Integrand size = 26, antiderivative size = 91 \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c x^2}+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} c^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {457, 98, 95, 214} \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {(a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c x^2} \]
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Rule 95
Rule 98
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c x^2}-\frac {(b c+a d) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,x^2\right )}{4 a c} \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c x^2}-\frac {(b c+a d) \text {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+b x^2}}{\sqrt {c+d x^2}}\right )}{2 a c} \\ & = -\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c x^2}+\frac {(b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} c^{3/2}} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{2 a c x^2}+\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{3/2} c^{3/2}} \]
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Time = 3.16 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.51
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}{2 a c \,x^{2}}+\frac {\left (a d +b c \right ) \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{4 a c \sqrt {a c}\, \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(137\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{2 a c \,x^{2}}+\frac {\left (a d +b c \right ) \ln \left (\frac {2 a c +\left (a d +b c \right ) x^{2}+2 \sqrt {a c}\, \sqrt {b d \,x^{4}+\left (a d +b c \right ) x^{2}+a c}}{x^{2}}\right )}{4 a c \sqrt {a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(143\) |
default | \(\frac {\left (\ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) a d \,x^{2}+\ln \left (\frac {a d \,x^{2}+c b \,x^{2}+2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+2 a c}{x^{2}}\right ) b c \,x^{2}-2 \sqrt {a c}\, \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\right ) \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{4 a c \sqrt {a c}\, x^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}\) | \(181\) |
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Time = 0.32 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.05 \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\left [\frac {\sqrt {a c} {\left (b c + a d\right )} x^{2} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {a c}}{x^{4}}\right ) - 4 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a c}{8 \, a^{2} c^{2} x^{2}}, -\frac {\sqrt {-a c} {\left (b c + a d\right )} x^{2} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-a c}}{2 \, {\left (a b c d x^{4} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}\right ) + 2 \, \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a c}{4 \, a^{2} c^{2} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (71) = 142\).
Time = 0.32 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.54 \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {b d} b^{4} d {\left (\frac {{\left (b c + a d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} a b^{3} c d} - \frac {2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b c - {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b x^{2} + a} \sqrt {b d} - \sqrt {b^{2} c + {\left (b x^{2} + a\right )} b d - a b d}\right )}^{4}\right )} a b^{2} c d}\right )}}{2 \, {\left | b \right |}} \]
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Time = 11.49 (sec) , antiderivative size = 481, normalized size of antiderivative = 5.29 \[ \int \frac {1}{x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\frac {\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )\,\left (\frac {c\,b^2}{8}+\frac {a\,d\,b}{8}\right )}{a^{3/2}\,c^{3/2}\,d\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}-\frac {b^2}{8\,a\,c\,d}+\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2\,\left (\frac {a^2\,d^2}{8}-\frac {3\,a\,b\,c\,d}{8}+\frac {b^2\,c^2}{8}\right )}{a^2\,c^2\,d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}}{\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^3}{{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^3}+\frac {b\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{d\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}-\frac {{\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}^2\,\left (a\,d+b\,c\right )}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )}^2}}+\frac {\ln \left (\frac {\sqrt {b\,x^2+a}-\sqrt {a}}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}+a^{3/2}\,\sqrt {c}\,d\right )}{4\,a^2\,c^2}-\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {b\,x^2+a}-\sqrt {a}\,\sqrt {d\,x^2+c}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )}{\sqrt {d\,x^2+c}-\sqrt {c}}\right )\,\left (\sqrt {a}\,b\,c^{3/2}+a^{3/2}\,\sqrt {c}\,d\right )}{4\,a^2\,c^2}-\frac {d\,\left (\sqrt {b\,x^2+a}-\sqrt {a}\right )}{8\,a\,c\,\left (\sqrt {d\,x^2+c}-\sqrt {c}\right )} \]
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