)}{a^3 c \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 c \sqrt {c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 46} \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {b^2 x \log (x)}{a^3 c \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 c \sqrt {c x^2}}+\frac {b}{a^2 c \sqrt {c x^2}}-\frac {1}{2 a c x \sqrt {c x^2}} \]
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Rule 15
Rule 46
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x^3 (a+b x)} \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}} \\ & = \frac {b}{a^2 c \sqrt {c x^2}}-\frac {1}{2 a c x \sqrt {c x^2}}+\frac {b^2 x \log (x)}{a^3 c \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 c \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=-\frac {a x (a-2 b x)-2 b^2 x^3 \log (x)+2 b^2 x^3 \log (a+b x)}{2 a^3 \left (c x^2\right )^{3/2}} \]
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Time = 0.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.55
method | result | size |
default | \(\frac {x \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}\right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}} a^{3}}\) | \(49\) |
risch | \(\frac {\frac {b x}{a^{2}}-\frac {1}{2 a}}{c x \sqrt {c \,x^{2}}}+\frac {x \,b^{2} \ln \left (-x \right )}{c \sqrt {c \,x^{2}}\, a^{3}}-\frac {b^{2} x \ln \left (b x +a \right )}{a^{3} c \sqrt {c \,x^{2}}}\) | \(75\) |
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\frac {{\left (2 \, b^{2} x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, a^{3} c^{2} x^{3}} \]
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\[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\int \frac {1}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=-\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} b^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{3} c^{\frac {3}{2}}} + \frac {b}{\sqrt {c x^{2}} a^{2} c} - \frac {1}{2 \, a c^{\frac {3}{2}} x^{2}} \]
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Exception generated. \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx=\int \frac {1}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \]
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