Optimal. Leaf size=85 \[ \frac {2}{3} A b^3 x^{3/2}+\frac {2}{5} b^2 x^{5/2} (3 A c+b B)+\frac {2}{9} c^2 x^{9/2} (A c+3 b B)+\frac {6}{7} b c x^{7/2} (A c+b B)+\frac {2}{11} B c^3 x^{11/2} \]
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Rubi [A] time = 0.04, antiderivative size = 85, 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} b^2 x^{5/2} (3 A c+b B)+\frac {2}{3} A b^3 x^{3/2}+\frac {2}{9} c^2 x^{9/2} (A c+3 b B)+\frac {6}{7} b c x^{7/2} (A c+b B)+\frac {2}{11} B c^3 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 )^3}{x^{5/2}} \, dx &=\int \left (A b^3 \sqrt {x}+b^2 (b B+3 A c) x^{3/2}+3 b c (b B+A c) x^{5/2}+c^2 (3 b B+A c) x^{7/2}+B c^3 x^{9/2}\right ) \, dx\\ &=\frac {2}{3} A b^3 x^{3/2}+\frac {2}{5} b^2 (b B+3 A c) x^{5/2}+\frac {6}{7} b c (b B+A c) x^{7/2}+\frac {2}{9} c^2 (3 b B+A c) x^{9/2}+\frac {2}{11} B c^3 x^{11/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{3/2} (b+c x)^4-\frac {1}{315} x^{3/2} \left (105 b^3+189 b^2 c x+135 b c^2 x^2+35 c^3 x^3\right ) (3 b B-11 A c)\right )}{11 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (1155 A b^3 x^{3/2}+2079 A b^2 c x^{5/2}+1485 A b c^2 x^{7/2}+385 A c^3 x^{9/2}+693 b^3 B x^{5/2}+1485 b^2 B c x^{7/2}+1155 b B c^2 x^{9/2}+315 B c^3 x^{11/2}\right )}{3465} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 76, normalized size = 0.89 \begin {gather*} \frac {2}{3465} \, {\left (315 \, B c^{3} x^{5} + 1155 \, A b^{3} x + 385 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 1485 \, {\left (B b^{2} c + A b c^{2}\right )} x^{3} + 693 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{2}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{11} \, B c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B b c^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A c^{3} x^{\frac {9}{2}} + \frac {6}{7} \, B b^{2} c x^{\frac {7}{2}} + \frac {6}{7} \, A b c^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + \frac {6}{5} \, A b^{2} c x^{\frac {5}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (315 B \,c^{3} x^{4}+385 A \,c^{3} x^{3}+1155 B b \,c^{2} x^{3}+1485 A b \,c^{2} x^{2}+1485 B \,b^{2} c \,x^{2}+2079 A \,b^{2} c x +693 B \,b^{3} x +1155 A \,b^{3}\right ) x^{\frac {3}{2}}}{3465} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{11} \, B c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} + \frac {2}{9} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {9}{2}} + \frac {6}{7} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{5/2}\,\left (\frac {2\,B\,b^3}{5}+\frac {6\,A\,c\,b^2}{5}\right )+x^{9/2}\,\left (\frac {2\,A\,c^3}{9}+\frac {2\,B\,b\,c^2}{3}\right )+\frac {2\,A\,b^3\,x^{3/2}}{3}+\frac {2\,B\,c^3\,x^{11/2}}{11}+\frac {6\,b\,c\,x^{7/2}\,\left (A\,c+B\,b\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.35, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {3}{2}}}{3} + \frac {6 A b^{2} c x^{\frac {5}{2}}}{5} + \frac {6 A b c^{2} x^{\frac {7}{2}}}{7} + \frac {2 A c^{3} x^{\frac {9}{2}}}{9} + \frac {2 B b^{3} x^{\frac {5}{2}}}{5} + \frac {6 B b^{2} c x^{\frac {7}{2}}}{7} + \frac {2 B b c^{2} x^{\frac {9}{2}}}{3} + \frac {2 B c^{3} x^{\frac {11}{2}}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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