Integrand size = 22, antiderivative size = 110 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {1}{2} \sqrt {a} (3 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {457, 79, 52, 65, 214} \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=-\frac {1}{2} \sqrt {a} (2 a B+3 A b) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )+\frac {\left (a+b x^2\right )^{3/2} (2 a B+3 A b)}{6 a}+\frac {1}{2} \sqrt {a+b x^2} (2 a B+3 A b)-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2} \]
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Rule 52
Rule 65
Rule 79
Rule 214
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2} (A+B x)}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {\left (\frac {3 A b}{2}+a B\right ) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,x^2\right )}{2 a} \\ & = \frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {1}{4} (3 A b+2 a B) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {1}{4} (a (3 A b+2 a B)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right ) \\ & = \frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}+\frac {(a (3 A b+2 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 b} \\ & = \frac {1}{2} (3 A b+2 a B) \sqrt {a+b x^2}+\frac {(3 A b+2 a B) \left (a+b x^2\right )^{3/2}}{6 a}-\frac {A \left (a+b x^2\right )^{5/2}}{2 a x^2}-\frac {1}{2} \sqrt {a} (3 A b+2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {\sqrt {a+b x^2} \left (-3 a A+6 A b x^2+8 a B x^2+2 b B x^4\right )}{6 x^2}-\frac {1}{2} \sqrt {a} (3 A b+2 a B) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \]
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Time = 2.83 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.72
method | result | size |
pseudoelliptic | \(-\frac {3 \left (x^{2} a \left (A b +\frac {2 B a}{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{\sqrt {a}}\right )-\frac {2 \left (\left (\frac {4 x^{2} B}{3}-\frac {A}{2}\right ) a^{\frac {3}{2}}+b \,x^{2} \sqrt {a}\, \left (\frac {x^{2} B}{3}+A \right )\right ) \sqrt {b \,x^{2}+a}}{3}\right )}{2 \sqrt {a}\, x^{2}}\) | \(79\) |
risch | \(-\frac {a A \sqrt {b \,x^{2}+a}}{2 x^{2}}+B \,b^{2} \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )+A b \sqrt {b \,x^{2}+a}+2 B a \sqrt {b \,x^{2}+a}-\frac {\sqrt {a}\, \left (3 A b +2 B a \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2}\) | \(118\) |
default | \(B \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )+A \left (-\frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{2 a \,x^{2}}+\frac {3 b \left (\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )\right )}{2 a}\right )\) | \(134\) |
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Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\left [\frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {a} x^{2} \log \left (-\frac {b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (2 \, B b x^{4} + 2 \, {\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt {b x^{2} + a}}{12 \, x^{2}}, \frac {3 \, {\left (2 \, B a + 3 \, A b\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, B b x^{4} + 2 \, {\left (4 \, B a + 3 \, A b\right )} x^{2} - 3 \, A a\right )} \sqrt {b x^{2} + a}}{6 \, x^{2}}\right ] \]
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Time = 14.58 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=- \frac {3 A \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2} - \frac {A a \sqrt {b} \sqrt {\frac {a}{b x^{2}} + 1}}{2 x} + \frac {A a \sqrt {b}}{x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {A b^{\frac {3}{2}} x}{\sqrt {\frac {a}{b x^{2}} + 1}} - B a^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )} + \frac {B a^{2}}{\sqrt {b} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {B a \sqrt {b} x}{\sqrt {\frac {a}{b x^{2}} + 1}} + B b \left (\begin {cases} \frac {a \sqrt {a + b x^{2}}}{3 b} + \frac {x^{2} \sqrt {a + b x^{2}}}{3} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{2}}{2} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=-B a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) - \frac {3}{2} \, A \sqrt {a} b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right ) + \frac {1}{3} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B + \sqrt {b x^{2} + a} B a + \frac {3}{2} \, \sqrt {b x^{2} + a} A b + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A b}{2 \, a} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A}{2 \, a x^{2}} \]
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Time = 0.32 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} B b + 6 \, \sqrt {b x^{2} + a} B a b + 6 \, \sqrt {b x^{2} + a} A b^{2} - \frac {3 \, \sqrt {b x^{2} + a} A a b}{x^{2}} + \frac {3 \, {\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}}}{6 \, b} \]
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Time = 5.98 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{x^3} \, dx=\frac {B\,{\left (b\,x^2+a\right )}^{3/2}}{3}-B\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )+A\,b\,\sqrt {b\,x^2+a}+B\,a\,\sqrt {b\,x^2+a}-\frac {A\,a\,\sqrt {b\,x^2+a}}{2\,x^2}-\frac {3\,A\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2} \]
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