Integrand size = 24, antiderivative size = 162 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=\frac {1}{2} a c (4 b c+5 a d) \sqrt {c+d x^2}+\frac {1}{6} a (4 b c+5 a d) \left (c+d x^2\right )^{3/2}+\frac {a (4 b c+5 a d) \left (c+d x^2\right )^{5/2}}{10 c}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}-\frac {1}{2} a c^{3/2} (4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {457, 91, 81, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}-\frac {1}{2} a c^{3/2} (5 a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )+\frac {a \left (c+d x^2\right )^{5/2} (5 a d+4 b c)}{10 c}+\frac {1}{6} a \left (c+d x^2\right )^{3/2} (5 a d+4 b c)+\frac {1}{2} a c \sqrt {c+d x^2} (5 a d+4 b c)+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d} \]
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Rule 52
Rule 65
Rule 81
Rule 91
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^{5/2}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} a (4 b c+5 a d)+b^2 c x\right ) (c+d x)^{5/2}}{x} \, dx,x,x^2\right )}{2 c} \\ & = \frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}+\frac {(a (4 b c+5 a d)) \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{x} \, dx,x,x^2\right )}{4 c} \\ & = \frac {a (4 b c+5 a d) \left (c+d x^2\right )^{5/2}}{10 c}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}+\frac {1}{4} (a (4 b c+5 a d)) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{6} a (4 b c+5 a d) \left (c+d x^2\right )^{3/2}+\frac {a (4 b c+5 a d) \left (c+d x^2\right )^{5/2}}{10 c}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}+\frac {1}{4} (a c (4 b c+5 a d)) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} a c (4 b c+5 a d) \sqrt {c+d x^2}+\frac {1}{6} a (4 b c+5 a d) \left (c+d x^2\right )^{3/2}+\frac {a (4 b c+5 a d) \left (c+d x^2\right )^{5/2}}{10 c}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}+\frac {1}{4} \left (a c^2 (4 b c+5 a d)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} a c (4 b c+5 a d) \sqrt {c+d x^2}+\frac {1}{6} a (4 b c+5 a d) \left (c+d x^2\right )^{3/2}+\frac {a (4 b c+5 a d) \left (c+d x^2\right )^{5/2}}{10 c}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}+\frac {\left (a c^2 (4 b c+5 a d)\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 d} \\ & = \frac {1}{2} a c (4 b c+5 a d) \sqrt {c+d x^2}+\frac {1}{6} a (4 b c+5 a d) \left (c+d x^2\right )^{3/2}+\frac {a (4 b c+5 a d) \left (c+d x^2\right )^{5/2}}{10 c}+\frac {b^2 \left (c+d x^2\right )^{7/2}}{7 d}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}-\frac {1}{2} a c^{3/2} (4 b c+5 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=\frac {\sqrt {c+d x^2} \left (30 b^2 x^2 \left (c+d x^2\right )^3+35 a^2 d \left (-3 c^2+14 c d x^2+2 d^2 x^4\right )+28 a b d x^2 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )}{210 d x^2}-\frac {1}{2} a c^{3/2} (4 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right ) \]
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Time = 3.00 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {-\frac {15 x^{2} \left (a d +\frac {4 b c}{5}\right ) d \,c^{2} a \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{2}+\left (7 \left (\frac {9}{49} b^{2} x^{4}+\frac {22}{35} a b \,x^{2}+a^{2}\right ) x^{2} d^{2} c^{\frac {3}{2}}-\frac {3 d \left (-\frac {6}{7} b^{2} x^{4}-\frac {92}{15} a b \,x^{2}+a^{2}\right ) c^{\frac {5}{2}}}{2}+x^{2} \left (\frac {3 b^{2} c^{\frac {7}{2}}}{7}+d^{3} x^{2} \sqrt {c}\, \left (\frac {3}{7} b^{2} x^{4}+\frac {6}{5} a b \,x^{2}+a^{2}\right )\right )\right ) \sqrt {d \,x^{2}+c}}{3 \sqrt {c}\, d \,x^{2}}\) | \(153\) |
default | \(\frac {b^{2} \left (d \,x^{2}+c \right )^{\frac {7}{2}}}{7 d}+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}+\frac {5 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )}{2 c}\right )+2 a b \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )\) | \(183\) |
risch | \(-\frac {c^{2} a^{2} \sqrt {d \,x^{2}+c}}{2 x^{2}}+\frac {b^{2} d^{2} x^{6} \sqrt {d \,x^{2}+c}}{7}+\frac {3 b^{2} d c \,x^{4} \sqrt {d \,x^{2}+c}}{7}+\frac {3 b^{2} c^{2} x^{2} \sqrt {d \,x^{2}+c}}{7}+\frac {b^{2} c^{3} \sqrt {d \,x^{2}+c}}{7 d}+\frac {2 x^{4} d^{2} \sqrt {d \,x^{2}+c}\, a b}{5}+\frac {22 c d \,x^{2} \sqrt {d \,x^{2}+c}\, a b}{15}+\frac {46 c^{2} \sqrt {d \,x^{2}+c}\, a b}{15}-\frac {5 c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a^{2} d}{2}-2 c^{\frac {5}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right ) a b +\frac {x^{2} d^{2} \sqrt {d \,x^{2}+c}\, a^{2}}{3}+\frac {7 c d \sqrt {d \,x^{2}+c}\, a^{2}}{3}\) | \(252\) |
