Optimal. Leaf size=39 \[ \frac {2}{5} x^{5/2} (a B+A b)+\frac {2}{3} a A x^{3/2}+\frac {2}{7} b B x^{7/2} \]
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Rubi [A] time = 0.01, antiderivative size = 39, 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} \[ \frac {2}{5} x^{5/2} (a B+A b)+\frac {2}{3} a A x^{3/2}+\frac {2}{7} b B x^{7/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \sqrt {x} (a+b x) (A+B x) \, dx &=\int \left (a A \sqrt {x}+(A b+a B) x^{3/2}+b B x^{5/2}\right ) \, dx\\ &=\frac {2}{3} a A x^{3/2}+\frac {2}{5} (A b+a B) x^{5/2}+\frac {2}{7} b B x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.85 \[ \frac {2}{105} x^{3/2} (7 a (5 A+3 B x)+3 b x (7 A+5 B x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 30, normalized size = 0.77 \[ \frac {2}{105} \, {\left (15 \, B b x^{3} + 35 \, A a x + 21 \, {\left (B a + A b\right )} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 29, normalized size = 0.74 \[ \frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{5} \, B a x^{\frac {5}{2}} + \frac {2}{5} \, A b x^{\frac {5}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.72 \[ \frac {2 \left (15 B b \,x^{2}+21 A b x +21 B a x +35 A a \right ) x^{\frac {3}{2}}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 27, normalized size = 0.69 \[ \frac {2}{7} \, B b x^{\frac {7}{2}} + \frac {2}{3} \, A a x^{\frac {3}{2}} + \frac {2}{5} \, {\left (B a + A b\right )} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 27, normalized size = 0.69 \[ \frac {2\,x^{3/2}\,\left (35\,A\,a+21\,A\,b\,x+21\,B\,a\,x+15\,B\,b\,x^2\right )}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.88, size = 37, normalized size = 0.95 \[ \frac {2 A a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {7}{2}}}{7} + \frac {2 x^{\frac {5}{2}} \left (A b + B a\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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