Integrand size = 22, antiderivative size = 109 \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a+c x^2}}{c e}-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}-\frac {d^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^2 \sqrt {c d^2+a e^2}} \]
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Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1668, 12, 858, 223, 212, 739} \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {d^2 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^2 \sqrt {a e^2+c d^2}}-\frac {d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}+\frac {\sqrt {a+c x^2}}{c e} \]
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Rule 12
Rule 212
Rule 223
Rule 739
Rule 858
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a+c x^2}}{c e}-\frac {\int \frac {c d e x}{(d+e x) \sqrt {a+c x^2}} \, dx}{c e^2} \\ & = \frac {\sqrt {a+c x^2}}{c e}-\frac {d \int \frac {x}{(d+e x) \sqrt {a+c x^2}} \, dx}{e} \\ & = \frac {\sqrt {a+c x^2}}{c e}-\frac {d \int \frac {1}{\sqrt {a+c x^2}} \, dx}{e^2}+\frac {d^2 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^2} \\ & = \frac {\sqrt {a+c x^2}}{c e}-\frac {d \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{e^2}-\frac {d^2 \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}}{c e}-\frac {d \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{\sqrt {c} e^2}-\frac {d^2 \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 = 0.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.17 \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\frac {e \sqrt {a+c x^2}}{c}-\frac {2 d^2 \arctan \left (\frac {\sqrt {-c d^2-a e^2} x}{\sqrt {a} (d+e x)-d \sqrt {a+c x^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {2 d \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a}-\sqrt {a+c x^2}}\right )}{\sqrt {c}}}{e^2} \]
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Time = 0.42 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.58
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}+a}}{c e}-\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(172\) |
risch | \(\frac {\sqrt {c \,x^{2}+a}}{c e}-\frac {d \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{2} \sqrt {c}}-\frac {d^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(172\) |
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Time = 0.47 (sec) , antiderivative size = 745, normalized size of antiderivative = 6.83 \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {\sqrt {c d^{2} + a e^{2}} c d^{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 ) + {\left (c d^{3} + a d e^{2}\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^{2} d^{2} e^{2} + a c e^{4}\right )}}, -\frac {2 \, \sqrt {-c d^{2} - a e^{2}} c d^{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}\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^{2} d^{2} e^{2} + a c e^{4}\right )}}, \frac {\sqrt {c d^{2} + a e^{2}} c d^{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^{3} + a d e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + 2 \, {\left (c d^{2} e + a e^{3}\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}, -\frac {\sqrt {-c d^{2} - a e^{2}} c d^{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}\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^{2} d^{2} e^{2} + a c e^{4}}\right ] \]
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\[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^{2}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
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Exception generated. \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^2}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^2}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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