Optimal. Leaf size=182 \[ \frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {2}{3} a \left (a b B+A \left (b^2+a c\right )\right ) x^{9/2}+\frac {2}{11} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{11/2}+\frac {2}{13} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{13/2}+\frac {2}{5} c \left (b^2 B+A b c+a B c\right ) x^{15/2}+\frac {2}{17} c^2 (3 b B+A c) x^{17/2}+\frac {2}{19} B c^3 x^{19/2} \]
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Rubi [A]
time = 0.07, antiderivative size = 182, 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 = {779}
\begin {gather*} \frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 x^{7/2} (a B+3 A b)+\frac {2}{5} c x^{15/2} \left (a B c+A b c+b^2 B\right )+\frac {2}{3} a x^{9/2} \left (A \left (a c+b^2\right )+a b B\right )+\frac {2}{13} x^{13/2} \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac {2}{11} x^{11/2} \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {2}{17} c^2 x^{17/2} (A c+3 b B)+\frac {2}{19} B c^3 x^{19/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 779
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A x^{3/2}+a^2 (3 A b+a B) x^{5/2}+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^{7/2}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{9/2}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{11/2}+3 c \left (b^2 B+A b c+a B c\right ) x^{13/2}+c^2 (3 b B+A c) x^{15/2}+B c^3 x^{17/2}\right ) \, dx\\ &=\frac {2}{5} a^3 A x^{5/2}+\frac {2}{7} a^2 (3 A b+a B) x^{7/2}+\frac {2}{3} a \left (a b B+A \left (b^2+a c\right )\right ) x^{9/2}+\frac {2}{11} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^{11/2}+\frac {2}{13} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^{13/2}+\frac {2}{5} c \left (b^2 B+A b c+a B c\right ) x^{15/2}+\frac {2}{17} c^2 (3 b B+A c) x^{17/2}+\frac {2}{19} B c^3 x^{19/2}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 178, normalized size = 0.98 \begin {gather*} \frac {2 x^{5/2} \left (138567 a^3 (7 A+5 B x)+20995 a^2 x (11 A (9 b+7 c x)+7 B x (11 b+9 c x))+2261 a 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 )+21 x^3 \left (19 A \left (1105 b^3+2805 b^2 c x+2431 b c^2 x^2+715 c^3 x^3\right )+11 B x \left (1615 b^3+4199 b^2 c x+3705 b c^2 x^2+1105 c^3 x^3\right )\right )\right )}{4849845} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 226, normalized size = 1.24
method | result | size |
gosper | \(\frac {2 x^{\frac {5}{2}} \left (255255 B \,c^{3} x^{7}+285285 A \,c^{3} x^{6}+855855 B b \,c^{2} x^{6}+969969 A b \,c^{2} x^{5}+969969 a B \,c^{2} x^{5}+969969 B \,b^{2} c \,x^{5}+1119195 a A \,c^{2} x^{4}+1119195 A \,b^{2} c \,x^{4}+2238390 B a b c \,x^{4}+373065 B \,b^{3} x^{4}+2645370 A a b c \,x^{3}+440895 A \,b^{3} x^{3}+1322685 a^{2} B c \,x^{3}+1322685 B a \,b^{2} x^{3}+1616615 a^{2} A c \,x^{2}+1616615 A a \,b^{2} x^{2}+1616615 a^{2} b B \,x^{2}+2078505 A \,a^{2} b x +692835 B \,a^{3} x +969969 A \,a^{3}\right )}{4849845}\) | \(192\) |
trager | \(\frac {2 x^{\frac {5}{2}} \left (255255 B \,c^{3} x^{7}+285285 A \,c^{3} x^{6}+855855 B b \,c^{2} x^{6}+969969 A b \,c^{2} x^{5}+969969 a B \,c^{2} x^{5}+969969 B \,b^{2} c \,x^{5}+1119195 a A \,c^{2} x^{4}+1119195 A \,b^{2} c \,x^{4}+2238390 B a b c \,x^{4}+373065 B \,b^{3} x^{4}+2645370 A a b c \,x^{3}+440895 A \,b^{3} x^{3}+1322685 a^{2} B c \,x^{3}+1322685 B a \,b^{2} x^{3}+1616615 a^{2} A c \,x^{2}+1616615 A a \,b^{2} x^{2}+1616615 a^{2} b B \,x^{2}+2078505 A \,a^{2} b x +692835 B \,a^{3} x +969969 A \,a^{3}\right )}{4849845}\) | \(192\) |
risch | \(\frac {2 x^{\frac {5}{2}} \left (255255 B \,c^{3} x^{7}+285285 A \,c^{3} x^{6}+855855 B b \,c^{2} x^{6}+969969 A b \,c^{2} x^{5}+969969 a B \,c^{2} x^{5}+969969 B \,b^{2} c \,x^{5}+1119195 a A \,c^{2} x^{4}+1119195 A \,b^{2} c \,x^{4}+2238390 B a b c \,x^{4}+373065 B \,b^{3} x^{4}+2645370 A a b c \,x^{3}+440895 A \,b^{3} x^{3}+1322685 a^{2} B c \,x^{3}+1322685 B a \,b^{2} x^{3}+1616615 a^{2} A c \,x^{2}+1616615 A a \,b^{2} x^{2}+1616615 a^{2} b B \,x^{2}+2078505 A \,a^{2} b x +692835 B \,a^{3} x +969969 A \,a^{3}\right )}{4849845}\) | \(192\) |
derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {19}{2}}}{19}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 A b \,c^{2}+B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (A \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{3} A \,x^{\frac {5}{2}}}{5}\) | \(226\) |
