Integrand size = 20, antiderivative size = 91 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=-\frac {c \sqrt {c x^2}}{a^2 x^2}-\frac {b c \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b c \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b c \sqrt {c x^2} \log (a+b x)}{a^3 x} \]
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Time = 0.02 (sec) , antiderivative size = 91, 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} \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=-\frac {2 b c \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b c \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {b c \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {c \sqrt {c x^2}}{a^2 x^2} \]
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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^2 (a+b x)^2} \, dx}{x} \\ & = \frac {\left (c \sqrt {c x^2}\right ) \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}{x} \\ & = -\frac {c \sqrt {c x^2}}{a^2 x^2}-\frac {b c \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b c \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b c \sqrt {c x^2} \log (a+b x)}{a^3 x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.65 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=\left (c x^2\right )^{3/2} \left (\frac {-a-2 b x}{a^2 x^4 (a+b x)}-\frac {2 b \log (x)}{a^3 x^3}+\frac {2 b \log (a+b x)}{a^3 x^3}\right ) \]
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Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (2 b^{2} \ln \left (x \right ) x^{2}-2 b^{2} \ln \left (b x +a \right ) x^{2}+2 a b \ln \left (x \right ) x -2 \ln \left (b x +a \right ) x a b +2 a b x +a^{2}\right )}{x^{4} a^{3} \left (b x +a \right )}\) | \(74\) |
risch | \(\frac {c \sqrt {c \,x^{2}}\, \left (-\frac {2 b x}{a^{2}}-\frac {1}{a}\right )}{x^{2} \left (b x +a \right )}-\frac {2 b c \ln \left (x \right ) \sqrt {c \,x^{2}}}{a^{3} x}+\frac {2 c \sqrt {c \,x^{2}}\, b \ln \left (-b x -a \right )}{x \,a^{3}}\) | \(79\) |
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Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.71 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=-\frac {{\left (2 \, a b c x + a^{2} c - 2 \, {\left (b^{2} c x^{2} + a b c x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b x^{3} + a^{4} x^{2}} \]
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\[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{5} \left (a + b x\right )^{2}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=\frac {2 \, b c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b c^{\frac {3}{2}} \log \left (x\right )}{a^{3}} - \frac {2 \, b c^{\frac {3}{2}} x + a c^{\frac {3}{2}}}{a^{2} b x^{2} + a^{3} x} \]
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Exception generated. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2}}{x^5 (a+b x)^2} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}}{x^5\,{\left (a+b\,x\right )}^2} \,d x \]
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