Optimal. Leaf size=53 \[ \frac {2}{3} x^{3/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{7} B c x^{7/2} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} \frac {2}{3} x^{3/2} (a B+A b)+2 a A \sqrt {x}+\frac {2}{5} x^{5/2} (A c+b B)+\frac {2}{7} B c x^{7/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{\sqrt {x}} \, dx &=\int \left (\frac {a A}{\sqrt {x}}+(A b+a B) \sqrt {x}+(b B+A c) x^{3/2}+B c x^{5/2}\right ) \, dx\\ &=2 a A \sqrt {x}+\frac {2}{3} (A b+a B) x^{3/2}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{7} B c x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 46, normalized size = 0.87 \begin {gather*} \frac {2}{105} \sqrt {x} (35 a (3 A+B x)+x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 59, normalized size = 1.11 \begin {gather*} \frac {2}{105} \left (105 a A \sqrt {x}+35 a B x^{3/2}+35 A b x^{3/2}+21 A c x^{5/2}+21 b B x^{5/2}+15 B c x^{7/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 39, normalized size = 0.74 \begin {gather*} \frac {2}{105} \, {\left (15 \, B c x^{3} + 21 \, {\left (B b + A c\right )} x^{2} + 105 \, A a + 35 \, {\left (B a + A b\right )} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 0.81 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + \frac {2}{3} \, B a x^{\frac {3}{2}} + \frac {2}{3} \, A b x^{\frac {3}{2}} + 2 \, A a \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.79 \begin {gather*} \frac {2 \left (15 B c \,x^{3}+21 A c \,x^{2}+21 B b \,x^{2}+35 A b x +35 B a x +105 A a \right ) \sqrt {x}}{105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 39, normalized size = 0.74 \begin {gather*} \frac {2}{7} \, B c x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} + 2 \, A a \sqrt {x} + \frac {2}{3} \, {\left (B a + A b\right )} 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 = 41, normalized size = 0.77 \begin {gather*} x^{3/2}\,\left (\frac {2\,A\,b}{3}+\frac {2\,B\,a}{3}\right )+x^{5/2}\,\left (\frac {2\,A\,c}{5}+\frac {2\,B\,b}{5}\right )+2\,A\,a\,\sqrt {x}+\frac {2\,B\,c\,x^{7/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 68, normalized size = 1.28 \begin {gather*} 2 A a \sqrt {x} + \frac {2 A b x^{\frac {3}{2}}}{3} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B a x^{\frac {3}{2}}}{3} + \frac {2 B b x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {7}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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