default | \(\frac {2 B \,c^{3} x^{\frac {19}{2}}}{19}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 A b \,c^{2}+B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (A \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{\frac {11}{2}}}{11}+\frac {2 \left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{\frac {9}{2}}}{9}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {7}{2}}}{7}+\frac {2 a^{3} A \,x^{\frac {5}{2}}}{5}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 166, normalized size = 0.91 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {2}{17} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {17}{2}} + \frac {2}{5} \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{13} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{\frac {13}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} + \frac {2}{11} \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{\frac {11}{2}} + \frac {2}{3} \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.48, size = 171, normalized size = 0.94 \begin {gather*} \frac {2}{4849845} \, {\left (255255 \, B c^{3} x^{9} + 285285 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + 969969 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{7} + 373065 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{6} + 969969 \, A a^{3} x^{2} + 440895 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{5} + 1616615 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} + 692835 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.65, size = 294, normalized size = 1.62 \begin {gather*} \frac {2 A a^{3} x^{\frac {5}{2}}}{5} + \frac {6 A a^{2} b x^{\frac {7}{2}}}{7} + \frac {2 A a^{2} c x^{\frac {9}{2}}}{3} + \frac {2 A a b^{2} x^{\frac {9}{2}}}{3} + \frac {12 A a b c x^{\frac {11}{2}}}{11} + \frac {6 A a c^{2} x^{\frac {13}{2}}}{13} + \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {6 A b^{2} c x^{\frac {13}{2}}}{13} + \frac {2 A b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {17}{2}}}{17} + \frac {2 B a^{3} x^{\frac {7}{2}}}{7} + \frac {2 B a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 B a^{2} c x^{\frac {11}{2}}}{11} + \frac {6 B a b^{2} x^{\frac {11}{2}}}{11} + \frac {12 B a b c x^{\frac {13}{2}}}{13} + \frac {2 B a c^{2} x^{\frac {15}{2}}}{5} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} + \frac {2 B b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B c^{3} x^{\frac {19}{2}}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.89, size = 193, normalized size = 1.06 \begin {gather*} \frac {2}{19} \, B c^{3} x^{\frac {19}{2}} + \frac {6}{17} \, B b c^{2} x^{\frac {17}{2}} + \frac {2}{17} \, A c^{3} x^{\frac {17}{2}} + \frac {2}{5} \, B b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, B a c^{2} x^{\frac {15}{2}} + \frac {2}{5} \, A b c^{2} x^{\frac {15}{2}} + \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {12}{13} \, B a b c x^{\frac {13}{2}} + \frac {6}{13} \, A b^{2} c x^{\frac {13}{2}} + \frac {6}{13} \, A a c^{2} x^{\frac {13}{2}} + \frac {6}{11} \, B a b^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {6}{11} \, B a^{2} c x^{\frac {11}{2}} + \frac {12}{11} \, A a b c x^{\frac {11}{2}} + \frac {2}{3} \, B a^{2} b x^{\frac {9}{2}} + \frac {2}{3} \, A a b^{2} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} c x^{\frac {9}{2}} + \frac {2}{7} \, B a^{3} x^{\frac {7}{2}} + \frac {6}{7} \, A a^{2} b x^{\frac {7}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 169, normalized size = 0.93 \begin {gather*} x^{11/2}\,\left (\frac {6\,B\,c\,a^2}{11}+\frac {6\,B\,a\,b^2}{11}+\frac {12\,A\,c\,a\,b}{11}+\frac {2\,A\,b^3}{11}\right )+x^{13/2}\,\left (\frac {2\,B\,b^3}{13}+\frac {6\,A\,b^2\,c}{13}+\frac {12\,B\,a\,b\,c}{13}+\frac {6\,A\,a\,c^2}{13}\right )+x^{7/2}\,\left (\frac {2\,B\,a^3}{7}+\frac {6\,A\,b\,a^2}{7}\right )+x^{17/2}\,\left (\frac {2\,A\,c^3}{17}+\frac {6\,B\,b\,c^2}{17}\right )+x^{9/2}\,\left (\frac {2\,B\,a^2\,b}{3}+\frac {2\,A\,c\,a^2}{3}+\frac {2\,A\,a\,b^2}{3}\right )+x^{15/2}\,\left (\frac {2\,B\,b^2\,c}{5}+\frac {2\,A\,b\,c^2}{5}+\frac {2\,B\,a\,c^2}{5}\right )+\frac {2\,A\,a^3\,x^{5/2}}{5}+\frac {2\,B\,c^3\,x^{19/2}}{19} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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