Integrand size = 24, antiderivative size = 222 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=\frac {5 d \left (8 b^2 c^2+a d (12 b c+a d)\right ) \sqrt {c+d x^2}}{16 c}+\frac {5 d \left (8 b^2 c^2+a d (12 b c+a d)\right ) \left (c+d x^2\right )^{3/2}}{48 c^2}-\frac {\left (8 b^2 c^2+a d (12 b c+a d)\right ) \left (c+d x^2\right )^{5/2}}{16 c^2 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}-\frac {5 d \left (8 b^2 c^2+a d (12 b c+a d)\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}} \]
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Time = 0.18 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {457, 91, 79, 43, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {5 d \left (a d (a d+12 b c)+8 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}}-\frac {\left (c+d x^2\right )^{5/2} \left (\frac {a d (a d+12 b c)}{c^2}+8 b^2\right )}{16 x^2}+\frac {5 d \left (c+d x^2\right )^{3/2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{48 c^2}+\frac {5 d \sqrt {c+d x^2} \left (a d (a d+12 b c)+8 b^2 c^2\right )}{16 c}-\frac {a \left (c+d x^2\right )^{7/2} (a d+12 b c)}{24 c^2 x^4} \]
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Rule 43
Rule 52
Rule 65
Rule 79
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^4} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} a (12 b c+a d)+3 b^2 c x\right ) (c+d x)^{5/2}}{x^3} \, dx,x,x^2\right )}{6 c} \\ & = -\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{16} \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \text {Subst}\left (\int \frac {(c+d x)^{5/2}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{32} \left (5 d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{x} \, dx,x,x^2\right ) \\ & = \frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{32} \left (5 c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right )\right ) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right ) \\ & = \frac {5}{16} c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \sqrt {c+d x^2}+\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{32} \left (5 d \left (8 b^2 c^2+12 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {5}{16} c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \sqrt {c+d x^2}+\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}+\frac {1}{16} \left (5 \left (8 b^2 c^2+12 a b c d+a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right ) \\ & = \frac {5}{16} c d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \sqrt {c+d x^2}+\frac {5}{48} d \left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{3/2}-\frac {\left (8 b^2+\frac {a d (12 b c+a d)}{c^2}\right ) \left (c+d x^2\right )^{5/2}}{16 x^2}-\frac {a^2 \left (c+d x^2\right )^{7/2}}{6 c x^6}-\frac {a (12 b c+a d) \left (c+d x^2\right )^{7/2}}{24 c^2 x^4}-\frac {5 d \left (8 b^2 c^2+12 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.68 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=-\frac {\sqrt {c+d x^2} \left (12 a b x^2 \left (2 c^2+9 c d x^2-8 d^2 x^4\right )-8 b^2 x^4 \left (-3 c^2+14 c d x^2+2 d^2 x^4\right )+a^2 \left (8 c^2+26 c d x^2+33 d^2 x^4\right )\right )}{48 x^6}-\frac {5 d \left (8 b^2 c^2+12 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{16 \sqrt {c}} \]
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Time = 2.94 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.65
method | result | size |
pseudoelliptic | \(-\frac {11 \left (\frac {5 d \,x^{6} \left (a^{2} d^{2}+12 a b c d +8 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{11}+\sqrt {d \,x^{2}+c}\, \left (\frac {26 \left (-\frac {56}{13} b^{2} x^{4}+\frac {54}{13} a b \,x^{2}+a^{2}\right ) x^{2} d \,c^{\frac {3}{2}}}{33}+\frac {8 \left (b^{2} x^{4}+a b \,x^{2}+\frac {1}{3} a^{2}\right ) c^{\frac {5}{2}}}{11}+d^{2} x^{4} \sqrt {c}\, \left (-\frac {16}{33} b^{2} x^{4}-\frac {32}{11} a b \,x^{2}+a^{2}\right )\right )\right )}{16 \sqrt {c}\, x^{6}}\) | \(144\) |
risch | \(-\frac {\sqrt {d \,x^{2}+c}\, \left (33 a^{2} d^{2} x^{4}+108 x^{4} a b c d +24 b^{2} c^{2} x^{4}+26 a^{2} c d \,x^{2}+24 a b \,c^{2} x^{2}+8 a^{2} c^{2}\right )}{48 x^{6}}+\frac {d \left (16 b^{2} d^{2} \left (\frac {x^{2} \sqrt {d \,x^{2}+c}}{3 d}-\frac {2 c \sqrt {d \,x^{2}+c}}{3 d^{2}}\right )+32 a b d \sqrt {d \,x^{2}+c}+48 b^{2} c \sqrt {d \,x^{2}+c}-\frac {\left (5 a^{2} d^{2}+60 a b c d +40 b^{2} c^{2}\right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{\sqrt {c}}\right )}{16}\) | \(201\) |
default | \(a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{6 c \,x^{6}}+\frac {d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{4 c \,x^{4}}+\frac {3 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}+\frac {5 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )}{2 c}\right )}{4 c}\right )}{6 c}\right )+b^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}+\frac {5 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )}{2 c}\right )+2 a b \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{4 c \,x^{4}}+\frac {3 d \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}+\frac {5 d \left (\frac {\left (d \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\right )\right )}{2 c}\right )}{4 c}\right )\) | \(356\) |
