Optimal. Leaf size=41 \[ \frac {B \sqrt {a+b x^2}}{b^2}-\frac {A b-a B}{b^2 \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \[ \frac {B \sqrt {a+b x^2}}{b^2}-\frac {A b-a B}{b^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^{3/2}}+\frac {B}{b \sqrt {a+b x}}\right ) \, dx,x,x^2\right )\\ &=-\frac {A b-a B}{b^2 \sqrt {a+b x^2}}+\frac {B \sqrt {a+b x^2}}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.73 \[ \frac {2 a B-A b+b B x^2}{b^2 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 40, normalized size = 0.98 \[ \frac {{\left (B b x^{2} + 2 \, B a - A b\right )} \sqrt {b x^{2} + a}}{b^{3} x^{2} + a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 36, normalized size = 0.88 \[ \frac {\sqrt {b x^{2} + a} B}{b^{2}} + \frac {B a - A b}{\sqrt {b x^{2} + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.73 \[ -\frac {-B b \,x^{2}+A b -2 B a}{\sqrt {b \,x^{2}+a}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 49, normalized size = 1.20 \[ \frac {B x^{2}}{\sqrt {b x^{2} + a} b} + \frac {2 \, B a}{\sqrt {b x^{2} + a} b^{2}} - \frac {A}{\sqrt {b x^{2} + a} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 30, normalized size = 0.73 \[ \frac {B\,a-A\,b+B\,\left (b\,x^2+a\right )}{b^2\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 66, normalized size = 1.61 \[ \begin {cases} - \frac {A}{b \sqrt {a + b x^{2}}} + \frac {2 B a}{b^{2} \sqrt {a + b x^{2}}} + \frac {B x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{2}}{2} + \frac {B x^{4}}{4}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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