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Time = 0.27 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.15 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=\left [\frac {105 \, {\left (4 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (30 \, b^{2} d^{3} x^{8} + 6 \, {\left (15 \, b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{6} - 105 \, a^{2} c^{2} d + 2 \, {\left (45 \, b^{2} c^{2} d + 154 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (15 \, b^{2} c^{3} + 322 \, a b c^{2} d + 245 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{420 \, d x^{2}}, \frac {105 \, {\left (4 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (30 \, b^{2} d^{3} x^{8} + 6 \, {\left (15 \, b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{6} - 105 \, a^{2} c^{2} d + 2 \, {\left (45 \, b^{2} c^{2} d + 154 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{4} + 2 \, {\left (15 \, b^{2} c^{3} + 322 \, a b c^{2} d + 245 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{210 \, d x^{2}}\right ] \]
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Time = 24.12 (sec) , antiderivative size = 570, normalized size of antiderivative = 3.52 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=- \frac {5 a^{2} c^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2} - \frac {a^{2} c^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 x} + \frac {2 a^{2} c^{2} \sqrt {d}}{x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {2 a^{2} c d^{\frac {3}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} + a^{2} d^{2} \left (\begin {cases} \frac {c \sqrt {c + d x^{2}}}{3 d} + \frac {x^{2} \sqrt {c + d x^{2}}}{3} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 2 a b c^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )} + \frac {2 a b c^{3}}{\sqrt {d} x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {2 a b c^{2} \sqrt {d} x}{\sqrt {\frac {c}{d x^{2}} + 1}} + 4 a b c d \left (\begin {cases} \frac {c \sqrt {c + d x^{2}}}{3 d} + \frac {x^{2} \sqrt {c + d x^{2}}}{3} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 a b d^{2} \left (\begin {cases} - \frac {2 c^{2} \sqrt {c + d x^{2}}}{15 d^{2}} + \frac {c x^{2} \sqrt {c + d x^{2}}}{15 d} + \frac {x^{4} \sqrt {c + d x^{2}}}{5} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + b^{2} c^{2} \left (\begin {cases} \frac {c \sqrt {c + d x^{2}}}{3 d} + \frac {x^{2} \sqrt {c + d x^{2}}}{3} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 2 b^{2} c d \left (\begin {cases} - \frac {2 c^{2} \sqrt {c + d x^{2}}}{15 d^{2}} + \frac {c x^{2} \sqrt {c + d x^{2}}}{15 d} + \frac {x^{4} \sqrt {c + d x^{2}}}{5} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + b^{2} d^{2} \left (\begin {cases} \frac {8 c^{3} \sqrt {c + d x^{2}}}{105 d^{3}} - \frac {4 c^{2} x^{2} \sqrt {c + d x^{2}}}{105 d^{2}} + \frac {c x^{4} \sqrt {c + d x^{2}}}{35 d} + \frac {x^{6} \sqrt {c + d x^{2}}}{7} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=-2 \, a b c^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {5}{2} \, a^{2} c^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) + \frac {2}{5} \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b + \frac {2}{3} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c + 2 \, \sqrt {d x^{2} + c} a b c^{2} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2}}{7 \, d} + \frac {5}{6} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d}{2 \, c} + \frac {5}{2} \, \sqrt {d x^{2} + c} a^{2} c d - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{2 \, c x^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=\frac {30 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} + 84 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d + 140 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c d + 420 \, \sqrt {d x^{2} + c} a b c^{2} d + 70 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{2} + 420 \, \sqrt {d x^{2} + c} a^{2} c d^{2} - \frac {105 \, \sqrt {d x^{2} + c} a^{2} c^{2} d}{x^{2}} + \frac {105 \, {\left (4 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}}}{210 \, d} \]
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Time = 5.83 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^3} \, dx=\sqrt {d\,x^2+c}\,\left (c^2\,\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d}-\frac {2\,b^2\,c}{d}\right )-2\,c\,\left (2\,c\,\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d}-\frac {2\,b^2\,c}{d}\right )-\frac {{\left (a\,d-b\,c\right )}^2}{d}+\frac {b^2\,c^2}{d}\right )\right )-\left (\frac {2\,b^2\,c-2\,a\,b\,d}{5\,d}-\frac {2\,b^2\,c}{5\,d}\right )\,{\left (d\,x^2+c\right )}^{5/2}-{\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {2\,c\,\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d}-\frac {2\,b^2\,c}{d}\right )}{3}-\frac {{\left (a\,d-b\,c\right )}^2}{3\,d}+\frac {b^2\,c^2}{3\,d}\right )+\frac {b^2\,{\left (d\,x^2+c\right )}^{7/2}}{7\,d}-\frac {a^2\,c^2\,\sqrt {d\,x^2+c}}{2\,x^2}+\frac {a\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,\left (5\,a\,d+4\,b\,c\right )\,1{}\mathrm {i}}{2} \]
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