Optimal. Leaf size=95 \[ \frac {3 A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}+\frac {A+B x}{a x^2 \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {823, 835, 807, 266, 63, 208} \[ -\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}+\frac {3 A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}+\frac {A+B x}{a x^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 835
Rubi steps
\begin {align*} \int \frac {A+B x}{x^3 \left (a+b x^2\right )^{3/2}} \, dx &=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {\int \frac {-3 a A b-2 a b B x}{x^3 \sqrt {a+b x^2}} \, dx}{a^2 b}\\ &=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}+\frac {\int \frac {4 a^2 b B-3 a A b^2 x}{x^2 \sqrt {a+b x^2}} \, dx}{2 a^3 b}\\ &=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}-\frac {(3 A b) \int \frac {1}{x \sqrt {a+b x^2}} \, dx}{2 a^2}\\ &=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}-\frac {(3 A b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{4 a^2}\\ &=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}-\frac {(3 A) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{2 a^2}\\ &=\frac {A+B x}{a x^2 \sqrt {a+b x^2}}-\frac {3 A \sqrt {a+b x^2}}{2 a^2 x^2}-\frac {2 B \sqrt {a+b x^2}}{a^2 x}+\frac {3 A b \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 75, normalized size = 0.79 \[ \frac {3 A b \sqrt {\frac {b x^2}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )-\frac {a (A+2 B x)}{x^2}-b (3 A+4 B x)}{2 a^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.23, size = 211, normalized size = 2.22 \[ \left [\frac {3 \, {\left (A b^{2} x^{4} + A a b x^{2}\right )} \sqrt {a} \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (4 \, B a b x^{3} + 3 \, A a b x^{2} + 2 \, B a^{2} x + A a^{2}\right )} \sqrt {b x^{2} + a}}{4 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}, -\frac {3 \, {\left (A b^{2} x^{4} + A a b x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{2} + a}}\right ) + {\left (4 \, B a b x^{3} + 3 \, A a b x^{2} + 2 \, B a^{2} x + A a^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a^{3} b x^{4} + a^{4} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 171, normalized size = 1.80 \[ -\frac {\frac {B b x}{a^{2}} + \frac {A b}{a^{2}}}{\sqrt {b x^{2} + a}} - \frac {3 \, A b \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a \sqrt {b} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b - 2 \, B a^{2} \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 1.06 \[ -\frac {2 B b x}{\sqrt {b \,x^{2}+a}\, a^{2}}+\frac {3 A b \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )}{2 a^{\frac {5}{2}}}-\frac {3 A b}{2 \sqrt {b \,x^{2}+a}\, a^{2}}-\frac {B}{\sqrt {b \,x^{2}+a}\, a x}-\frac {A}{2 \sqrt {b \,x^{2}+a}\, a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 89, normalized size = 0.94 \[ -\frac {2 \, B b x}{\sqrt {b x^{2} + a} a^{2}} + \frac {3 \, A b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | x \right |}}\right )}{2 \, a^{\frac {5}{2}}} - \frac {3 \, A b}{2 \, \sqrt {b x^{2} + a} a^{2}} - \frac {B}{\sqrt {b x^{2} + a} a x} - \frac {A}{2 \, \sqrt {b x^{2} + a} a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 94, normalized size = 0.99 \[ \frac {3\,A\,b\,\mathrm {atanh}\left (\frac {\sqrt {b\,x^2+a}}{\sqrt {a}}\right )}{2\,a^{5/2}}-\frac {3\,A\,b}{2\,a^2\,\sqrt {b\,x^2+a}}-\frac {A}{2\,a\,x^2\,\sqrt {b\,x^2+a}}-\frac {\sqrt {b\,x^2+a}\,\left (\frac {B}{a}+\frac {2\,B\,b\,x^2}{a^2}\right )}{b\,x^3+a\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.79, size = 124, normalized size = 1.31 \[ A \left (- \frac {1}{2 a \sqrt {b} x^{3} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {3 \sqrt {b}}{2 a^{2} x \sqrt {\frac {a}{b x^{2}} + 1}} + \frac {3 b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x} \right )}}{2 a^{\frac {5}{2}}}\right ) + B \left (- \frac {1}{a \sqrt {b} x^{2} \sqrt {\frac {a}{b x^{2}} + 1}} - \frac {2 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x^{2}} + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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