Integrand size = 19, antiderivative size = 83 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}-\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 83, 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 = {676, 674, 213} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=-\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}-\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]
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Rule 213
Rule 674
Rule 676
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac {1}{4} (3 c) \int \frac {\sqrt {b x+c x^2}}{x^{5/2}} \, dx \\ & = -\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac {1}{8} \left (3 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx \\ & = -\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}+\frac {1}{4} \left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right ) \\ & = -\frac {3 c \sqrt {b x+c x^2}}{4 x^{3/2}}-\frac {\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=-\frac {\sqrt {x (b+c x)} \left (\sqrt {b} \sqrt {b+c x} (2 b+5 c x)+3 c^2 x^2 \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{4 \sqrt {b} x^{5/2} \sqrt {b+c x}} \]
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Time = 2.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (5 c x +2 b \right )}{4 x^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}-\frac {3 c^{2} \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{4 \sqrt {b}\, \sqrt {x \left (c x +b \right )}}\) | \(68\) |
default | \(-\frac {\sqrt {x \left (c x +b \right )}\, \left (3 \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{2} x^{2}+5 c x \sqrt {b}\, \sqrt {c x +b}+2 b^{\frac {3}{2}} \sqrt {c x +b}\right )}{4 x^{\frac {5}{2}} \sqrt {c x +b}\, \sqrt {b}}\) | \(72\) |
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Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.84 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=\left [\frac {3 \, \sqrt {b} c^{2} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) - 2 \, {\left (5 \, b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b x^{3}}, \frac {3 \, \sqrt {-b} c^{2} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (5 \, b c x + 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b x^{3}}\right ] \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{\frac {9}{2}}}\, dx \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{x^{\frac {9}{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.77 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=\frac {\frac {3 \, c^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {5 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{3} - 3 \, \sqrt {c x + b} b c^{3}}{c^{2} x^{2}}}{4 \, c} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^{9/2}} \,d x \]
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