Optimal. Leaf size=61 \[ 2 a^2 A \sqrt {x}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{5} b (A b+2 a B) x^{5/2}+\frac {2}{7} b^2 B x^{7/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 61, 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^2 A \sqrt {x}+\frac {2}{5} b x^{5/2} (2 a B+A b)+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{7} b^2 B x^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{\sqrt {x}} \, dx &=\int \left (\frac {a^2 A}{\sqrt {x}}+a (2 A b+a B) \sqrt {x}+b (A b+2 a B) x^{3/2}+b^2 B x^{5/2}\right ) \, dx\\ &=2 a^2 A \sqrt {x}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{5} b (A b+2 a B) x^{5/2}+\frac {2}{7} b^2 B x^{7/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{105} \sqrt {x} \left (35 a^2 (3 A+B x)+14 a b x (5 A+3 B x)+3 b^2 x^2 (7 A+5 B x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 52, normalized size = 0.85
method | result | size |
trager | \(\left (\frac {2}{7} b^{2} B \,x^{3}+\frac {2}{5} A \,b^{2} x^{2}+\frac {4}{5} B a b \,x^{2}+\frac {4}{3} a A b x +\frac {2}{3} a^{2} B x +2 a^{2} A \right ) \sqrt {x}\) | \(51\) |
gosper | \(\frac {2 \sqrt {x}\, \left (15 b^{2} B \,x^{3}+21 A \,b^{2} x^{2}+42 B a b \,x^{2}+70 a A b x +35 a^{2} B x +105 a^{2} A \right )}{105}\) | \(52\) |
derivativedivides | \(\frac {2 b^{2} B \,x^{\frac {7}{2}}}{7}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {3}{2}}}{3}+2 a^{2} A \sqrt {x}\) | \(52\) |
default | \(\frac {2 b^{2} B \,x^{\frac {7}{2}}}{7}+\frac {2 \left (b^{2} A +2 a b B \right ) x^{\frac {5}{2}}}{5}+\frac {2 \left (2 a b A +a^{2} B \right ) x^{\frac {3}{2}}}{3}+2 a^{2} A \sqrt {x}\) | \(52\) |
risch | \(\frac {2 \sqrt {x}\, \left (15 b^{2} B \,x^{3}+21 A \,b^{2} x^{2}+42 B a b \,x^{2}+70 a A b x +35 a^{2} B x +105 a^{2} A \right )}{105}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + 2 \, A a^{2} \sqrt {x} + \frac {2}{5} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a 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 = 1.69, size = 51, normalized size = 0.84 \begin {gather*} \frac {2}{105} \, {\left (15 \, B b^{2} x^{3} + 105 \, A a^{2} + 21 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 35 \, {\left (B a^{2} + 2 \, A a 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 = 0.12, size = 78, normalized size = 1.28 \begin {gather*} 2 A a^{2} \sqrt {x} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {5}{2}}}{5} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.65, size = 53, normalized size = 0.87 \begin {gather*} \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{5} \, B a b x^{\frac {5}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} + 2 \, A a^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 51, normalized size = 0.84 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{5/2}\,\left (\frac {2\,A\,b^2}{5}+\frac {4\,B\,a\,b}{5}\right )+2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,b^2\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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