Integrand size = 22, antiderivative size = 109 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {3 a (4 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}} \]
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Time = 0.03 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {464, 201, 223, 212} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\frac {3 a (a B+4 A b) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}}+\frac {x \left (a+b x^2\right )^{3/2} (a B+4 A b)}{4 a}+\frac {3}{8} x \sqrt {a+b x^2} (a B+4 A b)-\frac {A \left (a+b x^2\right )^{5/2}}{a x} \]
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Rule 201
Rule 212
Rule 223
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {A \left (a+b x^2\right )^{5/2}}{a x}-\frac {(-4 A b-a B) \int \left (a+b x^2\right )^{3/2} \, dx}{a} \\ & = \frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {1}{4} (3 (4 A b+a B)) \int \sqrt {a+b x^2} \, dx \\ & = \frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {1}{8} (3 a (4 A b+a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx \\ & = \frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {1}{8} (3 a (4 A b+a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right ) \\ & = \frac {3}{8} (4 A b+a B) x \sqrt {a+b x^2}+\frac {(4 A b+a B) x \left (a+b x^2\right )^{3/2}}{4 a}-\frac {A \left (a+b x^2\right )^{5/2}}{a x}+\frac {3 a (4 A b+a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\frac {\sqrt {a+b x^2} \left (-8 a A+4 A b x^2+5 a B x^2+2 b B x^4\right )}{8 x}+\frac {3 a (4 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{4 \sqrt {b}} \]
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Time = 2.83 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{2}+a}\, \left (-2 b B \,x^{4}-4 A b \,x^{2}-5 B a \,x^{2}+8 A a \right )}{8 x}+\frac {3 a \left (4 A b +B a \right ) \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{8 \sqrt {b}}\) | \(72\) |
pseudoelliptic | \(\frac {\frac {3 a x \left (A b +\frac {B a}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{2}-\left (-\frac {x^{2} \left (\frac {x^{2} B}{2}+A \right ) b^{\frac {3}{2}}}{2}+a \sqrt {b}\, \left (-\frac {5 x^{2} B}{8}+A \right )\right ) \sqrt {b \,x^{2}+a}}{x \sqrt {b}}\) | \(79\) |
default | \(B \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )\) | \(132\) |
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Time = 0.26 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.67 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\left [\frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, B b^{2} x^{4} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{16 \, b x}, -\frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b^{2} x^{4} - 8 \, A a b + {\left (5 \, B a b + 4 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{8 \, b x}\right ] \]
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Time = 1.66 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=- \frac {A a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + A a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + A b \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + B a \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) + B b \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a x \sqrt {a + b x^{2}}}{8 b} + \frac {x^{3} \sqrt {a + b x^{2}}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{3}}{3} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B x + \frac {3}{8} \, \sqrt {b x^{2} + a} B a x + \frac {3}{2} \, \sqrt {b x^{2} + a} A b x + \frac {3 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {3}{2} \, A a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{x} \]
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Time = 0.32 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\frac {2 \, A a^{2} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{8} \, {\left (2 \, B b x^{2} + \frac {5 \, B a b^{2} + 4 \, A b^{3}}{b^{2}}\right )} \sqrt {b x^{2} + a} x - \frac {3 \, {\left (B a^{2} + 4 \, A a b\right )} \log \left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2}\right )}{16 \, \sqrt {b}} \]
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Time = 6.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^2} \, dx=\frac {B\,x\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \]
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