Optimal. Leaf size=63 \[ \frac {2}{9} a^2 A x^{9/2}+\frac {2}{13} b x^{13/2} (2 a B+A b)+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{15} b^2 B x^{15/2} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {76} \begin {gather*} \frac {2}{9} a^2 A x^{9/2}+\frac {2}{13} b x^{13/2} (2 a B+A b)+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{15} b^2 B x^{15/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int x^{7/2} (a+b x)^2 (A+B x) \, dx &=\int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{9/2}+b (A b+2 a B) x^{11/2}+b^2 B x^{13/2}\right ) \, dx\\ &=\frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{13} b (A b+2 a B) x^{13/2}+\frac {2}{15} b^2 B x^{15/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 x^{9/2} \left (65 a^2 (11 A+9 B x)+90 a b x (13 A+11 B x)+33 b^2 x^2 (15 A+13 B x)\right )}{6435} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (715 a^2 A x^{9/2}+585 a^2 B x^{11/2}+1170 a A b x^{11/2}+990 a b B x^{13/2}+495 A b^2 x^{13/2}+429 b^2 B x^{15/2}\right )}{6435} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{6435} \, {\left (429 \, B b^{2} x^{7} + 715 \, A a^{2} x^{4} + 495 \, {\left (2 \, B a b + A b^{2}\right )} x^{6} + 585 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B a b x^{\frac {13}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {4}{11} \, A a b x^{\frac {11}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 \left (429 B \,b^{2} x^{3}+495 A \,b^{2} x^{2}+990 B a b \,x^{2}+1170 A a b x +585 B \,a^{2} x +715 a^{2} A \right ) x^{\frac {9}{2}}}{6435} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {11}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 51, normalized size = 0.81 \begin {gather*} x^{11/2}\,\left (\frac {2\,B\,a^2}{11}+\frac {4\,A\,b\,a}{11}\right )+x^{13/2}\,\left (\frac {2\,A\,b^2}{13}+\frac {4\,B\,a\,b}{13}\right )+\frac {2\,A\,a^2\,x^{9/2}}{9}+\frac {2\,B\,b^2\,x^{15/2}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.52, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A a^{2} x^{\frac {9}{2}}}{9} + \frac {4 A a b x^{\frac {11}{2}}}{11} + \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} x^{\frac {11}{2}}}{11} + \frac {4 B a b x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} x^{\frac {15}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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