Integrand size = 24, antiderivative size = 206 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {483, 597, 12, 385, 211} \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}}-\frac {\sqrt {c+d x^2} (5 b c-2 a d)}{6 a^2 c x^3 (b c-a d)}+\frac {\sqrt {c+d x^2} \left (-4 a^2 d^2-8 a b c d+15 b^2 c^2\right )}{6 a^3 c^2 x (b c-a d)}+\frac {b \sqrt {c+d x^2}}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \]
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Rule 12
Rule 211
Rule 385
Rule 483
Rule 597
Rubi steps \begin{align*} \text {integral}& = \frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int \frac {-5 b c+2 a d-4 b d x^2}{x^4 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\int \frac {-15 b^2 c^2+8 a b c d+4 a^2 d^2-2 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^2 c (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int -\frac {3 b^2 c^2 (5 b c-6 a d)}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{6 a^3 c^2 (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\left (b^2 (5 b c-6 a d)\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx}{2 a^3 (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\left (b^2 (5 b c-6 a d)\right ) \text {Subst}\left (\int \frac {1}{a-(-b c+a d) x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right )}{2 a^3 (b c-a d)} \\ & = -\frac {(5 b c-2 a d) \sqrt {c+d x^2}}{6 a^2 c (b c-a d) x^3}+\frac {\left (15 b^2 c^2-8 a b c d-4 a^2 d^2\right ) \sqrt {c+d x^2}}{6 a^3 c^2 (b c-a d) x}+\frac {b \sqrt {c+d x^2}}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^2 (5 b c-6 a d) \tan ^{-1}\left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 a^{7/2} (b c-a d)^{3/2}} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=-\frac {\sqrt {c+d x^2} \left (15 b^3 c^2 x^4+2 a b^2 c x^2 \left (5 c-4 d x^2\right )+2 a^3 d \left (c-2 d x^2\right )-2 a^2 b \left (c^2+3 c d x^2+2 d^2 x^4\right )\right )}{6 a^3 c^2 (-b c+a d) x^3 \left (a+b x^2\right )}-\frac {b^2 (5 b c-6 a d) \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{2 a^{7/2} (b c-a d)^{3/2}} \]
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Time = 3.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(\frac {-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 x^{3}}-\frac {b^{2} c^{2} \left (\frac {b \sqrt {d \,x^{2}+c}\, x}{b \,x^{2}+a}-\frac {\left (6 a d -5 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}\right )}{2 \left (a d -b c \right )}}{a^{3} c^{2}}\) | \(131\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}-6 c b \,x^{2}+a c \right )}{3 c^{2} a^{3} x^{3}}+\frac {b^{2} \left (-\frac {\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{4 b}-\frac {\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}}{4 b}-\frac {5 \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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 \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 )}{4 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{3}}\) | \(856\) |
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^{2}}+\frac {2 b \sqrt {d \,x^{2}+c}}{a^{3} c x}-\frac {b \left (\frac {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}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{3}}-\frac {b \left (\frac {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}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \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 )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a^{3}}-\frac {5 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 )}{4 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {5 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 )}{4 a^{3} \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\) | \(885\) |
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Time = 0.43 (sec) , antiderivative size = 758, normalized size of antiderivative = 3.68 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \sqrt {-a b c + a^{2} d} \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 (2 \, a^{3} b^{2} c^{3} - 4 \, a^{4} b c^{2} d + 2 \, a^{5} c d^{2} - {\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{24 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{5} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \sqrt {a b c - a^{2} d} \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 (2 \, a^{3} b^{2} c^{3} - 4 \, a^{4} b c^{2} d + 2 \, a^{5} c d^{2} - {\left (15 \, a b^{4} c^{3} - 23 \, a^{2} b^{3} c^{2} d + 4 \, a^{3} b^{2} c d^{2} + 4 \, a^{4} b d^{3}\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2} + 2 \, a^{5} d^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, {\left ({\left (a^{4} b^{3} c^{4} - 2 \, a^{5} b^{2} c^{3} d + a^{6} b c^{2} d^{2}\right )} x^{5} + {\left (a^{5} b^{2} c^{4} - 2 \, a^{6} b c^{3} d + a^{7} c^{2} d^{2}\right )} x^{3}\right )}}\right ] \]
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\[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^{4} \left (a + b x^{2}\right )^{2} \sqrt {c + d x^{2}}}\, dx \]
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\[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{2} \sqrt {d x^{2} + c} x^{4}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (182) = 364\).
Time = 0.99 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.82 \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\frac {1}{6} \, d^{\frac {7}{2}} {\left (\frac {3 \, {\left (5 \, b^{3} c - 6 \, a b^{2} d\right )} \arctan \left (-\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {6 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{3} c - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b^{2} d - b^{3} c^{2}\right )}}{{\left (a^{3} b c d^{3} - a^{4} d^{4}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} - \frac {8 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + 3 \, b c^{2} + a c d\right )}}{{\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{3} d^{3}}\right )} \]
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Timed out. \[ \int \frac {1}{x^4 \left (a+b x^2\right )^2 \sqrt {c+d x^2}} \, dx=\int \frac {1}{x^4\,{\left (b\,x^2+a\right )}^2\,\sqrt {d\,x^2+c}} \,d x \]
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