Integrand size = 21, antiderivative size = 129 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\frac {\sqrt {b x+c x^2}}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e^2}+\frac {\sqrt {d} \sqrt {c d-b e} \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^2} \]
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Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {748, 857, 634, 212, 738} \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e)}{\sqrt {c} e^2}+\frac {\sqrt {d} \sqrt {c d-b e} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^2}+\frac {\sqrt {b x+c x^2}}{e} \]
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Rule 212
Rule 634
Rule 738
Rule 748
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x+c x^2}}{e}-\frac {\int \frac {b d+(2 c d-b e) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{2 e} \\ & = \frac {\sqrt {b x+c x^2}}{e}+\frac {(d (c d-b e)) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^2}-\frac {(2 c d-b e) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{2 e^2} \\ & = \frac {\sqrt {b x+c x^2}}{e}-\frac {(2 d (c d-b 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 )}{e^2}-\frac {(2 c d-b e) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{e^2} \\ & = \frac {\sqrt {b x+c x^2}}{e}-\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{\sqrt {c} e^2}+\frac {\sqrt {d} \sqrt {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 )}{e^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.29 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.12 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (c \sqrt {d} e \sqrt {x} \sqrt {b+c x}+2 \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+2 \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+2 \sqrt {c} \sqrt {d} (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )\right )}{c \sqrt {d} e^2 \sqrt {x (b+c x)}} \]
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Time = 2.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.77
method | result | size |
pseudoelliptic | \(-\frac {-\frac {2 d \left (b e -c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}-\sqrt {x \left (c x +b \right )}\, e -\frac {\left (b e -2 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{\sqrt {c}}}{e^{2}}\) | \(99\) |
risch | \(\frac {x \left (c x +b \right )}{e \sqrt {x \left (c x +b \right )}}+\frac {\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {2 d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{2 e}\) | \(206\) |
default | \(\frac {\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e}\) | \(284\) |
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Time = 0.33 (sec) , antiderivative size = 488, normalized size of antiderivative = 3.78 \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\left [\frac {2 \, \sqrt {c x^{2} + b x} c e - {\left (2 \, c d - b e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, \sqrt {c d^{2} - b d e} c \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right )}{2 \, c e^{2}}, \frac {2 \, \sqrt {c x^{2} + b x} c e + 4 \, \sqrt {-c d^{2} + b d e} c \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) - {\left (2 \, c d - b e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c e^{2}}, \frac {\sqrt {c x^{2} + b x} c e + {\left (2 \, c d - b e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c d^{2} - b d e} c \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right )}{c e^{2}}, \frac {\sqrt {c x^{2} + b x} c e + 2 \, \sqrt {-c d^{2} + b d e} c \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + {\left (2 \, c d - b e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right )}{c e^{2}}\right ] \]
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\[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {b x+c x^2}}{d+e x} \, dx=\int \frac {\sqrt {c\,x^2+b\,x}}{d+e\,x} \,d x \]
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