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 = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
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*}
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Mathematica [A] time = 0.02, size = 55, normalized size = 0.87 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (693 A b^2 x^{5/2}+990 A b c x^{7/2}+385 A c^2 x^{9/2}+495 b^2 B x^{7/2}+770 b B c x^{9/2}+315 B c^2 x^{11/2}\right )}{3465} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 56, normalized size = 0.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 0.84 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 \left (315 B \,c^{2} x^{3}+385 A \,c^{2} x^{2}+770 B b c \,x^{2}+990 A b c x +495 B \,b^{2} x +693 A \,b^{2}\right ) x^{\frac {5}{2}}}{3465} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 51, normalized size = 0.81 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{7/2}\,\left (\frac {2\,B\,b^2}{7}+\frac {4\,A\,c\,b}{7}\right )+x^{9/2}\,\left (\frac {2\,A\,c^2}{9}+\frac {4\,B\,b\,c}{9}\right )+\frac {2\,A\,b^2\,x^{5/2}}{5}+\frac {2\,B\,c^2\,x^{11/2}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.57, size = 80, normalized size = 1.27 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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