Integrand size = 22, antiderivative size = 168 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=-\frac {\sqrt {a+c x^2}}{2 a d x^2}+\frac {e \sqrt {a+c x^2}}{a d^2 x}+\frac {e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}+\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d}-\frac {e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3} \]
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Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {975, 272, 44, 65, 214, 270, 739, 212} \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d}-\frac {e^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3}+\frac {e^3 \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d^3 \sqrt {a e^2+c d^2}}+\frac {e \sqrt {a+c x^2}}{a d^2 x}-\frac {\sqrt {a+c x^2}}{2 a d x^2} \]
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Rule 44
Rule 65
Rule 212
Rule 214
Rule 270
Rule 272
Rule 739
Rule 975
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{d x^3 \sqrt {a+c x^2}}-\frac {e}{d^2 x^2 \sqrt {a+c x^2}}+\frac {e^2}{d^3 x \sqrt {a+c x^2}}-\frac {e^3}{d^3 (d+e x) \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {\int \frac {1}{x^3 \sqrt {a+c x^2}} \, dx}{d}-\frac {e \int \frac {1}{x^2 \sqrt {a+c x^2}} \, dx}{d^2}+\frac {e^2 \int \frac {1}{x \sqrt {a+c x^2}} \, dx}{d^3}-\frac {e^3 \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d^3} \\ & = \frac {e \sqrt {a+c x^2}}{a d^2 x}+\frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d}+\frac {e^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d^3}+\frac {e^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 )}{d^3} \\ & = -\frac {\sqrt {a+c x^2}}{2 a d x^2}+\frac {e \sqrt {a+c x^2}}{a d^2 x}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}-\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{4 a d}+\frac {e^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d^3} \\ & = -\frac {\sqrt {a+c x^2}}{2 a d x^2}+\frac {e \sqrt {a+c x^2}}{a d^2 x}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{2 a d} \\ & = -\frac {\sqrt {a+c x^2}}{2 a d x^2}+\frac {e \sqrt {a+c x^2}}{a d^2 x}+\frac {e^3 \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d^3 \sqrt {c d^2+a e^2}}+\frac {c \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d}-\frac {e^2 \tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d^3} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\frac {\sqrt {a} \left (d \left (c d^2+a e^2\right ) (-d+2 e x) \sqrt {a+c x^2}-4 a e^3 \sqrt {-c d^2-a e^2} x^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )\right )-2 \left (c^2 d^4-a c d^2 e^2-2 a^2 e^4\right ) x^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{2 a^{3/2} d^3 \left (c d^2+a e^2\right ) x^2} \]
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Time = 0.44 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (-2 e x +d \right )}{2 a \,d^{2} x^{2}}+\frac {\frac {2 e^{2} a \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}+\frac {\left (-2 e^{2} a +c \,d^{2}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d \sqrt {a}}}{2 a \,d^{2}}\) | \(209\) |
default | \(\frac {-\frac {\sqrt {c \,x^{2}+a}}{2 a \,x^{2}}+\frac {c \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d}-\frac {e^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d^{3} \sqrt {a}}+\frac {e \sqrt {c \,x^{2}+a}}{a \,d^{2} x}+\frac {e^{2} \ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{3} \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(235\) |
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Time = 0.57 (sec) , antiderivative size = 956, normalized size of antiderivative = 5.69 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {2 \, \sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{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^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {a} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}, \frac {4 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{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^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {a} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{4 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}, \frac {\sqrt {c d^{2} + a e^{2}} a^{2} e^{3} x^{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^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} a^{2} e^{3} x^{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^{2} d^{4} - a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (a c d^{4} + a^{2} d^{2} e^{2} - 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a}}{2 \, {\left (a^{2} c d^{5} + a^{3} d^{3} e^{2}\right )} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} x^{3}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=-c^{\frac {3}{2}} {\left (\frac {2 \, e^{3} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} - a e^{2}}}\right )}{\sqrt {-c d^{2} - a e^{2}} c^{\frac {3}{2}} d^{3}} + \frac {{\left (c d^{2} - 2 \, a e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a c^{\frac {3}{2}} d^{3}} - \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} \sqrt {c} d - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} a e + {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} a \sqrt {c} d + 2 \, a^{2} e}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{2} a c d^{2}}\right )} \]
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Timed out. \[ \int \frac {1}{x^3 (d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{x^3\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
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