Optimal. Leaf size=113 \[ \frac {2}{5} a^2 A x^{5/2}+\frac {2}{11} x^{11/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{9} x^{9/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{7} a x^{7/2} (a B+2 A b)+\frac {2}{13} c x^{13/2} (A c+2 b B)+\frac {2}{15} B c^2 x^{15/2} \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, 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} \[ \frac {2}{5} a^2 A x^{5/2}+\frac {2}{11} x^{11/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{9} x^{9/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{7} a x^{7/2} (a B+2 A b)+\frac {2}{13} c x^{13/2} (A c+2 b B)+\frac {2}{15} B c^2 x^{15/2} \]
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{3/2}+a (2 A b+a B) x^{5/2}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{7/2}+\left (b^2 B+2 A b c+2 a B c\right ) x^{9/2}+c (2 b B+A c) x^{11/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac {2}{5} a^2 A x^{5/2}+\frac {2}{7} a (2 A b+a B) x^{7/2}+\frac {2}{9} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{9/2}+\frac {2}{11} \left (b^2 B+2 A b c+2 a B c\right ) x^{11/2}+\frac {2}{13} c (2 b B+A c) x^{13/2}+\frac {2}{15} B c^2 x^{15/2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 102, normalized size = 0.90 \[ \frac {2 x^{5/2} \left (1287 a^2 (7 A+5 B x)+130 a x (11 A (9 b+7 c x)+7 B x (11 b+9 c x))+7 x^2 \left (5 A \left (143 b^2+234 b c x+99 c^2 x^2\right )+3 B x \left (195 b^2+330 b c x+143 c^2 x^2\right )\right )\right )}{45045} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 98, normalized size = 0.87 \[ \frac {2}{45045} \, {\left (3003 \, B c^{2} x^{7} + 3465 \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + 4095 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{5} + 9009 \, A a^{2} x^{2} + 5005 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{4} + 6435 \, {\left (B a^{2} + 2 \, A a b\right )} x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 103, normalized size = 0.91 \[ \frac {2}{15} \, B c^{2} x^{\frac {15}{2}} + \frac {4}{13} \, B b c x^{\frac {13}{2}} + \frac {2}{13} \, A c^{2} x^{\frac {13}{2}} + \frac {2}{11} \, B b^{2} x^{\frac {11}{2}} + \frac {4}{11} \, B a c x^{\frac {11}{2}} + \frac {4}{11} \, A b c x^{\frac {11}{2}} + \frac {4}{9} \, B a b x^{\frac {9}{2}} + \frac {2}{9} \, A b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A a c x^{\frac {9}{2}} + \frac {2}{7} \, B a^{2} x^{\frac {7}{2}} + \frac {4}{7} \, A a b x^{\frac {7}{2}} + \frac {2}{5} \, A a^{2} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 102, normalized size = 0.90 \[ \frac {2 \left (3003 B \,c^{2} x^{5}+3465 A \,c^{2} x^{4}+6930 x^{4} b B c +8190 x^{3} A b c +8190 B a c \,x^{3}+4095 B \,b^{2} x^{3}+10010 A a c \,x^{2}+5005 A \,b^{2} x^{2}+10010 B a b \,x^{2}+12870 A a b x +6435 B \,a^{2} x +9009 A \,a^{2}\right ) x^{\frac {5}{2}}}{45045} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 93, normalized size = 0.82 \[ \frac {2}{15} \, B c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {11}{2}} + \frac {2}{5} \, A a^{2} x^{\frac {5}{2}} + \frac {2}{9} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 93, normalized size = 0.82 \[ x^{7/2}\,\left (\frac {2\,B\,a^2}{7}+\frac {4\,A\,b\,a}{7}\right )+x^{13/2}\,\left (\frac {2\,A\,c^2}{13}+\frac {4\,B\,b\,c}{13}\right )+x^{9/2}\,\left (\frac {2\,A\,b^2}{9}+\frac {4\,B\,a\,b}{9}+\frac {4\,A\,a\,c}{9}\right )+x^{11/2}\,\left (\frac {2\,B\,b^2}{11}+\frac {4\,A\,c\,b}{11}+\frac {4\,B\,a\,c}{11}\right )+\frac {2\,A\,a^2\,x^{5/2}}{5}+\frac {2\,B\,c^2\,x^{15/2}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.77, size = 162, normalized size = 1.43 \[ \frac {2 A a^{2} x^{\frac {5}{2}}}{5} + \frac {4 A a b x^{\frac {7}{2}}}{7} + \frac {4 A a c x^{\frac {9}{2}}}{9} + \frac {2 A b^{2} x^{\frac {9}{2}}}{9} + \frac {4 A b c x^{\frac {11}{2}}}{11} + \frac {2 A c^{2} x^{\frac {13}{2}}}{13} + \frac {2 B a^{2} x^{\frac {7}{2}}}{7} + \frac {4 B a b x^{\frac {9}{2}}}{9} + \frac {4 B a c x^{\frac {11}{2}}}{11} + \frac {2 B b^{2} x^{\frac {11}{2}}}{11} + \frac {4 B b c x^{\frac {13}{2}}}{13} + \frac {2 B c^{2} x^{\frac {15}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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