Optimal. Leaf size=53 \[ \frac {2 \sqrt {a+b x} (2 A b-3 a B)}{3 a^2 \sqrt {x}}-\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} \frac {2 \sqrt {a+b x} (2 A b-3 a B)}{3 a^2 \sqrt {x}}-\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{5/2} \sqrt {a+b x}} \, dx &=-\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}}+\frac {\left (2 \left (-A b+\frac {3 a B}{2}\right )\right ) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a}\\ &=-\frac {2 A \sqrt {a+b x}}{3 a x^{3/2}}+\frac {2 (2 A b-3 a B) \sqrt {a+b x}}{3 a^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {a+b x} (a (A+3 B x)-2 A b x)}{3 a^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 35, normalized size = 0.66 \begin {gather*} -\frac {2 \sqrt {a+b x} (a A+3 a B x-2 A b x)}{3 a^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.48, size = 30, normalized size = 0.57 \begin {gather*} -\frac {2 \, {\left (A a + {\left (3 \, B a - 2 \, A b\right )} x\right )} \sqrt {b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.70, size = 73, normalized size = 1.38 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b {\left (\frac {{\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} {\left (b x + a\right )}}{a^{2}} - \frac {3 \, {\left (B a^{2} b^{2} - A a b^{3}\right )}}{a^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.57 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (-2 A x b +3 B a x +A a \right )}{3 a^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 62, normalized size = 1.17 \begin {gather*} -\frac {2 \, \sqrt {b x^{2} + a x} B}{a x} + \frac {4 \, \sqrt {b x^{2} + a x} A b}{3 \, a^{2} x} - \frac {2 \, \sqrt {b x^{2} + a x} A}{3 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 34, normalized size = 0.64 \begin {gather*} -\frac {\left (\frac {2\,A}{3\,a}-\frac {x\,\left (4\,A\,b-6\,B\,a\right )}{3\,a^2}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.30, size = 66, normalized size = 1.25 \begin {gather*} - \frac {2 A \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 a x} + \frac {4 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{2}} - \frac {2 B \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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