Integrand size = 26, antiderivative size = 141 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {e (B d-A e) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{d^{3/2} (c d-b e)^{3/2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {836, 12, 738, 212} \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {e (B d-A e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{d^{3/2} (c d-b e)^{3/2}}-\frac {2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt {b x+c x^2} (c d-b e)} \]
[In]
[Out]
Rule 12
Rule 212
Rule 738
Rule 836
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \int \frac {b^2 e (B d-A e)}{2 (d+e x) \sqrt {b x+c x^2}} \, dx}{b^2 d (c d-b e)} \\ & = -\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {(e (B d-A e)) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{d (c d-b e)} \\ & = -\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}+\frac {(2 e (B d-A e)) \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 (c d-b e)} \\ & = -\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{b^2 d (c d-b e) \sqrt {b x+c x^2}}-\frac {e (B d-A 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 )}{d^{3/2} (c d-b e)^{3/2}} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (\sqrt {d} \sqrt {-c d+b e} \left (b B c d x+A \left (-b c d+b^2 e-2 c^2 d x+b c e x\right )\right )+b^2 e (B d-A e) \sqrt {x} \sqrt {b+c x} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )\right )}{b^2 d^{3/2} (-c d+b e)^{3/2} \sqrt {x (b+c x)}} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 A \sqrt {x \left (c x +b \right )}}{d x}+\frac {2 \left (A e -B d \right ) b^{2} e \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}\, d \left (b e -c d \right )}+\frac {2 c \left (A c -B b \right ) x}{\left (b e -c d \right ) \sqrt {x \left (c x +b \right )}}}{b^{2}}\) | \(120\) |
risch | \(-\frac {2 A \left (c x +b \right )}{b^{2} d \sqrt {x \left (c x +b \right )}}-\frac {-\frac {b \left (A e -B d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {2 \left (A c -B b \right ) d \sqrt {c \left (x +\frac {b}{c}\right )^{2}-b \left (x +\frac {b}{c}\right )}}{\left (b e -c d \right ) b \left (x +\frac {b}{c}\right )}}{b d}\) | \(238\) |
default | \(-\frac {2 B \left (2 c x +b \right )}{e \,b^{2} \sqrt {c \,x^{2}+b x}}+\frac {\left (A e -B d \right ) \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}\) | \(375\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (127) = 254\).
Time = 0.40 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.85 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=\left [\frac {\sqrt {c d^{2} - b d e} {\left ({\left (B b^{2} c d e - A b^{2} c e^{2}\right )} x^{2} + {\left (B b^{3} d e - A b^{3} e^{2}\right )} x\right )} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} + {\left (A b^{2} c d e^{2} - {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{3} + {\left (B b^{2} c - 3 \, A b c^{2}\right )} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e + b^{4} c d^{2} e^{2}\right )} x^{2} + {\left (b^{3} c^{2} d^{4} - 2 \, b^{4} c d^{3} e + b^{5} d^{2} e^{2}\right )} x}, -\frac {2 \, {\left (\sqrt {-c d^{2} + b d e} {\left ({\left (B b^{2} c d e - A b^{2} c e^{2}\right )} x^{2} + {\left (B b^{3} d e - A b^{3} e^{2}\right )} x\right )} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + {\left (A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} + {\left (A b^{2} c d e^{2} - {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{3} + {\left (B b^{2} c - 3 \, A b c^{2}\right )} d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}\right )}}{{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e + b^{4} c d^{2} e^{2}\right )} x^{2} + {\left (b^{3} c^{2} d^{4} - 2 \, b^{4} c d^{3} e + b^{5} d^{2} e^{2}\right )} x}\right ] \]
[In]
[Out]
\[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (\frac {{\left (B b c d^{2} - 2 \, A c^{2} d^{2} + A b c d e\right )} x}{b^{2} c d^{3} - b^{3} d^{2} e} - \frac {A b c d^{2} - A b^{2} d e}{b^{2} c d^{3} - b^{3} d^{2} e}\right )}}{\sqrt {c x^{2} + b x}} + \frac {2 \, {\left (B d e - A e^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{{\left (c d^{2} - b d e\right )} \sqrt {-c d^{2} + b d e}} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]
[In]
[Out]