Integrand size = 24, antiderivative size = 103 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=\frac {4 a b-\frac {2 b^2 c}{d}-\frac {3 a^2 d}{c}}{2 c \sqrt {c+d x^2}}-\frac {a^2}{2 c x^2 \sqrt {c+d x^2}}-\frac {a (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{5/2}} \]
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Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {457, 91, 79, 65, 214} \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {3 a^2 d^2-4 a b c d+2 b^2 c^2}{2 c^2 d \sqrt {c+d x^2}}-\frac {a^2}{2 c x^2 \sqrt {c+d x^2}}-\frac {a (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{5/2}} \]
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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}{x^2 (c+d x)^{3/2}} \, dx,x,x^2\right ) \\ & = -\frac {a^2}{2 c x^2 \sqrt {c+d x^2}}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} a (4 b c-3 a d)+b^2 c x}{x (c+d x)^{3/2}} \, dx,x,x^2\right )}{2 c} \\ & = -\frac {2 b^2 c^2-4 a b c d+3 a^2 d^2}{2 c^2 d \sqrt {c+d x^2}}-\frac {a^2}{2 c x^2 \sqrt {c+d x^2}}+\frac {(a (4 b c-3 a d)) \text {Subst}\left (\int \frac {1}{x \sqrt {c+d x}} \, dx,x,x^2\right )}{4 c^2} \\ & = -\frac {2 b^2 c^2-4 a b c d+3 a^2 d^2}{2 c^2 d \sqrt {c+d x^2}}-\frac {a^2}{2 c x^2 \sqrt {c+d x^2}}+\frac {(a (4 b c-3 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^2 d} \\ & = -\frac {2 b^2 c^2-4 a b c d+3 a^2 d^2}{2 c^2 d \sqrt {c+d x^2}}-\frac {a^2}{2 c x^2 \sqrt {c+d x^2}}-\frac {a (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{5/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=\frac {-\frac {\sqrt {c} \left (2 b^2 c^2 x^2-4 a b c d x^2+a^2 d \left (c+3 d x^2\right )\right )}{d x^2 \sqrt {c+d x^2}}+a (-4 b c+3 a d) \text {arctanh}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 c^{5/2}} \]
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Time = 2.91 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(\frac {\frac {3 \sqrt {d \,x^{2}+c}\, a d \,x^{2} \left (a d -\frac {4 b c}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{\sqrt {c}}\right )}{2}-\frac {a d \left (-4 b \,x^{2}+a \right ) c^{\frac {3}{2}}}{2}-\frac {3 x^{2} \left (a^{2} d^{2} \sqrt {c}+\frac {2 b^{2} c^{\frac {5}{2}}}{3}\right )}{2}}{d \,c^{\frac {5}{2}} x^{2} \sqrt {d \,x^{2}+c}}\) | \(98\) |
risch | \(-\frac {a^{2} \sqrt {d \,x^{2}+c}}{2 c^{2} x^{2}}-\frac {-\frac {a^{2} d^{2}-2 b^{2} c^{2}}{d \sqrt {d \,x^{2}+c}}+a c \left (3 a d -4 b c \right ) \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c^{2}}\) | \(112\) |
default | \(-\frac {b^{2}}{d \sqrt {d \,x^{2}+c}}+a^{2} \left (-\frac {1}{2 c \,x^{2} \sqrt {d \,x^{2}+c}}-\frac {3 d \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+2 a b \left (\frac {1}{c \sqrt {d \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {d \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}\right )\) | \(135\) |
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Time = 0.28 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.83 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} + 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (a^{2} c^{2} d + {\left (2 \, b^{2} c^{3} - 4 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{4 \, {\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}}, \frac {{\left ({\left (4 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x^{4} + {\left (4 \, a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (a^{2} c^{2} d + {\left (2 \, b^{2} c^{3} - 4 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{2 \, {\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{2}}{x^{3} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=-\frac {2 \, a b \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{c^{\frac {3}{2}}} + \frac {3 \, a^{2} d \operatorname {arsinh}\left (\frac {c}{\sqrt {c d} {\left | x \right |}}\right )}{2 \, c^{\frac {5}{2}}} + \frac {2 \, a b}{\sqrt {d x^{2} + c} c} - \frac {b^{2}}{\sqrt {d x^{2} + c} d} - \frac {3 \, a^{2} d}{2 \, \sqrt {d x^{2} + c} c^{2}} - \frac {a^{2}}{2 \, \sqrt {d x^{2} + c} c x^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{2 \, \sqrt {-c} c^{2}} - \frac {2 \, {\left (d x^{2} + c\right )} b^{2} c^{2} - 2 \, b^{2} c^{3} - 4 \, {\left (d x^{2} + c\right )} a b c d + 4 \, a b c^{2} d + 3 \, {\left (d x^{2} + c\right )} a^{2} d^{2} - 2 \, a^{2} c d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )}^{\frac {3}{2}} - \sqrt {d x^{2} + c} c\right )} c^{2} d} \]
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Time = 5.85 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+b x^2\right )^2}{x^3 \left (c+d x^2\right )^{3/2}} \, dx=\frac {\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{c}-\frac {\left (d\,x^2+c\right )\,\left (3\,a^2\,d^2-4\,a\,b\,c\,d+2\,b^2\,c^2\right )}{2\,c^2}}{d\,{\left (d\,x^2+c\right )}^{3/2}-c\,d\,\sqrt {d\,x^2+c}}+\frac {a\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )\,\left (3\,a\,d-4\,b\,c\right )}{2\,c^{5/2}} \]
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