Optimal. Leaf size=50 \[ \frac {-A-B x}{3 b \left (a+b x^2\right )^{3/2}}+\frac {B x}{3 a b \sqrt {a+b x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {778, 191} \begin {gather*} \frac {B x}{3 a b \sqrt {a+b x^2}}-\frac {A+B x}{3 b \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 778
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\left (a+b x^2\right )^{5/2}} \, dx &=-\frac {A+B x}{3 b \left (a+b x^2\right )^{3/2}}+\frac {B \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {A+B x}{3 b \left (a+b x^2\right )^{3/2}}+\frac {B x}{3 a b \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.64 \begin {gather*} \frac {b B x^3-a A}{3 a b \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 32, normalized size = 0.64 \begin {gather*} \frac {b B x^3-a A}{3 a b \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 49, normalized size = 0.98 \begin {gather*} \frac {{\left (B b x^{3} - A a\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 26, normalized size = 0.52 \begin {gather*} \frac {\frac {B x^{3}}{a} - \frac {A}{b}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.58 \begin {gather*} -\frac {-B b \,x^{3}+A a}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 51, normalized size = 1.02 \begin {gather*} -\frac {B x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {B x}{3 \, \sqrt {b x^{2} + a} a b} - \frac {A}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 34, normalized size = 0.68 \begin {gather*} \frac {B\,x^3}{3\,a\,{\left (b\,x^2+a\right )}^{3/2}}-\frac {A}{3\,b\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.98, size = 95, normalized size = 1.90 \begin {gather*} A \left (\begin {cases} - \frac {1}{3 a b \sqrt {a + b x^{2}} + 3 b^{2} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {B x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {3}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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