Integrand size = 21, antiderivative size = 183 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-b e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{16 d^{5/2} (c d-b e)^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {744, 734, 738, 212} \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=-\frac {b^2 (2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{16 d^{5/2} (c d-b e)^{5/2}}+\frac {\sqrt {b x+c x^2} (2 c d-b e) (x (2 c d-b e)+b d)}{8 d^2 (d+e x)^2 (c d-b e)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (d+e x)^3 (c d-b e)} \]
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Rule 212
Rule 734
Rule 738
Rule 744
Rubi steps \begin{align*} \text {integral}& = -\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {(2 c d-b e) \int \frac {\sqrt {b x+c x^2}}{(d+e x)^3} \, dx}{2 d (c d-b e)} \\ & = \frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {\left (b^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{16 d^2 (c d-b e)^2} \\ & = \frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}+\frac {\left (b^2 (2 c d-b e)\right ) \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 )}{8 d^2 (c d-b e)^2} \\ & = \frac {(2 c d-b e) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{8 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e \left (b x+c x^2\right )^{3/2}}{3 d (c d-b e) (d+e x)^3}-\frac {b^2 (2 c d-b e) \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 )}{16 d^{5/2} (c d-b e)^{5/2}} \\ \end{align*}
Time = 10.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\frac {\sqrt {x (b+c x)} \left (\frac {e x^{3/2} (b+c x)}{(d+e x)^3}-\frac {3 (2 c d-b e) \left (\sqrt {d} \sqrt {c d-b e} \sqrt {x} \sqrt {b+c x} (2 c d x+b (d-e x))-b^2 (d+e x)^2 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )}{8 d^{3/2} (c d-b e)^{3/2} \sqrt {b+c x} (d+e x)^2}\right )}{3 d (-c d+b e) \sqrt {x}} \]
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Time = 2.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(-\frac {b^{2} \left (e x +d \right )^{3} \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\left (\left (-4 c^{2} x -2 b c \right ) d^{3}+e \left (-\frac {4}{3} c^{2} x^{2}+\frac {14}{3} b c x +b^{2}\right ) d^{2}-\frac {8 x \left (-\frac {c x}{2}+b \right ) e^{2} b d}{3}-b^{2} e^{3} x^{2}\right ) \sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}}{8 \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{3} \left (b e -c d \right )^{2} d^{2}}\) | \(167\) |
default | \(\text {Expression too large to display}\) | \(1085\) |
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Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (161) = 322\).
Time = 0.57 (sec) , antiderivative size = 969, normalized size of antiderivative = 5.30 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\left [-\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} d^{3} e + {\left (2 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{3} + 3 \, {\left (2 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (2 \, b^{2} c d^{3} e - b^{3} d^{2} e^{2}\right )} x\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 (6 \, b c^{2} d^{5} - 9 \, b^{2} c d^{4} e + 3 \, b^{3} d^{3} e^{2} + {\left (4 \, c^{3} d^{4} e - 8 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - 3 \, b^{3} d e^{4}\right )} x^{2} + 2 \, {\left (6 \, c^{3} d^{5} - 13 \, b c^{2} d^{4} e + 11 \, b^{2} c d^{3} e^{2} - 4 \, b^{3} d^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, {\left (c^{3} d^{9} - 3 \, b c^{2} d^{8} e + 3 \, b^{2} c d^{7} e^{2} - b^{3} d^{6} e^{3} + {\left (c^{3} d^{6} e^{3} - 3 \, b c^{2} d^{5} e^{4} + 3 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, b c^{2} d^{6} e^{3} + 3 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, b c^{2} d^{7} e^{2} + 3 \, b^{2} c d^{6} e^{3} - b^{3} d^{5} e^{4}\right )} x\right )}}, -\frac {3 \, {\left (2 \, b^{2} c d^{4} - b^{3} d^{3} e + {\left (2 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{3} + 3 \, {\left (2 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (2 \, b^{2} c d^{3} e - b^{3} d^{2} e^{2}\right )} x\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 (6 \, b c^{2} d^{5} - 9 \, b^{2} c d^{4} e + 3 \, b^{3} d^{3} e^{2} + {\left (4 \, c^{3} d^{4} e - 8 \, b c^{2} d^{3} e^{2} + 7 \, b^{2} c d^{2} e^{3} - 3 \, b^{3} d e^{4}\right )} x^{2} + 2 \, {\left (6 \, c^{3} d^{5} - 13 \, b c^{2} d^{4} e + 11 \, b^{2} c d^{3} e^{2} - 4 \, b^{3} d^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, {\left (c^{3} d^{9} - 3 \, b c^{2} d^{8} e + 3 \, b^{2} c d^{7} e^{2} - b^{3} d^{6} e^{3} + {\left (c^{3} d^{6} e^{3} - 3 \, b c^{2} d^{5} e^{4} + 3 \, b^{2} c d^{4} e^{5} - b^{3} d^{3} e^{6}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, b c^{2} d^{6} e^{3} + 3 \, b^{2} c d^{5} e^{4} - b^{3} d^{4} e^{5}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, b c^{2} d^{7} e^{2} + 3 \, b^{2} c d^{6} e^{3} - b^{3} d^{5} e^{4}\right )} x\right )}}\right ] \]
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\[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{\left (d + e x\right )^{4}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (161) = 322\).
Time = 0.30 (sec) , antiderivative size = 831, normalized size of antiderivative = 4.54 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\frac {{\left (2 \, b^{2} c d - b^{3} e\right )} \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 )}{8 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} + \frac {6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{2} c d e^{4} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b^{3} e^{5} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{\frac {7}{2}} d^{4} e - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b c^{\frac {5}{2}} d^{3} e^{2} + 78 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{2} c^{\frac {3}{2}} d^{2} e^{3} - 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{3} \sqrt {c} d e^{4} + 32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{4} d^{5} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{3} d^{4} e - 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} c^{2} d^{3} e^{2} + 74 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{3} c d^{2} e^{3} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{4} d e^{4} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {7}{2}} d^{5} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c^{\frac {5}{2}} d^{4} e - 6 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{3} c^{\frac {3}{2}} d^{3} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{4} \sqrt {c} d^{2} e^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c^{3} d^{5} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} c^{2} d^{4} e + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{4} c d^{3} e^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{5} d^{2} e^{3} + 4 \, b^{3} c^{\frac {5}{2}} d^{5} - 4 \, b^{4} c^{\frac {3}{2}} d^{4} e + 3 \, b^{5} \sqrt {c} d^{3} e^{2}}{24 \, {\left (c^{2} d^{4} e^{2} - 2 \, b c d^{3} e^{3} + b^{2} d^{2} e^{4}\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 )}^{3}} \]
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Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^4} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{{\left (d+e\,x\right )}^4} \,d x \]
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