Optimal. Leaf size=63 \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{11} B c^2 x^{11/2} \]
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Rubi [A] time = 0.0299997, antiderivative size = 63, normalized size of antiderivative = 1., 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 = {765} \[ \frac{2}{5} A b^2 x^{5/2}+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{7} b x^{7/2} (2 A c+b B)+\frac{2}{11} B c^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^2}{\sqrt{x}} \, dx &=\int \left (A b^2 x^{3/2}+b (b B+2 A c) x^{5/2}+c (2 b B+A c) x^{7/2}+B c^2 x^{9/2}\right ) \, dx\\ &=\frac{2}{5} A b^2 x^{5/2}+\frac{2}{7} b (b B+2 A c) x^{7/2}+\frac{2}{9} c (2 b B+A c) x^{9/2}+\frac{2}{11} B c^2 x^{11/2}\\ \end{align*}
Mathematica [A] time = 0.0165894, size = 55, normalized size = 0.87 \[ \frac{2 x^{5/2} \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )}{3465} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{630\,B{c}^{2}{x}^{3}+770\,A{c}^{2}{x}^{2}+1540\,B{x}^{2}bc+1980\,Abcx+990\,{b}^{2}Bx+1386\,A{b}^{2}}{3465}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0969, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75695, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c^{2} x^{5} + 693 \, A b^{2} x^{2} + 385 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \,{\left (B b^{2} + 2 \, A b c\right )} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.47915, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + \frac{4 A b c x^{\frac{7}{2}}}{7} + \frac{2 A c^{2} x^{\frac{9}{2}}}{9} + \frac{2 B b^{2} x^{\frac{7}{2}}}{7} + \frac{4 B b c x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{11}{2}}}{11} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12664, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{2}} + \frac{4}{9} \, B b c x^{\frac{9}{2}} + \frac{2}{9} \, A c^{2} x^{\frac{9}{2}} + \frac{2}{7} \, B b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A b c x^{\frac{7}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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