Integrand size = 21, antiderivative size = 207 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 c^2 d^2-4 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {3 e^2 (2 c d-b e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{2 d^{5/2} (c d-b e)^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 207, 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 = {754, 820, 738, 212} \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {3 e^2 (2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{5/2} (c d-b e)^{5/2}}-\frac {e \sqrt {b x+c x^2} \left (3 b^2 e^2-4 b c d e+4 c^2 d^2\right )}{b^2 d^2 (d+e x) (c d-b e)^2}-\frac {2 (c x (2 c d-b e)+b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (d+e x) (c d-b e)} \]
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Rule 212
Rule 738
Rule 754
Rule 820
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {2 \int \frac {\frac {1}{2} b e (2 c d-3 b e)+c e (2 c d-b e) x}{(d+e x)^2 \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 c^2 d^2-4 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (3 e^2 (2 c d-b e)\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 d^2 (c d-b e)^2} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 c^2 d^2-4 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}-\frac {\left (3 e^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 )}{d^2 (c d-b e)^2} \\ & = -\frac {2 (b (c d-b e)+c (2 c d-b e) x)}{b^2 d (c d-b e) (d+e x) \sqrt {b x+c x^2}}-\frac {e \left (4 c^2 d^2-4 b c d e+3 b^2 e^2\right ) \sqrt {b x+c x^2}}{b^2 d^2 (c d-b e)^2 (d+e x)}+\frac {3 e^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 )}{2 d^{5/2} (c d-b e)^{5/2}} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {x \left (-\frac {\sqrt {d} (b+c x) \left (4 c^3 d^2 x (d+e x)+b^3 e^2 (2 d+3 e x)+2 b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-4 d^2-2 d e x+3 e^2 x^2\right )\right )}{b^2 (c d-b e)^2 (d+e x)}-\frac {3 e^2 (2 c d-b e) \sqrt {x} (b+c x)^{3/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}\right )}{d^{5/2} (x (b+c x))^{3/2}} \]
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Time = 2.49 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-\frac {3 \sqrt {x \left (c x +b \right )}\, b^{2} e^{2} \left (e x +d \right ) \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{2}+\left (c^{2} \left (2 c x +b \right ) d^{3}-2 \left (-c^{2} x^{2}+\frac {1}{2} b c x +b^{2}\right ) c e \,d^{2}+e^{2} b \left (c x +b \right ) \left (-2 c x +b \right ) d +\frac {3 b^{2} e^{3} x \left (c x +b \right )}{2}\right ) \sqrt {d \left (b e -c d \right )}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}\, d^{2} \left (e x +d \right ) \left (b e -c d \right )^{2} b^{2}}\) | \(187\) |
risch | \(-\frac {2 \left (c x +b \right )}{b^{2} d^{2} \sqrt {x \left (c x +b \right )}}-\frac {2 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-b \left (\frac {b}{c}+x \right )}}{b^{2} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d^{2} \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right )}+\frac {3 b \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {3 e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) c}{d \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\) | \(437\) |
default | \(\frac {\frac {e^{2}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {3 e \left (b e -2 c d \right ) \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e \left (b e -2 c d \right ) \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}+\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{2}}\) | \(560\) |
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Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (189) = 378\).
