Integrand size = 19, antiderivative size = 117 \[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}-\frac {15 c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}} \]
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Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {686, 680, 674, 213} \[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {15 c^2 \text {arctanh}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}} \]
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Rule 213
Rule 674
Rule 680
Rule 686
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}-\frac {(5 c) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{4 b} \\ & = -\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {\left (15 c^2\right ) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{8 b^2} \\ & = -\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {\left (15 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b^3} \\ & = -\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {\left (15 c^2\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b^3} \\ & = -\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}-\frac {15 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.72 \[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {b} \left (-2 b^2+5 b c x+15 c^2 x^2\right )-15 c^2 x^2 \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{4 b^{7/2} x^{3/2} \sqrt {x (b+c x)}} \]
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Time = 2.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65
method | result | size |
default | \(-\frac {\sqrt {x \left (c x +b \right )}\, \left (15 \sqrt {c x +b}\, \operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) c^{2} x^{2}-5 b^{\frac {3}{2}} c x -15 c^{2} x^{2} \sqrt {b}+2 b^{\frac {5}{2}}\right )}{4 x^{\frac {5}{2}} \left (c x +b \right ) b^{\frac {7}{2}}}\) | \(76\) |
risch | \(-\frac {\left (c x +b \right ) \left (-7 c x +2 b \right )}{4 b^{3} x^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}+\frac {c^{2} \left (\frac {16}{\sqrt {c x +b}}-\frac {30 \,\operatorname {arctanh}\left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{8 b^{3} \sqrt {x \left (c x +b \right )}}\) | \(86\) |
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Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.86 \[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\left [\frac {15 \, {\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (15 \, b c^{2} x^{2} + 5 \, b^{2} c x - 2 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}, \frac {15 \, {\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (15 \, b c^{2} x^{2} + 5 \, b^{2} c x - 2 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}\right ] \]
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\[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{\frac {3}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{\frac {3}{2}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\frac {15 \, c^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{4 \, \sqrt {-b} b^{3}} + \frac {2 \, c^{2}}{\sqrt {c x + b} b^{3}} + \frac {7 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{2} - 9 \, \sqrt {c x + b} b c^{2}}{4 \, b^{3} c^{2} x^{2}} \]
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Timed out. \[ \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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