Integrand size = 20, antiderivative size = 88 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=-\frac {c \sqrt {c x^2}}{2 a x^3}+\frac {b c \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x} \]
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Time = 0.02 (sec) , antiderivative size = 88, 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} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=\frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b c \sqrt {c x^2}}{a^2 x^2}-\frac {c \sqrt {c x^2}}{2 a x^3} \]
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Rule 15
Rule 46
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^3 (a+b x)} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \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}{x} \\ & = -\frac {c \sqrt {c x^2}}{2 a x^3}+\frac {b c \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=\frac {\left (c x^2\right )^{3/2} \left (-a (a-2 b x)+2 b^2 x^2 \log (x)-2 b^2 x^2 \log (a+b x)\right )}{2 a^3 x^5} \]
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Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b x -a^{2}\right )}{2 a^{3} x^{5}}\) | \(51\) |
risch | \(\frac {c \sqrt {c \,x^{2}}\, \left (\frac {b x}{a^{2}}-\frac {1}{2 a}\right )}{x^{3}}+\frac {c \sqrt {c \,x^{2}}\, b^{2} \ln \left (-x \right )}{x \,a^{3}}-\frac {b^{2} c \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{a^{3} x}\) | \(73\) |
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Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.53 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=\frac {{\left (2 \, b^{2} c x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b c x - a^{2} c\right )} \sqrt {c x^{2}}}{2 \, a^{3} x^{3}} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{6} \left (a + b x\right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.59 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=-\frac {b^{2} c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} c^{\frac {3}{2}} \log \left (x\right )}{a^{3}} + \frac {2 \, b c^{\frac {3}{2}} x - a c^{\frac {3}{2}}}{2 \, a^{2} x^{2}} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^6 (a+b x)} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^6\,\left (a+b\,x\right )} \,d x \]
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