Integrand size = 26, antiderivative size = 170 \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=-\frac {3 (b c-a d)}{2 a^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{2 a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {3 \sqrt {c} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{5/2} e^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1981, 1980, 296, 331, 214} \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {3 \sqrt {c} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{5/2} e^{3/2}}-\frac {3 (b c-a d)}{2 a^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{2 a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )} \]
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Rule 214
Rule 296
Rule 331
Rule 1980
Rule 1981
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \left (\frac {e (a+b x)}{c+d x}\right )^{3/2}} \, dx,x,x^2\right ) \\ & = ((b c-a d) e) \text {Subst}\left (\int \frac {1}{x^2 \left (-a e+c x^2\right )^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right ) \\ & = \frac {b c-a d}{2 a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {1}{x^2 \left (-a e+c x^2\right )} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 a} \\ & = -\frac {3 (b c-a d)}{2 a^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{2 a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}-\frac {(3 c (b c-a d)) \text {Subst}\left (\int \frac {1}{-a e+c x^2} \, dx,x,\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}\right )}{2 a^2 e} \\ & = -\frac {3 (b c-a d)}{2 a^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {b c-a d}{2 a \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (a e-\frac {c e \left (a+b x^2\right )}{c+d x^2}\right )}+\frac {3 \sqrt {c} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {a} \sqrt {e}}\right )}{2 a^{5/2} e^{3/2}} \\ \end{align*}
Time = 2.52 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {-\sqrt {a} \sqrt {c+d x^2} \left (3 b c x^2+a \left (c-2 d x^2\right )\right )+3 \sqrt {c} (b c-a d) x^2 \sqrt {a+b x^2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{5/2} e x^2 \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {c+d x^2}} \]
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Time = 0.20 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.42
method | result | size |
risch | \(-\frac {c \left (b \,x^{2}+a \right )}{2 a^{2} x^{2} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}+\frac {\left (-\frac {3 c \left (a d -b c \right ) \ln \left (\frac {2 a c e +\left (e d a +e b c \right ) x^{2}+2 \sqrt {a c e}\, \sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}}{x^{2}}\right )}{2 \sqrt {a c e}}+\frac {\left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (d \,x^{2}+c \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{2 a^{2} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(242\) |
default | \(\frac {\left (2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} d \,x^{6}-3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a^{2} b c d \,x^{4}+3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a \,b^{2} c^{2} x^{4}+4 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b d \,x^{4}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, b^{2} c \,x^{4}-3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a^{3} c d \,x^{2}+3 \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}+2 a c}{x^{2}}\right ) a^{2} b \,c^{2} x^{2}+4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a^{2} d \,x^{2}-4 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {a c}\, a b c \,x^{2}-2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, b \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a^{2} d \,x^{2}+2 \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {a c}\, a b c \,x^{2}-2 \left (b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c \right )^{\frac {3}{2}} \sqrt {a c}\, a \right ) \left (b \,x^{2}+a \right )}{4 \sqrt {a c}\, x^{2} a^{3} \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \left (d \,x^{2}+c \right ) {\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}}}\) | \(641\) |
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Time = 1.29 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.76 \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (b^{2} c - a b d\right )} e x^{4} + {\left (a b c - a^{2} d\right )} e x^{2}\right )} \sqrt {\frac {c}{a e}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (a b c d + a^{2} d^{2}\right )} x^{4} + 2 \, a^{2} c^{2} + {\left (a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {\frac {c}{a e}}}{x^{4}}\right ) + 4 \, {\left ({\left (3 \, b c d - 2 \, a d^{2}\right )} x^{4} + a c^{2} + {\left (3 \, b c^{2} - a c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{8 \, {\left (a^{2} b e^{2} x^{4} + a^{3} e^{2} x^{2}\right )}}, -\frac {3 \, {\left ({\left (b^{2} c - a b d\right )} e x^{4} + {\left (a b c - a^{2} d\right )} e x^{2}\right )} \sqrt {-\frac {c}{a e}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}} \sqrt {-\frac {c}{a e}}}{2 \, {\left (b c x^{2} + a c\right )}}\right ) + 2 \, {\left ({\left (3 \, b c d - 2 \, a d^{2}\right )} x^{4} + a c^{2} + {\left (3 \, b c^{2} - a c d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{4 \, {\left (a^{2} b e^{2} x^{4} + a^{3} e^{2} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{x^3 \left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \]
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