Optimal. Leaf size=85 \[ \frac {2}{5} A b^3 x^{5/2}+\frac {2}{7} b^2 x^{7/2} (3 A c+b B)+\frac {2}{11} c^2 x^{11/2} (A c+3 b B)+\frac {2}{3} b c x^{9/2} (A c+b B)+\frac {2}{13} B c^3 x^{13/2} \]
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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}{7} b^2 x^{7/2} (3 A c+b B)+\frac {2}{5} A b^3 x^{5/2}+\frac {2}{11} c^2 x^{11/2} (A c+3 b B)+\frac {2}{3} b c x^{9/2} (A c+b B)+\frac {2}{13} B c^3 x^{13/2} \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^{3/2}} \, dx &=\int \left (A b^3 x^{3/2}+b^2 (b B+3 A c) x^{5/2}+3 b c (b B+A c) x^{7/2}+c^2 (3 b B+A c) x^{9/2}+B c^3 x^{11/2}\right ) \, dx\\ &=\frac {2}{5} A b^3 x^{5/2}+\frac {2}{7} b^2 (b B+3 A c) x^{7/2}+\frac {2}{3} b c (b B+A c) x^{9/2}+\frac {2}{11} c^2 (3 b B+A c) x^{11/2}+\frac {2}{13} B c^3 x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.82 \begin {gather*} \frac {2 \left (B x^{5/2} (b+c x)^4-\frac {x^{5/2} \left (231 b^3+495 b^2 c x+385 b c^2 x^2+105 c^3 x^3\right ) (5 b B-13 A c)}{1155}\right )}{13 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 (3003 A b^3 x^{5/2}+6435 A b^2 c x^{7/2}+5005 A b c^2 x^{9/2}+1365 A c^3 x^{11/2}+2145 b^3 B x^{7/2}+5005 b^2 B c x^{9/2}+4095 b B c^2 x^{11/2}+1155 B c^3 x^{13/2}\right )}{15015} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{15015} \, {\left (1155 \, B c^{3} x^{6} + 3003 \, A b^{3} x^{2} + 1365 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{5} + 5005 \, {\left (B b^{2} c + A b c^{2}\right )} x^{4} + 2145 \, {\left (B b^{3} + 3 \, A b^{2} 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 = 0.15, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{13} \, B c^{3} x^{\frac {13}{2}} + \frac {6}{11} \, B b c^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A c^{3} x^{\frac {11}{2}} + \frac {2}{3} \, B b^{2} c x^{\frac {9}{2}} + \frac {2}{3} \, A b c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B b^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A b^{2} c x^{\frac {7}{2}} + \frac {2}{5} \, A b^{3} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 76, normalized size = 0.89 \begin {gather*} \frac {2 \left (1155 B \,c^{3} x^{4}+1365 A \,c^{3} x^{3}+4095 B b \,c^{2} x^{3}+5005 A b \,c^{2} x^{2}+5005 B \,b^{2} c \,x^{2}+6435 A \,b^{2} c x +2145 B \,b^{3} x +3003 A \,b^{3}\right ) x^{\frac {5}{2}}}{15015} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{13} \, B c^{3} x^{\frac {13}{2}} + \frac {2}{5} \, A b^{3} x^{\frac {5}{2}} + \frac {2}{11} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {11}{2}} + \frac {2}{3} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B b^{3} + 3 \, A b^{2} 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 = 0.03, size = 69, normalized size = 0.81 \begin {gather*} x^{7/2}\,\left (\frac {2\,B\,b^3}{7}+\frac {6\,A\,c\,b^2}{7}\right )+x^{11/2}\,\left (\frac {2\,A\,c^3}{11}+\frac {6\,B\,b\,c^2}{11}\right )+\frac {2\,A\,b^3\,x^{5/2}}{5}+\frac {2\,B\,c^3\,x^{13/2}}{13}+\frac {2\,b\,c\,x^{9/2}\,\left (A\,c+B\,b\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.73, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {5}{2}}}{5} + \frac {6 A b^{2} c x^{\frac {7}{2}}}{7} + \frac {2 A b c^{2} x^{\frac {9}{2}}}{3} + \frac {2 A c^{3} x^{\frac {11}{2}}}{11} + \frac {2 B b^{3} x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} c x^{\frac {9}{2}}}{3} + \frac {6 B b c^{2} x^{\frac {11}{2}}}{11} + \frac {2 B c^{3} x^{\frac {13}{2}}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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