Optimal. Leaf size=35 \[ -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 b B \sqrt {x} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{3 x^{3/2}}+2 b B \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{x^{5/2}} \, dx &=\int \left (\frac {a A}{x^{5/2}}+\frac {A b+a B}{x^{3/2}}+\frac {b B}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 a A}{3 x^{3/2}}-\frac {2 (A b+a B)}{\sqrt {x}}+2 b B \sqrt {x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2 (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 31, normalized size = 0.89 \begin {gather*} \frac {2 \left (-a A-3 a B x-3 A b x+3 b B x^2\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 27, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (3 \, B b x^{2} - A a - 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.20, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B b \sqrt {x} - \frac {2 \, {\left (3 \, B a x + 3 \, A b x + A a\right )}}{3 \, x^{\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 = 27, normalized size = 0.77 \begin {gather*} -\frac {2 \left (-3 B b \,x^{2}+3 A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B b \sqrt {x} - \frac {2 \, {\left (A a + 3 \, {\left (B a + A b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 27, normalized size = 0.77 \begin {gather*} -\frac {2\,A\,a+6\,A\,b\,x+6\,B\,a\,x-6\,B\,b\,x^2}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 41, normalized size = 1.17 \begin {gather*} - \frac {2 A a}{3 x^{\frac {3}{2}}} - \frac {2 A b}{\sqrt {x}} - \frac {2 B a}{\sqrt {x}} + 2 B b \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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