Integrand size = 24, antiderivative size = 147 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} b \sqrt {d} (3 b c+4 a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {473, 464, 283, 201, 223, 212} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}+\frac {1}{2} b \sqrt {d} (4 a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )-\frac {b \left (c+d x^2\right )^{3/2} (4 a d+3 b c)}{3 c x}+\frac {b d x \sqrt {c+d x^2} (4 a d+3 b c)}{2 c}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3} \]
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Rule 201
Rule 212
Rule 223
Rule 283
Rule 464
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}+\frac {\int \frac {\left (10 a b c+5 b^2 c x^2\right ) \left (c+d x^2\right )^{3/2}}{x^4} \, dx}{5 c} \\ & = -\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {(b (3 b c+4 a d)) \int \frac {\left (c+d x^2\right )^{3/2}}{x^2} \, dx}{3 c} \\ & = -\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {(b d (3 b c+4 a d)) \int \sqrt {c+d x^2} \, dx}{c} \\ & = \frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} (b d (3 b c+4 a d)) \int \frac {1}{\sqrt {c+d x^2}} \, dx \\ & = \frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} (b d (3 b c+4 a d)) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right ) \\ & = \frac {b d (3 b c+4 a d) x \sqrt {c+d x^2}}{2 c}-\frac {b (3 b c+4 a d) \left (c+d x^2\right )^{3/2}}{3 c x}-\frac {a^2 \left (c+d x^2\right )^{5/2}}{5 c x^5}-\frac {2 a b \left (c+d x^2\right )^{5/2}}{3 c x^3}+\frac {1}{2} b \sqrt {d} (3 b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right ) \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {c+d x^2} \left (15 b^2 c x^4 \left (2 c-d x^2\right )+6 a^2 \left (c+d x^2\right )^2+20 a b c x^2 \left (c+4 d x^2\right )\right )}{30 c x^5}-\frac {1}{2} b \sqrt {d} (3 b c+4 a d) \log \left (-\sqrt {d} x+\sqrt {c+d x^2}\right ) \]
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Time = 2.96 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.82
method | result | size |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-15 b^{2} c d \,x^{6}+6 a^{2} d^{2} x^{4}+80 x^{4} a b c d +30 b^{2} c^{2} x^{4}+12 a^{2} c d \,x^{2}+20 a b \,c^{2} x^{2}+6 a^{2} c^{2}\right )}{30 x^{5} c}+\frac {\left (4 a d +3 b c \right ) b \sqrt {d}\, \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2}\) | \(120\) |
pseudoelliptic | \(-\frac {-10 x^{5} b \left (a d +\frac {3 b c}{4}\right ) d c \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )+\sqrt {d \,x^{2}+c}\, \left (2 x^{2} \left (-\frac {5}{4} b^{2} x^{4}+\frac {20}{3} a b \,x^{2}+a^{2}\right ) c \,d^{\frac {3}{2}}+d^{\frac {5}{2}} a^{2} x^{4}+c^{2} \sqrt {d}\, \left (5 b^{2} x^{4}+\frac {10}{3} a b \,x^{2}+a^{2}\right )\right )}{5 \sqrt {d}\, x^{5} c}\) | \(121\) |
default | \(b^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{c x}+\frac {4 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{c}\right )-\frac {a^{2} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5 c \,x^{5}}+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}+\frac {2 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{c x}+\frac {4 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{c}\right )}{3 c}\right )\) | \(204\) |
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Time = 0.28 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\left [\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (15 \, b^{2} c d x^{6} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \, {\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, c x^{5}}, -\frac {15 \, {\left (3 \, b^{2} c^{2} + 4 \, a b c d\right )} \sqrt {-d} x^{5} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (15 \, b^{2} c d x^{6} - 2 \, {\left (15 \, b^{2} c^{2} + 40 \, a b c d + 3 \, a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 4 \, {\left (5 \, a b c^{2} + 3 \, a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, c x^{5}}\right ] \]
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Time = 3.00 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.36 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=- \frac {a^{2} c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{4}} - \frac {2 a^{2} d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 x^{2}} - \frac {a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{5 c} - \frac {2 a b \sqrt {c} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {2 a b c \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {2 a b d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + 2 a b d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} - \frac {2 a b d^{2} x}{\sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {3}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} \sqrt {c} d x}{\sqrt {1 + \frac {d x^{2}}{c}}} + b^{2} c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} + b^{2} d \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + d x^{2}} + 2 d x \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {d x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {c + d x^{2}}}{2} & \text {for}\: d \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\frac {3}{2} \, \sqrt {d x^{2} + c} b^{2} d x + \frac {2 \, \sqrt {d x^{2} + c} a b d^{2} x}{c} + \frac {3}{2} \, b^{2} c \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + 2 \, a b d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{x} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d}{3 \, c x} - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{3 \, c x^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2}}{5 \, c x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (123) = 246\).
Time = 0.35 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\frac {1}{2} \, \sqrt {d x^{2} + c} b^{2} d x - \frac {1}{4} \, {\left (3 \, b^{2} c \sqrt {d} + 4 \, a b d^{\frac {3}{2}}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right ) + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c^{2} \sqrt {d} + 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b c d^{\frac {3}{2}} + 15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{3} \sqrt {d} - 180 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c^{2} d^{\frac {3}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{4} \sqrt {d} + 220 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} d^{\frac {3}{2}} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{5} \sqrt {d} - 140 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} d^{\frac {3}{2}} + 15 \, b^{2} c^{6} \sqrt {d} + 40 \, a b c^{5} d^{\frac {3}{2}} + 3 \, a^{2} c^{4} d^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{5}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{3/2}}{x^6} \,d x \]
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