Optimal. Leaf size=63 \[ \frac {2}{3} a^2 A x^{3/2}+\frac {2}{9} a (2 A b+a B) x^{9/2}+\frac {2}{15} b (A b+2 a B) x^{15/2}+\frac {2}{21} b^2 B x^{21/2} \]
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Rubi [A]
time = 0.02, 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 = {459}
\begin {gather*} \frac {2}{3} a^2 A x^{3/2}+\frac {2}{15} b x^{15/2} (2 a B+A b)+\frac {2}{9} a x^{9/2} (a B+2 A b)+\frac {2}{21} b^2 B x^{21/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b x^3\right )^2 \left (A+B x^3\right ) \, dx &=\int \left (a^2 A \sqrt {x}+a (2 A b+a B) x^{7/2}+b (A b+2 a B) x^{13/2}+b^2 B x^{19/2}\right ) \, dx\\ &=\frac {2}{3} a^2 A x^{3/2}+\frac {2}{9} a (2 A b+a B) x^{9/2}+\frac {2}{15} b (A b+2 a B) x^{15/2}+\frac {2}{21} b^2 B x^{21/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 59, normalized size = 0.94 \begin {gather*} \frac {2}{315} x^{3/2} \left (35 a^2 \left (3 A+B x^3\right )+14 a b x^3 \left (5 A+3 B x^3\right )+3 b^2 x^6 \left (7 A+5 B x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 52, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\) | \(52\) |
default | \(\frac {2 b^{2} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{2} A \,x^{\frac {3}{2}}}{3}\) | \(52\) |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (15 b^{2} B \,x^{9}+21 A \,b^{2} x^{6}+42 B a b \,x^{6}+70 a A b \,x^{3}+35 a^{2} B \,x^{3}+105 a^{2} A \right )}{315}\) | \(56\) |
trager | \(\frac {2 x^{\frac {3}{2}} \left (15 b^{2} B \,x^{9}+21 A \,b^{2} x^{6}+42 B a b \,x^{6}+70 a A b \,x^{3}+35 a^{2} B \,x^{3}+105 a^{2} A \right )}{315}\) | \(56\) |
risch | \(\frac {2 x^{\frac {3}{2}} \left (15 b^{2} B \,x^{9}+21 A \,b^{2} x^{6}+42 B a b \,x^{6}+70 a A b \,x^{3}+35 a^{2} B \,x^{3}+105 a^{2} A \right )}{315}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{21} \, B b^{2} x^{\frac {21}{2}} + \frac {2}{15} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {15}{2}} + \frac {2}{9} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.47, size = 54, normalized size = 0.86 \begin {gather*} \frac {2}{315} \, {\left (15 \, B b^{2} x^{10} + 21 \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + 35 \, {\left (B a^{2} + 2 \, A a b\right )} x^{4} + 105 \, A a^{2} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.77, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A a^{2} x^{\frac {3}{2}}}{3} + \frac {4 A a b x^{\frac {9}{2}}}{9} + \frac {2 A b^{2} x^{\frac {15}{2}}}{15} + \frac {2 B a^{2} x^{\frac {9}{2}}}{9} + \frac {4 B a b x^{\frac {15}{2}}}{15} + \frac {2 B b^{2} x^{\frac {21}{2}}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.57, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{21} \, B b^{2} x^{\frac {21}{2}} + \frac {4}{15} \, B a b x^{\frac {15}{2}} + \frac {2}{15} \, A b^{2} x^{\frac {15}{2}} + \frac {2}{9} \, B a^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A a b x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} x^{\frac {3}{2}} \end {gather*}
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 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,a^2}{9}+\frac {4\,A\,b\,a}{9}\right )+x^{15/2}\,\left (\frac {2\,A\,b^2}{15}+\frac {4\,B\,a\,b}{15}\right )+\frac {2\,A\,a^2\,x^{3/2}}{3}+\frac {2\,B\,b^2\,x^{21/2}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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