Optimal. Leaf size=63 \[ -\frac {2 a^2 A}{3 x^{3/2}}+\frac {2}{9} b x^{9/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{15} b^2 B x^{15/2} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {448} \[ -\frac {2 a^2 A}{3 x^{3/2}}+\frac {2}{9} b x^{9/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{15} b^2 B x^{15/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2 \left (A+B x^3\right )}{x^{5/2}} \, dx &=\int \left (\frac {a^2 A}{x^{5/2}}+a (2 A b+a B) \sqrt {x}+b (A b+2 a B) x^{7/2}+b^2 B x^{13/2}\right ) \, dx\\ &=-\frac {2 a^2 A}{3 x^{3/2}}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{9} b (A b+2 a B) x^{9/2}+\frac {2}{15} b^2 B x^{15/2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 57, normalized size = 0.90 \[ \frac {-30 a^2 \left (A-B x^3\right )+20 a b x^3 \left (3 A+B x^3\right )+2 b^2 x^6 \left (5 A+3 B x^3\right )}{45 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 53, normalized size = 0.84 \[ \frac {2 \, {\left (3 \, B b^{2} x^{9} + 5 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3} - 15 \, A a^{2}\right )}}{45 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 0.84 \[ \frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {4}{9} \, B a b x^{\frac {9}{2}} + \frac {2}{9} \, A b^{2} x^{\frac {9}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} - \frac {2 \, A a^{2}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 0.89 \[ -\frac {2 \left (-3 b^{2} B \,x^{9}-5 A \,b^{2} x^{6}-10 B a b \,x^{6}-30 A a b \,x^{3}-15 B \,a^{2} x^{3}+15 a^{2} A \right )}{45 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 51, normalized size = 0.81 \[ \frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {2}{9} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {9}{2}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {3}{2}} - \frac {2 \, A a^{2}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.81 \[ x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{9/2}\,\left (\frac {2\,A\,b^2}{9}+\frac {4\,B\,a\,b}{9}\right )-\frac {2\,A\,a^2}{3\,x^{3/2}}+\frac {2\,B\,b^2\,x^{15/2}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.74, size = 80, normalized size = 1.27 \[ - \frac {2 A a^{2}}{3 x^{\frac {3}{2}}} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {2 A b^{2} x^{\frac {9}{2}}}{9} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {9}{2}}}{9} + \frac {2 B b^{2} x^{\frac {15}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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