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 = 0.0237014, antiderivative size = 53, normalized size of antiderivative = 1., 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} \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{7} B c x^{7/2} \]
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*}
Mathematica [A] time = 0.0474787, size = 46, normalized size = 0.87 \[ \frac{2}{105} \sqrt{x} (35 a (3 A+B x)+x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 42, normalized size = 0.8 \begin{align*}{\frac{30\,Bc{x}^{3}+42\,Ac{x}^{2}+42\,Bb{x}^{2}+70\,Abx+70\,aBx+210\,aA}{105}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0119, size = 53, normalized size = 1. \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19683, size = 105, normalized size = 1.98 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.811051, size = 68, normalized size = 1.28 \begin{align*} 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16998, size = 58, normalized size = 1.09 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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