Optimal. Leaf size=40 \[ -\frac {2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac {2 B}{b^2 \sqrt {a+b x}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} -\frac {2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac {2 B}{b^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {A+B x}{(a+b x)^{5/2}} \, dx &=\int \left (\frac {A b-a B}{b (a+b x)^{5/2}}+\frac {B}{b (a+b x)^{3/2}}\right ) \, dx\\ &=-\frac {2 (A b-a B)}{3 b^2 (a+b x)^{3/2}}-\frac {2 B}{b^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.72 \begin {gather*} -\frac {2 (2 a B+A b+3 b B x)}{3 b^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 32, normalized size = 0.80 \begin {gather*} -\frac {2 (3 B (a+b x)-a B+A b)}{3 b^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 46, normalized size = 1.15 \begin {gather*} -\frac {2 \, {\left (3 \, B b x + 2 \, B a + A b\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 28, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (b x + a\right )} B - B a + A b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.65 \begin {gather*} -\frac {2 \left (3 B b x +A b +2 B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 28, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (b x + a\right )} B - B a + A b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 29, normalized size = 0.72 \begin {gather*} -\frac {2\,A\,b-2\,B\,a+6\,B\,\left (a+b\,x\right )}{3\,b^2\,{\left (a+b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.16, size = 124, normalized size = 3.10 \begin {gather*} \begin {cases} - \frac {2 A b}{3 a b^{2} \sqrt {a + b x} + 3 b^{3} x \sqrt {a + b x}} - \frac {4 B a}{3 a b^{2} \sqrt {a + b x} + 3 b^{3} x \sqrt {a + b x}} - \frac {6 B b x}{3 a b^{2} \sqrt {a + b x} + 3 b^{3} x \sqrt {a + b x}} & \text {for}\: b \neq 0 \\\frac {A x + \frac {B x^{2}}{2}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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