Integrand size = 19, antiderivative size = 81 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=-\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 \sqrt {a}} \]
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Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2045, 2033, 212} \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 \sqrt {a}}-\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5} \]
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Rule 212
Rule 2033
Rule 2045
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}+\frac {1}{4} (3 b) \int \frac {\sqrt {a x^2+b x^3}}{x^3} \, dx \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}+\frac {1}{8} \left (3 b^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}-\frac {1}{4} \left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right ) \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{4 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{2 x^5}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 \sqrt {a}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {x^2 (a+b x)} \left (\sqrt {a} \sqrt {a+b x} (2 a+5 b x)+3 b^2 x^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{4 \sqrt {a} x^3 \sqrt {a+b x}} \]
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Time = 1.94 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {\left (5 b x +2 a \right ) \sqrt {x^{2} \left (b x +a \right )}}{4 x^{3}}-\frac {3 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{4 \sqrt {a}\, x \sqrt {b x +a}}\) | \(67\) |
default | \(-\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{2} x^{2}+5 \sqrt {a}\, \left (b x +a \right )^{\frac {3}{2}}-3 \sqrt {b x +a}\, a^{\frac {3}{2}}\right )}{4 x^{5} \left (b x +a \right )^{\frac {3}{2}} \sqrt {a}}\) | \(74\) |
pseudoelliptic | \(-\frac {-\frac {15 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) x^{5} b^{5}}{128}+\sqrt {b x +a}\, \left (\frac {15 \sqrt {a}\, b^{4} x^{4}}{128}-\frac {5 a^{\frac {3}{2}} b^{3} x^{3}}{64}+\frac {a^{\frac {5}{2}} b^{2} x^{2}}{16}+\frac {11 a^{\frac {7}{2}} b x}{8}+a^{\frac {9}{2}}\right )}{5 a^{\frac {7}{2}} x^{5}}\) | \(82\) |
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Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} x^{3} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, \sqrt {b x^{3} + a x^{2}} {\left (5 \, a b x + 2 \, a^{2}\right )}}{8 \, a x^{3}}, \frac {3 \, \sqrt {-a} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) - \sqrt {b x^{3} + a x^{2}} {\left (5 \, a b x + 2 \, a^{2}\right )}}{4 \, a x^{3}}\right ] \]
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\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
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\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=\frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} - \frac {5 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {b x + a} a b^{3} \mathrm {sgn}\left (x\right )}{b^{2} x^{2}}}{4 \, b} \]
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Timed out. \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^6} \,d x \]
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