Optimal. Leaf size=51 \[ \frac {2 A x}{3 a^2 \sqrt {a+b x^2}}+\frac {A b x-a B}{3 a b \left (a+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {639, 191} \[ \frac {2 A x}{3 a^2 \sqrt {a+b x^2}}-\frac {a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 639
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+b x^2\right )^{5/2}} \, dx &=-\frac {a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {(2 A) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {a B-A b x}{3 a b \left (a+b x^2\right )^{3/2}}+\frac {2 A x}{3 a^2 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 43, normalized size = 0.84 \[ \frac {-a^2 B+3 a A b x+2 A b^2 x^3}{3 a^2 b \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 62, normalized size = 1.22 \[ \frac {{\left (2 \, A b^{2} x^{3} + 3 \, A a b x - B a^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 37, normalized size = 0.73 \[ \frac {{\left (\frac {2 \, A b x^{2}}{a^{2}} + \frac {3 \, A}{a}\right )} x - \frac {B}{b}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.78 \[ \frac {2 A \,x^{3} b^{2}+3 A x a b -B \,a^{2}}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 48, normalized size = 0.94 \[ \frac {2 \, A x}{3 \, \sqrt {b x^{2} + a} a^{2}} + \frac {A x}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {B}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 41, normalized size = 0.80 \[ \frac {2\,A\,b\,x\,\left (b\,x^2+a\right )-B\,a^2+A\,a\,b\,x}{3\,a^2\,b\,{\left (b\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.20, size = 146, normalized size = 2.86 \[ A \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {2 b x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 3 a^{\frac {5}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} - \frac {1}{3 a b \sqrt {a + b x^{2}} + 3 b^{2} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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