Integrand size = 24, antiderivative size = 109 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=\frac {a (4 b c+a d) \sqrt {c+d x^2}}{2 c}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}-\frac {a (4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}} \]
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Time = 0.06 (sec) , antiderivative size = 109, 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 = {457, 91, 81, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=-\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}-\frac {a (a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}}+\frac {a \sqrt {c+d x^2} (a d+4 b c)}{2 c}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d} \]
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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 \sqrt {c+d x}}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac {\text {Subst}\left (\int \frac {\left (\frac {1}{2} a (4 b c+a d)+b^2 c x\right ) \sqrt {c+d x}}{x} \, dx,x,x^2\right )}{2 c} \\ & = \frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac {(a (4 b c+a d)) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{x} \, dx,x,x^2\right )}{4 c} \\ & = \frac {a (4 b c+a d) \sqrt {c+d x^2}}{2 c}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac {1}{4} (a (4 b c+a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {a (4 b c+a d) \sqrt {c+d x^2}}{2 c}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}+\frac {(a (4 b c+a d)) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 d} \\ & = \frac {a (4 b c+a d) \sqrt {c+d x^2}}{2 c}+\frac {b^2 \left (c+d x^2\right )^{3/2}}{3 d}-\frac {a^2 \left (c+d x^2\right )^{3/2}}{2 c x^2}-\frac {a (4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=\frac {1}{6} \left (\frac {\sqrt {c+d x^2} \left (-3 a^2 d+12 a b d x^2+2 b^2 x^2 \left (c+d x^2\right )\right )}{d x^2}-\frac {3 a (4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) \]
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Time = 2.96 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.80
method | result | size |
pseudoelliptic | \(-\frac {a d \,x^{2} \left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+\sqrt {d \,x^{2}+c}\, \left (-\frac {2 c^{\frac {3}{2}} b^{2} x^{2}}{3}+d \sqrt {c}\, \left (-\frac {2}{3} b^{2} x^{4}-4 a b \,x^{2}+a^{2}\right )\right )}{2 \sqrt {c}\, d \,x^{2}}\) | \(87\) |
risch | \(-\frac {a^{2} \sqrt {d \,x^{2}+c}}{2 x^{2}}+b^{2} d \left (\frac {x^{2} \sqrt {d \,x^{2}+c}}{3 d}-\frac {2 c \sqrt {d \,x^{2}+c}}{3 d^{2}}\right )+\frac {b^{2} c \sqrt {d \,x^{2}+c}}{d}+2 a b \sqrt {d \,x^{2}+c}-\frac {a \left (a d +4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 \sqrt {c}}\) | \(124\) |
default | \(\frac {b^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{3 d}+a^{2} \left (-\frac {\left (d \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}+\frac {d \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )}{2 c}\right )+2 a b \left (\sqrt {d \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )\right )\) | \(127\) |
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Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=\left [\frac {3 \, {\left (4 \, a b c d + a^{2} d^{2}\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (2 \, b^{2} c d x^{4} - 3 \, a^{2} c d + 2 \, {\left (b^{2} c^{2} + 6 \, a b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{12 \, c d x^{2}}, \frac {3 \, {\left (4 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} c d x^{4} - 3 \, a^{2} c d + 2 \, {\left (b^{2} c^{2} + 6 \, a b c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{6 \, c d x^{2}}\right ] \]
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Time = 18.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=- \frac {a^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 x} - \frac {a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2 \sqrt {c}} - 2 a b \sqrt {c} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )} + \frac {2 a b c}{\sqrt {d} x \sqrt {\frac {c}{d x^{2}} + 1}} + \frac {2 a b \sqrt {d} x}{\sqrt {\frac {c}{d x^{2}} + 1}} + b^{2} \left (\begin {cases} \frac {c \sqrt {c + d x^{2}}}{3 d} + \frac {x^{2} \sqrt {c + d x^{2}}}{3} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=-2 \, a b \sqrt {c} \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right ) - \frac {a^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, \sqrt {c}} + 2 \, \sqrt {d x^{2} + c} a b + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2}}{3 \, d} + \frac {\sqrt {d x^{2} + c} a^{2} d}{2 \, c} - \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2}}{2 \, c x^{2}} \]
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Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=\frac {2 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} + 12 \, \sqrt {d x^{2} + c} a b d - \frac {3 \, \sqrt {d x^{2} + c} a^{2} d}{x^{2}} + \frac {3 \, {\left (4 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}}}{6 \, d} \]
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Time = 5.56 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x^2}}{x^3} \, dx=\frac {b^2\,{\left (d\,x^2+c\right )}^{3/2}}{3\,d}-\left (\frac {2\,b^2\,c-2\,a\,b\,d}{d}-\frac {2\,b^2\,c}{d}\right )\,\sqrt {d\,x^2+c}-\frac {a^2\,\sqrt {d\,x^2+c}}{2\,x^2}+\frac {a\,\mathrm {atan}\left (\frac {\sqrt {d\,x^2+c}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,\left (a\,d+4\,b\,c\right )\,1{}\mathrm {i}}{2\,\sqrt {c}} \]
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