Integrand size = 24, antiderivative size = 223 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=\frac {5}{16} \left (b^2 c^2+4 a d (3 b c+2 a d)\right ) x \sqrt {c+d x^2}+\frac {5 \left (b^2 c^2+4 a d (3 b c+2 a d)\right ) x \left (c+d x^2\right )^{3/2}}{24 c}+\frac {\left (b^2 c^2+4 a d (3 b c+2 a d)\right ) x \left (c+d x^2\right )^{5/2}}{6 c^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {5 c \left (b^2 c^2+4 a d (3 b c+2 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 \sqrt {d}} \]
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Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {473, 464, 201, 223, 212} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac {5 c \left (4 a d (2 a d+3 b c)+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 \sqrt {d}}+\frac {1}{6} x \left (c+d x^2\right )^{5/2} \left (\frac {4 a d (2 a d+3 b c)}{c^2}+b^2\right )+\frac {5 x \left (c+d x^2\right )^{3/2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )}{24 c}+\frac {5}{16} x \sqrt {c+d x^2} \left (4 a d (2 a d+3 b c)+b^2 c^2\right )-\frac {2 a \left (c+d x^2\right )^{7/2} (2 a d+3 b c)}{3 c^2 x} \]
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Rule 201
Rule 212
Rule 223
Rule 464
Rule 473
Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}+\frac {\int \frac {\left (2 a (3 b c+2 a d)+3 b^2 c x^2\right ) \left (c+d x^2\right )^{5/2}}{x^2} \, dx}{3 c} \\ & = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) \int \left (c+d x^2\right )^{5/2} \, dx \\ & = \frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{6} \left (5 c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right )\right ) \int \left (c+d x^2\right )^{3/2} \, dx \\ & = \frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{8} \left (5 \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \int \sqrt {c+d x^2} \, dx \\ & = \frac {5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}+\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{16} \left (5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {c+d x^2}} \, dx \\ & = \frac {5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}+\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {1}{16} \left (5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-d x^2} \, dx,x,\frac {x}{\sqrt {c+d x^2}}\right ) \\ & = \frac {5}{16} \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) x \sqrt {c+d x^2}+\frac {5}{24} c \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{3/2}+\frac {1}{6} \left (b^2+\frac {4 a d (3 b c+2 a d)}{c^2}\right ) x \left (c+d x^2\right )^{5/2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{3 c x^3}-\frac {2 a (3 b c+2 a d) \left (c+d x^2\right )^{7/2}}{3 c^2 x}+\frac {5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{16 \sqrt {d}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {c+d x^2} \left (-8 a^2 \left (2 c^2+14 c d x^2-3 d^2 x^4\right )+12 a b x^2 \left (-8 c^2+9 c d x^2+2 d^2 x^4\right )+b^2 x^4 \left (33 c^2+26 c d x^2+8 d^2 x^4\right )\right )}{48 x^3}+\frac {5 c \left (b^2 c^2+12 a b c d+8 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{8 \sqrt {d}} \]
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Time = 2.98 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {-\frac {15 \left (a^{2} d^{2}+\frac {3}{2} a b c d +\frac {1}{8} b^{2} c^{2}\right ) x^{3} c \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{2}+\sqrt {d \,x^{2}+c}\, \left (7 x^{2} c \left (-\frac {13}{56} b^{2} x^{4}-\frac {27}{28} a b \,x^{2}+a^{2}\right ) d^{\frac {3}{2}}-\frac {3 x^{4} \left (\frac {1}{3} b^{2} x^{4}+a b \,x^{2}+a^{2}\right ) d^{\frac {5}{2}}}{2}+c^{2} \sqrt {d}\, \left (-\frac {33}{16} b^{2} x^{4}+6 a b \,x^{2}+a^{2}\right )\right )}{3 \sqrt {d}\, x^{3}}\) | \(146\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (-8 b^{2} d^{2} x^{8}-24 a b \,d^{2} x^{6}-26 b^{2} c d \,x^{6}-24 a^{2} d^{2} x^{4}-108 x^{4} a b c d -33 b^{2} c^{2} x^{4}+112 a^{2} c d \,x^{2}+96 a b \,c^{2} x^{2}+16 a^{2} c^{2}\right )}{48 x^{3}}+\frac {5 c \left (8 a^{2} d^{2}+12 a b c d +b^{2} c^{2}\right ) \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{16 \sqrt {d}}\) | \(151\) |
default | \(b^{2} \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{3 c \,x^{3}}+\frac {4 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{c x}+\frac {6 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{c}\right )}{3 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{c x}+\frac {6 d \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {d \,x^{2}+c}}{2}+\frac {c \ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{c}\right )\) | \(287\) |
