Optimal. Leaf size=37 \[ -\frac {2 (a B+A b)}{3 x^{3/2}}-\frac {2 a A}{5 x^{5/2}}-\frac {2 b B}{\sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 37, 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)}{3 x^{3/2}}-\frac {2 a A}{5 x^{5/2}}-\frac {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^{7/2}} \, dx &=\int \left (\frac {a A}{x^{7/2}}+\frac {A b+a B}{x^{5/2}}+\frac {b B}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 a A}{5 x^{5/2}}-\frac {2 (A b+a B)}{3 x^{3/2}}-\frac {2 b B}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.81 \begin {gather*} -\frac {2 (a (3 A+5 B x)+5 b x (A+3 B x))}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 31, normalized size = 0.84 \begin {gather*} -\frac {2 \left (3 a A+5 a B x+5 A b x+15 b B x^2\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 27, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (15 \, B b x^{2} + 3 \, A a + 5 \, {\left (B a + A b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 27, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (15 \, B b x^{2} + 5 \, B a x + 5 \, A b x + 3 \, A a\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.76 \begin {gather*} -\frac {2 \left (15 B b \,x^{2}+5 A b x +5 B a x +3 A a \right )}{15 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 27, normalized size = 0.73 \begin {gather*} -\frac {2 \, {\left (15 \, B b x^{2} + 3 \, A a + 5 \, {\left (B a + A b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 28, normalized size = 0.76 \begin {gather*} -\frac {2\,B\,b\,x^2+\left (\frac {2\,A\,b}{3}+\frac {2\,B\,a}{3}\right )\,x+\frac {2\,A\,a}{5}}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.71, size = 46, normalized size = 1.24 \begin {gather*} - \frac {2 A a}{5 x^{\frac {5}{2}}} - \frac {2 A b}{3 x^{\frac {3}{2}}} - \frac {2 B a}{3 x^{\frac {3}{2}}} - \frac {2 B b}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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