Integrand size = 24, antiderivative size = 110 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}} \]
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Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {491, 597, 12, 385, 211} \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {b^2 \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}}+\frac {\sqrt {c+d x^2} (2 a d+3 b c)}{3 a^2 c^2 x}-\frac {\sqrt {c+d x^2}}{3 a c x^3} \]
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Rule 12
Rule 211
Rule 385
Rule 491
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {\int \frac {-3 b c-2 a d-2 b d x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a c} \\ & = -\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}-\frac {\int -\frac {3 b^2 c^2}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{3 a^2 c^2} \\ & = -\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{a^2} \\ & = -\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{a^2} \\ & = -\frac {\sqrt {c+d x^2}}{3 a c x^3}+\frac {(3 b c+2 a d) \sqrt {c+d x^2}}{3 a^2 c^2 x}+\frac {b^2 \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\sqrt {c+d x^2} \left (-a c+3 b c x^2+2 a d x^2\right )}{3 a^2 c^2 x^3}-\frac {b^2 \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2} \sqrt {b c-a d}} \]
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Time = 3.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 x^{3}}+\frac {b^{2} c^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{a^{2} c^{2}}\) | \(87\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-3 c b \,x^{2}+a c \right )}{3 c^{2} a^{2} x^{3}}+\frac {b^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{2}}\) | \(352\) |
default | \(\frac {-\frac {\sqrt {d \,x^{2}+c}}{3 c \,x^{3}}+\frac {2 d \sqrt {d \,x^{2}+c}}{3 c^{2} x}}{a}+\frac {b \sqrt {d \,x^{2}+c}}{a^{2} c x}-\frac {b^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {b^{2} \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 a^{2} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(378\) |
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Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (92) = 184\).
Time = 0.31 (sec) , antiderivative size = 414, normalized size of antiderivative = 3.76 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [-\frac {3 \, \sqrt {-a b c + a^{2} d} b^{2} c^{2} x^{3} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left ({\left (b c - 2 \, a d\right )} x^{3} - a c x\right )} \sqrt {-a b c + a^{2} d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a^{2} b c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{3}}, \frac {3 \, \sqrt {a b c - a^{2} d} b^{2} c^{2} x^{3} \arctan \left (\frac {\sqrt {a b c - a^{2} d} {\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{2 \, {\left ({\left (a b c d - a^{2} d^{2}\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (a^{2} b c^{2} - a^{3} c d - {\left (3 \, a b^{2} c^{2} - a^{2} b c d - 2 \, a^{3} d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a^{3} b c^{3} - a^{4} c^{2} d\right )} x^{3}}\right ] \]
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\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{2}\right ) \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (92) = 184\).
Time = 1.02 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.77 \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {1}{3} \, d^{\frac {5}{2}} {\left (\frac {3 \, b^{2} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} a^{2} d^{2}} + \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + 2 \, a c d\right )}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2} d^{2}}\right )} \]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^4\,\left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}} \,d x \]
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