Optimal. Leaf size=85 \[ \frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^2 (3 A b+a B) x^{9/2}+\frac {6}{11} a b (A b+a B) x^{11/2}+\frac {2}{13} b^2 (A b+3 a B) x^{13/2}+\frac {2}{15} b^3 B x^{15/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, 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 = {77}
\begin {gather*} \frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^2 x^{9/2} (a B+3 A b)+\frac {2}{13} b^2 x^{13/2} (3 a B+A b)+\frac {6}{11} a b x^{11/2} (a B+A b)+\frac {2}{15} b^3 B x^{15/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^{5/2} (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A x^{5/2}+a^2 (3 A b+a B) x^{7/2}+3 a b (A b+a B) x^{9/2}+b^2 (A b+3 a B) x^{11/2}+b^3 B x^{13/2}\right ) \, dx\\ &=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{9} a^2 (3 A b+a B) x^{9/2}+\frac {6}{11} a b (A b+a B) x^{11/2}+\frac {2}{13} b^2 (A b+3 a B) x^{13/2}+\frac {2}{15} b^3 B x^{15/2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 71, normalized size = 0.84 \begin {gather*} \frac {2 x^{7/2} \left (715 a^3 (9 A+7 B x)+1365 a^2 b x (11 A+9 B x)+945 a b^2 x^2 (13 A+11 B x)+231 b^3 x^3 (15 A+13 B x)\right )}{45045} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 76, normalized size = 0.89
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (3003 b^{3} B \,x^{4}+3465 A \,b^{3} x^{3}+10395 B a \,b^{2} x^{3}+12285 a A \,b^{2} x^{2}+12285 B \,a^{2} b \,x^{2}+15015 a^{2} A b x +5005 a^{3} B x +6435 a^{3} A \right )}{45045}\) | \(76\) |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {15}{2}}}{15}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\) | \(76\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (3003 b^{3} B \,x^{4}+3465 A \,b^{3} x^{3}+10395 B a \,b^{2} x^{3}+12285 a A \,b^{2} x^{2}+12285 B \,a^{2} b \,x^{2}+15015 a^{2} A b x +5005 a^{3} B x +6435 a^{3} A \right )}{45045}\) | \(76\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (3003 b^{3} B \,x^{4}+3465 A \,b^{3} x^{3}+10395 B a \,b^{2} x^{3}+12285 a A \,b^{2} x^{2}+12285 B \,a^{2} b \,x^{2}+15015 a^{2} A b x +5005 a^{3} B x +6435 a^{3} A \right )}{45045}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2}{13} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {13}{2}} + \frac {6}{11} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, B b^{3} x^{7} + 6435 \, A a^{3} x^{3} + 3465 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 12285 \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + 5005 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.50, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {2 A a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 A a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 A b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B a^{3} x^{\frac {9}{2}}}{9} + \frac {6 B a^{2} b x^{\frac {11}{2}}}{11} + \frac {6 B a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B a^{2} b x^{\frac {11}{2}} + \frac {6}{11} \, A a b^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} b x^{\frac {9}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,a^3}{9}+\frac {2\,A\,b\,a^2}{3}\right )+x^{13/2}\,\left (\frac {2\,A\,b^3}{13}+\frac {6\,B\,a\,b^2}{13}\right )+\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,b^3\,x^{15/2}}{15}+\frac {6\,a\,b\,x^{11/2}\,\left (A\,b+B\,a\right )}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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