Optimal. Leaf size=83 \[ 2 a^3 A \sqrt {x}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{7} b^2 (A b+3 a B) x^{7/2}+\frac {2}{9} b^3 B x^{9/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 83, 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*} 2 a^3 A \sqrt {x}+\frac {2}{3} a^2 x^{3/2} (a B+3 A b)+\frac {2}{7} b^2 x^{7/2} (3 a B+A b)+\frac {6}{5} a b x^{5/2} (a B+A b)+\frac {2}{9} b^3 B x^{9/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{\sqrt {x}} \, dx &=\int \left (\frac {a^3 A}{\sqrt {x}}+a^2 (3 A b+a B) \sqrt {x}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{5/2}+b^3 B x^{7/2}\right ) \, dx\\ &=2 a^3 A \sqrt {x}+\frac {2}{3} a^2 (3 A b+a B) x^{3/2}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{7} b^2 (A b+3 a B) x^{7/2}+\frac {2}{9} b^3 B x^{9/2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 70, normalized size = 0.84 \begin {gather*} \frac {2}{315} \sqrt {x} \left (105 a^3 (3 A+B x)+63 a^2 b x (5 A+3 B x)+27 a b^2 x^2 (7 A+5 B x)+5 b^3 x^3 (9 A+7 B x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 76, normalized size = 0.92
method | result | size |
trager | \(\left (\frac {2}{9} b^{3} B \,x^{4}+\frac {2}{7} A \,b^{3} x^{3}+\frac {6}{7} B a \,b^{2} x^{3}+\frac {6}{5} a A \,b^{2} x^{2}+\frac {6}{5} B \,a^{2} b \,x^{2}+2 a^{2} A b x +\frac {2}{3} a^{3} B x +2 a^{3} A \right ) \sqrt {x}\) | \(75\) |
gosper | \(\frac {2 \sqrt {x}\, \left (35 b^{3} B \,x^{4}+45 A \,b^{3} x^{3}+135 B a \,b^{2} x^{3}+189 a A \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+315 a^{2} A b x +105 a^{3} B x +315 a^{3} A \right )}{315}\) | \(76\) |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {9}{2}}}{9}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {3}{2}}}{3}+2 a^{3} A \sqrt {x}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {9}{2}}}{9}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {7}{2}}}{7}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {3}{2}}}{3}+2 a^{3} A \sqrt {x}\) | \(76\) |
risch | \(\frac {2 \sqrt {x}\, \left (35 b^{3} B \,x^{4}+45 A \,b^{3} x^{3}+135 B a \,b^{2} x^{3}+189 a A \,b^{2} x^{2}+189 B \,a^{2} b \,x^{2}+315 a^{2} A b x +105 a^{3} B x +315 a^{3} A \right )}{315}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{9} \, B b^{3} x^{\frac {9}{2}} + 2 \, A a^{3} \sqrt {x} + \frac {2}{7} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {7}{2}} + \frac {6}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.17, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{315} \, {\left (35 \, B b^{3} x^{4} + 315 \, A a^{3} + 45 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 189 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 110, normalized size = 1.33 \begin {gather*} 2 A a^{3} \sqrt {x} + 2 A a^{2} b x^{\frac {3}{2}} + \frac {6 A a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 A b^{3} x^{\frac {7}{2}}}{7} + \frac {2 B a^{3} x^{\frac {3}{2}}}{3} + \frac {6 B a^{2} b x^{\frac {5}{2}}}{5} + \frac {6 B a b^{2} x^{\frac {7}{2}}}{7} + \frac {2 B b^{3} x^{\frac {9}{2}}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.99, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{9} \, B b^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B a b^{2} x^{\frac {7}{2}} + \frac {2}{7} \, A b^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B a^{2} b x^{\frac {5}{2}} + \frac {6}{5} \, A a b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}} + 2 \, A a^{3} \sqrt {x} \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 = 0.83 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,a^3}{3}+2\,A\,b\,a^2\right )+x^{7/2}\,\left (\frac {2\,A\,b^3}{7}+\frac {6\,B\,a\,b^2}{7}\right )+2\,A\,a^3\,\sqrt {x}+\frac {2\,B\,b^3\,x^{9/2}}{9}+\frac {6\,a\,b\,x^{5/2}\,\left (A\,b+B\,a\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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