Optimal. Leaf size=39 \[ \frac {2}{3} x^{3/2} (a B+A b)-\frac {2 a A}{3 x^{3/2}}+\frac {2}{9} b B x^{9/2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ \frac {2}{3} x^{3/2} (a B+A b)-\frac {2 a A}{3 x^{3/2}}+\frac {2}{9} b B x^{9/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right ) \left (A+B x^3\right )}{x^{5/2}} \, dx &=\int \left (\frac {a A}{x^{5/2}}+(A b+a B) \sqrt {x}+b B x^{7/2}\right ) \, dx\\ &=-\frac {2 a A}{3 x^{3/2}}+\frac {2}{3} (A b+a B) x^{3/2}+\frac {2}{9} b B x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 0.87 \[ \frac {2 \left (-3 a A+3 a B x^3+3 A b x^3+b B x^6\right )}{9 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 28, normalized size = 0.72 \[ \frac {2 \, {\left (B b x^{6} + 3 \, {\left (B a + A b\right )} x^{3} - 3 \, A a\right )}}{9 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 29, normalized size = 0.74 \[ \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} - \frac {2 \, A a}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 32, normalized size = 0.82 \[ -\frac {2 \left (-B b \,x^{6}-3 A b \,x^{3}-3 B a \,x^{3}+3 A a \right )}{9 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 27, normalized size = 0.69 \[ \frac {2}{9} \, B b x^{\frac {9}{2}} + \frac {2}{3} \, {\left (B a + A b\right )} x^{\frac {3}{2}} - \frac {2 \, A a}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 31, normalized size = 0.79 \[ \frac {6\,A\,b\,x^3-6\,A\,a+6\,B\,a\,x^3+2\,B\,b\,x^6}{9\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.74, size = 46, normalized size = 1.18 \[ - \frac {2 A a}{3 x^{\frac {3}{2}}} + \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 B a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {9}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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