Integrand size = 19, antiderivative size = 110 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=-\frac {\sqrt {a+c x^2}}{e (d+e x)}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^2}+\frac {c d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \sqrt {c d^2+a e^2}} \]
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Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {747, 858, 223, 212, 739} \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=\frac {c d \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \sqrt {a e^2+c d^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^2}-\frac {\sqrt {a+c x^2}}{e (d+e x)} \]
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Rule 212
Rule 223
Rule 739
Rule 747
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+c x^2}}{e (d+e x)}+\frac {c \int \frac {x}{(d+e x) \sqrt {a+c x^2}} \, dx}{e} \\ & = -\frac {\sqrt {a+c x^2}}{e (d+e x)}+\frac {c \int \frac {1}{\sqrt {a+c x^2}} \, dx}{e^2}-\frac {(c d) \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^2} \\ & = -\frac {\sqrt {a+c x^2}}{e (d+e x)}+\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{e^2}+\frac {(c d) \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{e^2} \\ & = -\frac {\sqrt {a+c x^2}}{e (d+e x)}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{e^2}+\frac {c d \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \sqrt {c d^2+a e^2}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=-\frac {\frac {e \sqrt {a+c x^2}}{d+e x}-\frac {2 c d \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\sqrt {c} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{e^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(535\) vs. \(2(96)=192\).
Time = 2.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.87
method | result | size |
default | \(\frac {-\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{\left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}-\frac {c d e \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}-\frac {\sqrt {c}\, d \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{e}-\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\right )}{e^{2} a +c \,d^{2}}+\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )-\frac {2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a +c \,d^{2}\right )}{e^{2}}-\frac {4 c^{2} d^{2}}{e^{2}}\right ) \ln \left (\frac {-\frac {c d}{e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{e^{2} a +c \,d^{2}}}{e^{2}}\) | \(536\) |
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Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (97) = 194\).
Time = 0.58 (sec) , antiderivative size = 884, normalized size of antiderivative = 8.04 \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=\left [\frac {{\left (c d^{3} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (c d e x + c d^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c d^{3} e^{2} + a d e^{4} + {\left (c d^{2} e^{3} + a e^{5}\right )} x\right )}}, \frac {2 \, {\left (c d e x + c d^{2}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (c d^{3} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c d^{3} e^{2} + a d e^{4} + {\left (c d^{2} e^{3} + a e^{5}\right )} x\right )}}, -\frac {2 \, {\left (c d^{3} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (c d e x + c d^{2}\right )} \sqrt {c d^{2} + a e^{2}} \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c d^{3} e^{2} + a d e^{4} + {\left (c d^{2} e^{3} + a e^{5}\right )} x\right )}}, \frac {{\left (c d e x + c d^{2}\right )} \sqrt {-c d^{2} - a e^{2}} \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) - {\left (c d^{3} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{c d^{3} e^{2} + a d e^{4} + {\left (c d^{2} e^{3} + a e^{5}\right )} x}\right ] \]
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\[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {a + c x^{2}}}{\left (d + e x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {a+c x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {c\,x^2+a}}{{\left (d+e\,x\right )}^2} \,d x \]
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