Integrand size = 24, antiderivative size = 80 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=\frac {b^2 \sqrt {c+d x^2}}{d}-\frac {a^2 \sqrt {c+d x^2}}{2 c x^2}-\frac {a (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 91, 81, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=-\frac {a^2 \sqrt {c+d x^2}}{2 c x^2}-\frac {a (4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{3/2}}+\frac {b^2 \sqrt {c+d x^2}}{d} \]
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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}{x^2 \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = -\frac {a^2 \sqrt {c+d x^2}}{2 c x^2}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} a (4 b c-a d)+b^2 c x}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c} \\ & = \frac {b^2 \sqrt {c+d x^2}}{d}-\frac {a^2 \sqrt {c+d x^2}}{2 c x^2}+\frac {(a (4 b c-a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c} \\ & = \frac {b^2 \sqrt {c+d x^2}}{d}-\frac {a^2 \sqrt {c+d x^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 c d} \\ & = \frac {b^2 \sqrt {c+d x^2}}{d}-\frac {a^2 \sqrt {c+d x^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 c^{3/2}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=\frac {\left (-a^2 d+2 b^2 c x^2\right ) \sqrt {c+d x^2}}{2 c d x^2}+\frac {a (-4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{3/2}} \]
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Time = 2.91 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(-\frac {-a d \,x^{2} \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )+\sqrt {d \,x^{2}+c}\, \left (-2 c^{\frac {3}{2}} b^{2} x^{2}+\sqrt {c}\, a^{2} d \right )}{2 c^{\frac {3}{2}} d \,x^{2}}\) | \(72\) |
risch | \(-\frac {a^{2} \sqrt {d \,x^{2}+c}}{2 c \,x^{2}}-\frac {-\frac {2 b^{2} c \sqrt {d \,x^{2}+c}}{d}-\frac {a \left (a d -4 b c \right ) \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{\sqrt {c}}}{2 c}\) | \(83\) |
default | \(\frac {b^{2} \sqrt {d \,x^{2}+c}}{d}+a^{2} \left (-\frac {\sqrt {d \,x^{2}+c}}{2 c \,x^{2}}+\frac {d \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{2 c^{\frac {3}{2}}}\right )-\frac {2 a b \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{\sqrt {c}}\) | \(99\) |
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Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.19 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=\left [-\frac {{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt {c} x^{2} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, b^{2} c^{2} x^{2} - a^{2} c d\right )} \sqrt {d x^{2} + c}}{4 \, c^{2} d x^{2}}, \frac {{\left (4 \, a b c d - a^{2} d^{2}\right )} \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + {\left (2 \, b^{2} c^{2} x^{2} - a^{2} c d\right )} \sqrt {d x^{2} + c}}{2 \, c^{2} d x^{2}}\right ] \]
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Time = 15.94 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=- \frac {a^{2} \sqrt {d} \sqrt {\frac {c}{d x^{2}} + 1}}{2 c x} + \frac {a^{2} d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{2 c^{\frac {3}{2}}} - \frac {2 a b \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} x} \right )}}{\sqrt {c}} + b^{2} \left (\begin {cases} \frac {\sqrt {c + d x^{2}}}{d} & \text {for}\: d \neq 0 \\\frac {x^{2}}{2 \sqrt {c}} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=-\frac {2 \, a b \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{\sqrt {c}} + \frac {a^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {d x^{2} + c} b^{2}}{d} - \frac {\sqrt {d x^{2} + c} a^{2}}{2 \, c x^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=\frac {2 \, \sqrt {d x^{2} + c} b^{2} - \frac {\sqrt {d x^{2} + c} a^{2} d}{c x^{2}} + \frac {{\left (4 \, a b c d - a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c}}{2 \, d} \]
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Time = 5.44 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \sqrt {c+d x^2}} \, dx=\frac {b^2\,\sqrt {d\,x^2+c}}{d}+\frac {a\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (a\,d-4\,b\,c\right )}{2\,c^{3/2}}-\frac {a^2\,\sqrt {d\,x^2+c}}{2\,c\,x^2} \]
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