Integrand size = 20, antiderivative size = 108 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-a^{3/2} B \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {827, 829, 858, 223, 212, 272, 65, 214} \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=a^{3/2} (-B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {3}{2} a A \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-\frac {\left (a+b x^2\right )^{3/2} (3 A-B x)}{3 x}+\frac {1}{2} \sqrt {a+b x^2} (2 a B+3 A b x) \]
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 272
Rule 827
Rule 829
Rule 858
Rubi steps \begin{align*} \text {integral}& = -\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}-\frac {1}{2} \int \frac {(-2 a B-6 A b x) \sqrt {a+b x^2}}{x} \, dx \\ & = \frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}-\frac {\int \frac {-4 a^2 b B-6 a A b^2 x}{x \sqrt {a+b x^2}} \, dx}{4 b} \\ & = \frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {1}{2} (3 a A b) \int \frac {1}{\sqrt {a+b x^2}} \, dx+\left (a^2 B\right ) \int \frac {1}{x \sqrt {a+b x^2}} \, dx \\ & = \frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {1}{2} (3 a A b) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )+\frac {1}{2} \left (a^2 B\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+\frac {\left (a^2 B\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{b} \\ & = \frac {1}{2} (2 a B+3 A b x) \sqrt {a+b x^2}-\frac {(3 A-B x) \left (a+b x^2\right )^{3/2}}{3 x}+\frac {3}{2} a A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )-a^{3/2} B \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\frac {\sqrt {a+b x^2} \left (b x^2 (3 A+2 B x)+a (-6 A+8 B x)\right )}{6 x}+2 a^{3/2} B \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )-\frac {3}{2} a A \sqrt {b} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right ) \]
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Time = 3.41 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {a A \sqrt {b \,x^{2}+a}}{x}+\frac {B b \,x^{2} \sqrt {b \,x^{2}+a}}{3}+\frac {4 a B \sqrt {b \,x^{2}+a}}{3}+\frac {b A x \sqrt {b \,x^{2}+a}}{2}+\frac {3 a \sqrt {b}\, A \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2}-a^{\frac {3}{2}} B \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\) | \(113\) |
default | \(B \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{a x}+\frac {4 b \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{a}\right )\) | \(133\) |
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Time = 0.30 (sec) , antiderivative size = 411, normalized size of antiderivative = 3.81 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\left [\frac {9 \, A a \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 6 \, B a^{\frac {3}{2}} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x}, -\frac {9 \, A a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 3 \, B a^{\frac {3}{2}} x \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x}, \frac {12 \, B \sqrt {-a} a x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + 9 \, A a \sqrt {b} x \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x}, -\frac {9 \, A a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - 6 \, B \sqrt {-a} a x \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b x^{3} + 3 \, A b x^{2} + 8 \, B a x - 6 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x}\right ] \]
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Time = 2.28 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.25 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=- \frac {A a^{\frac {3}{2}}}{x \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {A \sqrt {a} b x}{\sqrt {1 + \frac {b x^{2}}{a}}} + A a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )} + A b \left (\begin {cases} \frac {a \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {a + b x^{2}}}{2} & \text {for}\: b \neq 0 \\\sqrt {a} x & \text {otherwise} \end {cases}\right ) - B a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B a \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + B b \left (\begin {cases} \frac {a \sqrt {a + b x^{2}}}{3 b} + \frac {x^{2} \sqrt {a + b x^{2}}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\frac {3}{2} \, \sqrt {b x^{2} + a} A b x + \frac {3}{2} \, A a \sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right ) - B a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B + \sqrt {b x^{2} + a} B a - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{x} \]
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Time = 0.30 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.15 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\frac {2 \, B a^{2} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {3}{2} \, A a \sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right ) + \frac {2 \, A a^{2} \sqrt {b}}{{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a} + \frac {1}{6} \, \sqrt {b x^{2} + a} {\left (8 \, B a + {\left (2 \, B b x + 3 \, A b\right )} x\right )} \]
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Time = 6.73 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a+b x^2\right )^{3/2}}{x^2} \, dx=\frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{3}-B\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+B\,a\,\sqrt {b\,x^2+a}-\frac {A\,{\left (b\,x^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {b\,x^2}{a}\right )}{x\,{\left (\frac {b\,x^2}{a}+1\right )}^{3/2}} \]
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