Integrand size = 22, antiderivative size = 152 \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\left (2 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac {d^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \sqrt {c d^2+a e^2}} \]
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Time = 0.18 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1668, 858, 223, 212, 739} \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (2 c d^2-a e^2\right )}{2 c^{3/2} e^3}+\frac {d^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{e^3 \sqrt {a e^2+c d^2}}-\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {\sqrt {a+c x^2} (d+e x)}{2 c e^2} \]
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Rule 212
Rule 223
Rule 739
Rule 858
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\int \frac {-a d e^2-e \left (c d^2+a e^2\right ) x-3 c d e^2 x^2}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c e^3} \\ & = -\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\int \frac {-a c d e^4+c e^3 \left (2 c d^2-a e^2\right ) x}{(d+e x) \sqrt {a+c x^2}} \, dx}{2 c^2 e^5} \\ & = -\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}-\frac {d^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{e^3}+\frac {\left (2 c d^2-a e^2\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c e^3} \\ & = -\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {d^3 \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^3}+\frac {\left (2 c d^2-a e^2\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c e^3} \\ & = -\frac {3 d \sqrt {a+c x^2}}{2 c e^2}+\frac {(d+e x) \sqrt {a+c x^2}}{2 c e^2}+\frac {\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac {d^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{e^3 \sqrt {c d^2+a e^2}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {\frac {e (-2 d+e x) \sqrt {a+c x^2}}{c}+\frac {4 d^3 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {\left (-2 c d^2+a e^2\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{c^{3/2}}}{2 e^3} \]
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Time = 0.42 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32
method | result | size |
risch | \(-\frac {\left (-e x +2 d \right ) \sqrt {c \,x^{2}+a}}{2 c \,e^{2}}-\frac {\frac {\left (e^{2} a -2 c \,d^{2}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e \sqrt {c}}-\frac {2 c \,d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}}{2 e^{2} c}\) | \(201\) |
default | \(\frac {\frac {x \sqrt {c \,x^{2}+a}}{2 c}-\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}}{e}+\frac {d^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{e^{3} \sqrt {c}}-\frac {d \sqrt {c \,x^{2}+a}}{c \,e^{2}}+\frac {d^{3} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{4} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(216\) |
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Time = 2.41 (sec) , antiderivative size = 924, normalized size of antiderivative = 6.08 \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {\sqrt {c d^{2} + a e^{2}} c^{2} d^{3} \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 (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} c^{2} d^{3} \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 (2 \, c^{2} d^{4} + a c d^{2} e^{2} - a^{2} e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, c^{2} d^{3} e + 2 \, a c d e^{3} - {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}\right ] \]
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\[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^{3}}{\sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
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Exception generated. \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {x^3}{\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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