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Time = 0.28 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=\left [\frac {15 \, {\left (8 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {c} x^{6} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (16 \, b^{2} c d^{2} x^{8} + 16 \, {\left (7 \, b^{2} c^{2} d + 6 \, a b c d^{2}\right )} x^{6} - 8 \, a^{2} c^{3} - 3 \, {\left (8 \, b^{2} c^{3} + 36 \, a b c^{2} d + 11 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (12 \, a b c^{3} + 13 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{96 \, c x^{6}}, \frac {15 \, {\left (8 \, b^{2} c^{2} d + 12 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {-c} x^{6} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (16 \, b^{2} c d^{2} x^{8} + 16 \, {\left (7 \, b^{2} c^{2} d + 6 \, a b c d^{2}\right )} x^{6} - 8 \, a^{2} c^{3} - 3 \, {\left (8 \, b^{2} c^{3} + 36 \, a b c^{2} d + 11 \, a^{2} c d^{2}\right )} x^{4} - 2 \, {\left (12 \, a b c^{3} + 13 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{48 \, c x^{6}}\right ] \]
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Time = 89.31 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=- \frac {a^{2} c^{3}}{6 \sqrt {d} x^{7} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {17 a^{2} c^{2} \sqrt {d}}{24 x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {35 a^{2} c d^{\frac {3}{2}}}{48 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {a^{2} d^{\frac {5}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{2 x} - \frac {3 a^{2} d^{\frac {5}{2}}}{16 x \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {5 a^{2} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{16 \sqrt {c}} - \frac {15 a b \sqrt {c} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{4} - \frac {a b c^{3}}{2 \sqrt {d} x^{5} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {3 a b c^{2} \sqrt {d}}{4 x^{3} \sqrt {\frac {c}{d x^{2}} + 1}} - \frac {2 a b c d^{\frac {3}{2}} \sqrt {\frac {c}{d x^{2}} + 1}}{x} + \frac {7 a b c d^{\frac {3}{2}}}{4 x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {2 a b d^{\frac {5}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} - \frac {5 b^{2} c^{\frac {3}{2}} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2} - \frac {b^{2} c^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 x} + \frac {2 b^{2} c^{2} \sqrt {d}}{x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {2 b^{2} c d^{\frac {3}{2}} x}{\sqrt {\frac {c}{d x^{2}} + 1}} + b^{2} d^{2} \left (\begin {cases} \frac {c \sqrt {c + d x^{2}}}{3 d} + \frac {x^{2} \sqrt {c + d x^{2}}}{3} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=-\frac {5}{2} \, b^{2} c^{\frac {3}{2}} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {15}{4} \, a b \sqrt {c} d^{2} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {5 \, a^{2} d^{3} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{16 \, \sqrt {c}} + \frac {5}{6} \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} d}{2 \, c} + \frac {5}{2} \, \sqrt {d x^{2} + c} b^{2} c d + \frac {15}{4} \, \sqrt {d x^{2} + c} a b d^{2} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b d^{2}}{4 \, c^{2}} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b d^{2}}{4 \, c} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{3}}{16 \, c^{3}} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} d^{3}}{48 \, c^{2}} + \frac {5 \, \sqrt {d x^{2} + c} a^{2} d^{3}}{16 \, c} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2}}{2 \, c x^{2}} - \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a b d}{4 \, c^{2} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d^{2}}{16 \, c^{3} x^{2}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b}{2 \, c x^{4}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} d}{24 \, c^{2} x^{4}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2}}{6 \, c x^{6}} \]
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Time = 0.32 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=\frac {16 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} d^{2} + 96 \, \sqrt {d x^{2} + c} b^{2} c d^{2} + 96 \, \sqrt {d x^{2} + c} a b d^{3} + \frac {15 \, {\left (8 \, b^{2} c^{2} d^{2} + 12 \, a b c d^{3} + a^{2} d^{4}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {24 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{2} d^{2} - 48 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt {d x^{2} + c} b^{2} c^{4} d^{2} + 108 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c d^{3} - 192 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{2} d^{3} + 84 \, \sqrt {d x^{2} + c} a b c^{3} d^{3} + 33 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} d^{4} - 40 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c d^{4} + 15 \, \sqrt {d x^{2} + c} a^{2} c^{2} d^{4}}{d^{3} x^{6}}}{48 \, d} \]
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Time = 7.17 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^7} \, dx=\frac {\sqrt {d\,x^2+c}\,\left (\frac {5\,a^2\,c^2\,d^3}{16}+\frac {7\,a\,b\,c^3\,d^2}{4}+\frac {b^2\,c^4\,d}{2}\right )-{\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {5\,a^2\,c\,d^3}{6}+4\,a\,b\,c^2\,d^2+b^2\,c^3\,d\right )+{\left (d\,x^2+c\right )}^{5/2}\,\left (\frac {11\,a^2\,d^3}{16}+\frac {9\,a\,b\,c\,d^2}{4}+\frac {b^2\,c^2\,d}{2}\right )}{3\,c\,{\left (d\,x^2+c\right )}^2-3\,c^2\,\left (d\,x^2+c\right )-{\left (d\,x^2+c\right )}^3+c^3}+\left (2\,b\,d\,\left (a\,d-b\,c\right )+4\,b^2\,c\,d\right )\,\sqrt {d\,x^2+c}+\frac {b^2\,d\,{\left (d\,x^2+c\right )}^{3/2}}{3}+\frac {d\,\mathrm {atan}\left (\frac {d\,\sqrt {d\,x^2+c}\,\left (a^2\,d^2+12\,a\,b\,c\,d+8\,b^2\,c^2\right )\,5{}\mathrm {i}}{8\,\sqrt {c}\,\left (\frac {5\,a^2\,d^3}{8}+\frac {15\,a\,b\,c\,d^2}{2}+5\,b^2\,c^2\,d\right )}\right )\,\left (a^2\,d^2+12\,a\,b\,c\,d+8\,b^2\,c^2\right )\,5{}\mathrm {i}}{16\,\sqrt {c}} \]
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