Optimal. Leaf size=113 \[ \frac {2}{9} a^2 A x^{9/2}+\frac {2}{15} x^{15/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{13} x^{13/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{17} c x^{17/2} (A c+2 b B)+\frac {2}{19} B c^2 x^{19/2} \]
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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} \begin {gather*} \frac {2}{9} a^2 A x^{9/2}+\frac {2}{15} x^{15/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{13} x^{13/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{11} a x^{11/2} (a B+2 A b)+\frac {2}{17} c x^{17/2} (A c+2 b B)+\frac {2}{19} B c^2 x^{19/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int x^{7/2} (A+B x) \left (a+b x+c x^2\right )^2 \, dx &=\int \left (a^2 A x^{7/2}+a (2 A b+a B) x^{9/2}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{11/2}+\left (b^2 B+2 A b c+2 a B c\right ) x^{13/2}+c (2 b B+A c) x^{15/2}+B c^2 x^{17/2}\right ) \, dx\\ &=\frac {2}{9} a^2 A x^{9/2}+\frac {2}{11} a (2 A b+a B) x^{11/2}+\frac {2}{13} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{13/2}+\frac {2}{15} \left (b^2 B+2 A b c+2 a B c\right ) x^{15/2}+\frac {2}{17} c (2 b B+A c) x^{17/2}+\frac {2}{19} B c^2 x^{19/2}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 102, normalized size = 0.90 \begin {gather*} \frac {2 x^{9/2} \left (20995 a^2 (11 A+9 B x)+1938 a x (15 A (13 b+11 c x)+11 B x (15 b+13 c x))+33 x^2 \left (19 A \left (255 b^2+442 b c x+195 c^2 x^2\right )+13 B x \left (323 b^2+570 b c x+255 c^2 x^2\right )\right )\right )}{2078505} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 131, normalized size = 1.16 \begin {gather*} \frac {2 \left (230945 a^2 A x^{9/2}+188955 a^2 B x^{11/2}+377910 a A b x^{11/2}+319770 a A c x^{13/2}+319770 a b B x^{13/2}+277134 a B c x^{15/2}+159885 A b^2 x^{13/2}+277134 A b c x^{15/2}+122265 A c^2 x^{17/2}+138567 b^2 B x^{15/2}+244530 b B c x^{17/2}+109395 B c^2 x^{19/2}\right )}{2078505} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 98, normalized size = 0.87 \begin {gather*} \frac {2}{2078505} \, {\left (109395 \, B c^{2} x^{9} + 122265 \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + 138567 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{7} + 230945 \, A a^{2} x^{4} + 159885 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{6} + 188955 \, {\left (B a^{2} + 2 \, A a b\right )} x^{5}\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 = 103, normalized size = 0.91 \begin {gather*} \frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {4}{17} \, B b c x^{\frac {17}{2}} + \frac {2}{17} \, A c^{2} x^{\frac {17}{2}} + \frac {2}{15} \, B b^{2} x^{\frac {15}{2}} + \frac {4}{15} \, B a c x^{\frac {15}{2}} + \frac {4}{15} \, A b c x^{\frac {15}{2}} + \frac {4}{13} \, B a b x^{\frac {13}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {4}{13} \, A a c x^{\frac {13}{2}} + \frac {2}{11} \, B a^{2} x^{\frac {11}{2}} + \frac {4}{11} \, A a b x^{\frac {11}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 102, normalized size = 0.90 \begin {gather*} \frac {2 \left (109395 B \,c^{2} x^{5}+122265 A \,c^{2} x^{4}+244530 x^{4} b B c +277134 x^{3} A b c +277134 B a c \,x^{3}+138567 B \,b^{2} x^{3}+319770 A a c \,x^{2}+159885 A \,b^{2} x^{2}+319770 B a b \,x^{2}+377910 A a b x +188955 B \,a^{2} x +230945 A \,a^{2}\right ) x^{\frac {9}{2}}}{2078505} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 93, normalized size = 0.82 \begin {gather*} \frac {2}{19} \, B c^{2} x^{\frac {19}{2}} + \frac {2}{17} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{15} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {15}{2}} + \frac {2}{9} \, A a^{2} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {13}{2}} + \frac {2}{11} \, {\left (B a^{2} + 2 \, A a b\right )} x^{\frac {11}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 93, normalized size = 0.82 \begin {gather*} x^{11/2}\,\left (\frac {2\,B\,a^2}{11}+\frac {4\,A\,b\,a}{11}\right )+x^{17/2}\,\left (\frac {2\,A\,c^2}{17}+\frac {4\,B\,b\,c}{17}\right )+x^{13/2}\,\left (\frac {2\,A\,b^2}{13}+\frac {4\,B\,a\,b}{13}+\frac {4\,A\,a\,c}{13}\right )+x^{15/2}\,\left (\frac {2\,B\,b^2}{15}+\frac {4\,A\,c\,b}{15}+\frac {4\,B\,a\,c}{15}\right )+\frac {2\,A\,a^2\,x^{9/2}}{9}+\frac {2\,B\,c^2\,x^{19/2}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.11, size = 162, normalized size = 1.43 \begin {gather*} \frac {2 A a^{2} x^{\frac {9}{2}}}{9} + \frac {4 A a b x^{\frac {11}{2}}}{11} + \frac {4 A a c x^{\frac {13}{2}}}{13} + \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {4 A b c x^{\frac {15}{2}}}{15} + \frac {2 A c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B a^{2} x^{\frac {11}{2}}}{11} + \frac {4 B a b x^{\frac {13}{2}}}{13} + \frac {4 B a c x^{\frac {15}{2}}}{15} + \frac {2 B b^{2} x^{\frac {15}{2}}}{15} + \frac {4 B b c x^{\frac {17}{2}}}{17} + \frac {2 B c^{2} x^{\frac {19}{2}}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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