Optimal. Leaf size=111 \[ 2 a^2 A \sqrt{x}+\frac{2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{11} B c^2 x^{11/2} \]
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Rubi [A] time = 0.0563842, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {765} \[ 2 a^2 A \sqrt{x}+\frac{2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{3} a x^{3/2} (a B+2 A b)+\frac{2}{9} c x^{9/2} (A c+2 b B)+\frac{2}{11} B c^2 x^{11/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{\sqrt{x}} \, dx &=\int \left (\frac{a^2 A}{\sqrt{x}}+a (2 A b+a B) \sqrt{x}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{3/2}+\left (b^2 B+2 A b c+2 a B c\right ) x^{5/2}+c (2 b B+A c) x^{7/2}+B c^2 x^{9/2}\right ) \, dx\\ &=2 a^2 A \sqrt{x}+\frac{2}{3} a (2 A b+a B) x^{3/2}+\frac{2}{5} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{5/2}+\frac{2}{7} \left (b^2 B+2 A b c+2 a B c\right ) 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 = 0.123766, size = 100, normalized size = 0.9 \[ \frac{2 \sqrt{x} \left (1155 a^2 (3 A+B x)+66 a x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))+x^2 \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )\right )}{3465} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 102, normalized size = 0.9 \begin{align*}{\frac{630\,B{c}^{2}{x}^{5}+770\,A{c}^{2}{x}^{4}+1540\,B{x}^{4}bc+1980\,A{x}^{3}bc+1980\,aBc{x}^{3}+990\,{b}^{2}B{x}^{3}+2772\,aAc{x}^{2}+1386\,A{b}^{2}{x}^{2}+2772\,B{x}^{2}ab+4620\,aAbx+2310\,{a}^{2}Bx+6930\,A{a}^{2}}{3465}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0844, size = 126, normalized size = 1.14 \begin{align*} \frac{2}{11} \, B c^{2} x^{\frac{11}{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 \,{\left (B a + A b\right )} c\right )} x^{\frac{7}{2}} + 2 \, A a^{2} \sqrt{x} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B a^{2} + 2 \, 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 = 1.01699, size = 235, normalized size = 2.12 \begin{align*} \frac{2}{3465} \,{\left (315 \, B c^{2} x^{5} + 385 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 3465 \, A a^{2} + 693 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 1155 \,{\left (B a^{2} + 2 \, 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 = 2.10094, size = 160, normalized size = 1.44 \begin{align*} 2 A a^{2} \sqrt{x} + \frac{4 A a b x^{\frac{3}{2}}}{3} + \frac{4 A a c x^{\frac{5}{2}}}{5} + \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 a^{2} x^{\frac{3}{2}}}{3} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{4 B a c x^{\frac{7}{2}}}{7} + \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 = 1.19812, size = 139, normalized size = 1.25 \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} \, B a c x^{\frac{7}{2}} + \frac{4}{7} \, A b c 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{4}{5} \, A a c 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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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