Time = 0.44 (sec) , antiderivative size = 914, normalized size of antiderivative = 4.42 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (2 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} - b^{4} e^{4}\right )} x^{2} + {\left (2 \, b^{3} c d^{2} e^{2} - b^{4} d e^{3}\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 (2 \, b c^{3} d^{5} - 6 \, b^{2} c^{2} d^{4} e + 6 \, b^{3} c d^{3} e^{2} - 2 \, b^{4} d^{2} e^{3} + {\left (4 \, c^{4} d^{4} e - 8 \, b c^{3} d^{3} e^{2} + 7 \, b^{2} c^{2} d^{2} e^{3} - 3 \, b^{3} c d e^{4}\right )} x^{2} + {\left (4 \, c^{4} d^{5} - 6 \, b c^{3} d^{4} e + 5 \, b^{3} c d^{2} e^{3} - 3 \, b^{4} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{2 \, {\left ({\left (b^{2} c^{4} d^{6} e - 3 \, b^{3} c^{3} d^{5} e^{2} + 3 \, b^{4} c^{2} d^{4} e^{3} - b^{5} c d^{3} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{7} - 2 \, b^{3} c^{3} d^{6} e + 2 \, b^{5} c d^{4} e^{3} - b^{6} d^{3} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{7} - 3 \, b^{4} c^{2} d^{6} e + 3 \, b^{5} c d^{5} e^{2} - b^{6} d^{4} e^{3}\right )} x\right )}}, \frac {3 \, {\left ({\left (2 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d^{2} e^{2} + b^{3} c d e^{3} - b^{4} e^{4}\right )} x^{2} + {\left (2 \, b^{3} c d^{2} e^{2} - b^{4} d e^{3}\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 (2 \, b c^{3} d^{5} - 6 \, b^{2} c^{2} d^{4} e + 6 \, b^{3} c d^{3} e^{2} - 2 \, b^{4} d^{2} e^{3} + {\left (4 \, c^{4} d^{4} e - 8 \, b c^{3} d^{3} e^{2} + 7 \, b^{2} c^{2} d^{2} e^{3} - 3 \, b^{3} c d e^{4}\right )} x^{2} + {\left (4 \, c^{4} d^{5} - 6 \, b c^{3} d^{4} e + 5 \, b^{3} c d^{2} e^{3} - 3 \, b^{4} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{{\left (b^{2} c^{4} d^{6} e - 3 \, b^{3} c^{3} d^{5} e^{2} + 3 \, b^{4} c^{2} d^{4} e^{3} - b^{5} c d^{3} e^{4}\right )} x^{3} + {\left (b^{2} c^{4} d^{7} - 2 \, b^{3} c^{3} d^{6} e + 2 \, b^{5} c d^{4} e^{3} - b^{6} d^{3} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{7} - 3 \, b^{4} c^{2} d^{6} e + 3 \, b^{5} c d^{5} e^{2} - b^{6} d^{4} e^{3}\right )} x}\right ] \]
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\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (189) = 378\).
Time = 0.60 (sec) , antiderivative size = 821, normalized size of antiderivative = 3.97 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {{\left (6 \, b^{2} c d e^{5} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) - 3 \, b^{3} e^{6} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c} {\left | e \right |} \right |}\right ) + 8 \, \sqrt {c d^{2} - b d e} c^{\frac {5}{2}} d^{2} e^{2} {\left | e \right |} - 8 \, \sqrt {c d^{2} - b d e} b c^{\frac {3}{2}} d e^{3} {\left | e \right |} + 6 \, \sqrt {c d^{2} - b d e} b^{2} \sqrt {c} e^{4} {\left | e \right |}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c d^{2} - b d e} b^{2} c^{2} d^{4} {\left | e \right |} - 2 \, \sqrt {c d^{2} - b d e} b^{3} c d^{3} e {\left | e \right |} + \sqrt {c d^{2} - b d e} b^{4} d^{2} e^{2} {\left | e \right |}} - \frac {2 \, {\left (\frac {4 \, c^{3} d^{2} e^{7} - 4 \, b c^{2} d e^{8} + 3 \, b^{2} c e^{9}}{b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {\frac {4 \, c^{3} d^{3} e^{8} - 6 \, b c^{2} d^{2} e^{9} + 8 \, b^{2} c d e^{10} - 3 \, b^{3} e^{11}}{b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {b^{2} c d^{2} e^{11} - b^{3} d e^{12}}{{\left (b^{2} c^{2} d^{4} e^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 2 \, b^{3} c d^{3} e^{6} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + b^{4} d^{2} e^{7} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )} e}}{{\left (e x + d\right )} e}\right )}}{\sqrt {c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}}} - \frac {3 \, {\left (2 \, c d e^{5} - b e^{6}\right )} \log \left ({\left | 2 \, c d e - b e^{2} - 2 \, \sqrt {c d^{2} - b d e} {\left (\sqrt {c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c d^{2} e^{2} - b d e^{3}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |} \right |}\right )}{{\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} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}}{2 \, e^{2}} \]
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Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \]
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