Integrand size = 21, antiderivative size = 127 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {734, 738, 212} \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \]
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Rule 212
Rule 734
Rule 738
Rubi steps \begin{align*} \text {integral}& = \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d (c d-b e)} \\ & = \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{4 d (c d-b e)} \\ & = \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{4 d (c d-b e) (d+e x)^2}-\frac {b^2 \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \\ \end{align*}
Time = 10.23 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {\sqrt {d} (2 c d x+b (d-e x))}{(c d-b e) (d+e x)^2}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{(c d-b e)^{3/2} \sqrt {x} \sqrt {b+c x}}\right )}{4 d^{3/2}} \]
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Time = 2.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(-\frac {b^{2} \left (\frac {\sqrt {x \left (c x +b \right )}\, \left (-b e x +2 c d x +b d \right )}{b^{2} \left (e x +d \right )^{2}}+\frac {\arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}\right )}{4 \left (b e -c d \right ) d}\) | \(94\) |
default | \(\text {Expression too large to display}\) | \(986\) |
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Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (109) = 218\).
Time = 0.29 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.77 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\left [-\frac {{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (b c d^{3} - b^{2} d^{2} e + {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}}, -\frac {{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (b c d^{3} - b^{2} d^{2} e + {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{2} d^{6} - 2 \, b c d^{5} e + b^{2} d^{4} e^{2} + {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, b c d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x\right )}}\right ] \]
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\[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{3}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (109) = 218\).
Time = 0.31 (sec) , antiderivative size = 407, normalized size of antiderivative = 3.20 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=-\frac {b^{2} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c d^{2} - b d e\right )} \sqrt {-c d^{2} + b d e}} + \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2} + 2 \, b^{2} c^{\frac {3}{2}} d^{3} - b^{3} \sqrt {c} d^{2} e}{4 \, {\left (c d^{2} e^{2} - b d e^{3}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \]
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Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^3} \,d x \]
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