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Time = 0.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=\left [\frac {15 \, {\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {d} x^{3} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (8 \, b^{2} d^{3} x^{8} + 2 \, {\left (13 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{6} - 16 \, a^{2} c^{2} d + 3 \, {\left (11 \, b^{2} c^{2} d + 36 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{4} - 16 \, {\left (6 \, a b c^{2} d + 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, d x^{3}}, -\frac {15 \, {\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (8 \, b^{2} d^{3} x^{8} + 2 \, {\left (13 \, b^{2} c d^{2} + 12 \, a b d^{3}\right )} x^{6} - 16 \, a^{2} c^{2} d + 3 \, {\left (11 \, b^{2} c^{2} d + 36 \, a b c d^{2} + 8 \, a^{2} d^{3}\right )} x^{4} - 16 \, {\left (6 \, a b c^{2} d + 7 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, d x^{3}}\right ] \]
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Time = 3.12 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.36 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=- \frac {2 a^{2} c^{\frac {3}{2}} d}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {2 a^{2} \sqrt {c} d^{2} x}{\sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a^{2} c^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{3 x^{2}} - \frac {a^{2} c d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{3} + 2 a^{2} c d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} + a^{2} d^{2} \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 ) - \frac {2 a b c^{\frac {5}{2}}}{x \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {2 a b c^{\frac {3}{2}} d x}{\sqrt {1 + \frac {d x^{2}}{c}}} + 2 a b c^{2} \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )} + 4 a b c 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 ) + 2 a b d^{2} \left (\begin {cases} - \frac {c^{2} \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 )}{8 d} + \frac {c x \sqrt {c + d x^{2}}}{8 d} + \frac {x^{3} \sqrt {c + d x^{2}}}{4} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) + b^{2} c^{2} \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 ) + 2 b^{2} c d \left (\begin {cases} - \frac {c^{2} \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 )}{8 d} + \frac {c x \sqrt {c + d x^{2}}}{8 d} + \frac {x^{3} \sqrt {c + d x^{2}}}{4} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) + b^{2} d^{2} \left (\begin {cases} \frac {c^{3} \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 )}{16 d^{2}} - \frac {c^{2} x \sqrt {c + d x^{2}}}{16 d^{2}} + \frac {c x^{3} \sqrt {c + d x^{2}}}{24 d} + \frac {x^{5} \sqrt {c + d x^{2}}}{6} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=\frac {1}{6} \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} x + \frac {5}{24} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c x + \frac {5}{16} \, \sqrt {d x^{2} + c} b^{2} c^{2} x + \frac {5}{2} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d x + \frac {15}{4} \, \sqrt {d x^{2} + c} a b c d x + \frac {5}{2} \, \sqrt {d x^{2} + c} a^{2} d^{2} x + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2} x}{3 \, c} + \frac {5 \, b^{2} c^{3} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{16 \, \sqrt {d}} + \frac {15}{4} \, a b c^{2} \sqrt {d} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) + \frac {5}{2} \, a^{2} c d^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right ) - \frac {2 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b}{x} - \frac {4 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d}{3 \, c x} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{3 \, c x^{3}} \]
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Time = 0.42 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=\frac {1}{48} \, {\left (2 \, {\left (4 \, b^{2} d^{2} x^{2} + \frac {13 \, b^{2} c d^{5} + 12 \, a b d^{6}}{d^{4}}\right )} x^{2} + \frac {3 \, {\left (11 \, b^{2} c^{2} d^{4} + 36 \, a b c d^{5} + 8 \, a^{2} d^{6}\right )}}{d^{4}}\right )} \sqrt {d x^{2} + c} x - \frac {5 \, {\left (b^{2} c^{3} + 12 \, a b c^{2} d + 8 \, a^{2} c d^{2}\right )} \log \left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2}\right )}{32 \, \sqrt {d}} + \frac {2 \, {\left (6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{3} \sqrt {d} + 9 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c^{2} d^{\frac {3}{2}} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{4} \sqrt {d} - 12 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{3} d^{\frac {3}{2}} + 6 \, a b c^{5} \sqrt {d} + 7 \, a^{2} c^{4} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3}} \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^4} \, dx=\int \frac {{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2}}{x^4} \,d x \]
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