Optimal. Leaf size=55 \[ \frac {1}{6} A b^2 x^6+\frac {1}{7} b (b B+2 A c) x^7+\frac {1}{8} c (2 b B+A c) x^8+\frac {1}{9} B c^2 x^9 \]
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Rubi [A]
time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {779}
\begin {gather*} \frac {1}{6} A b^2 x^6+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{9} B c^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 779
Rubi steps
\begin {align*} \int x^3 (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^5+b (b B+2 A c) x^6+c (2 b B+A c) x^7+B c^2 x^8\right ) \, dx\\ &=\frac {1}{6} A b^2 x^6+\frac {1}{7} b (b B+2 A c) x^7+\frac {1}{8} c (2 b B+A c) x^8+\frac {1}{9} B c^2 x^9\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{6} A b^2 x^6+\frac {1}{7} b (b B+2 A c) x^7+\frac {1}{8} c (2 b B+A c) x^8+\frac {1}{9} B c^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 52, normalized size = 0.95
method | result | size |
gosper | \(\frac {x^{6} \left (56 B \,c^{2} x^{3}+63 A \,c^{2} x^{2}+126 b B \,x^{2} c +144 A b c x +72 b^{2} B x +84 b^{2} A \right )}{504}\) | \(52\) |
default | \(\frac {B \,c^{2} x^{9}}{9}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{8}}{8}+\frac {\left (2 A b c +b^{2} B \right ) x^{7}}{7}+\frac {A \,b^{2} x^{6}}{6}\) | \(52\) |
norman | \(\frac {B \,c^{2} x^{9}}{9}+\left (\frac {1}{8} A \,c^{2}+\frac {1}{4} b B c \right ) x^{8}+\left (\frac {2}{7} A b c +\frac {1}{7} b^{2} B \right ) x^{7}+\frac {A \,b^{2} x^{6}}{6}\) | \(52\) |
risch | \(\frac {1}{9} B \,c^{2} x^{9}+\frac {1}{8} A \,c^{2} x^{8}+\frac {1}{4} x^{8} b B c +\frac {2}{7} x^{7} A b c +\frac {1}{7} b^{2} B \,x^{7}+\frac {1}{6} A \,b^{2} x^{6}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{8} \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{2} + 2 \, A b c\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.34, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{8} \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{2} + 2 \, A b c\right )} x^{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 54, normalized size = 0.98 \begin {gather*} \frac {A b^{2} x^{6}}{6} + \frac {B c^{2} x^{9}}{9} + x^{8} \left (\frac {A c^{2}}{8} + \frac {B b c}{4}\right ) + x^{7} \cdot \left (\frac {2 A b c}{7} + \frac {B b^{2}}{7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.12, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} + \frac {1}{4} \, B b c x^{8} + \frac {1}{8} \, A c^{2} x^{8} + \frac {1}{7} \, B b^{2} x^{7} + \frac {2}{7} \, A b c x^{7} + \frac {1}{6} \, A b^{2} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 51, normalized size = 0.93 \begin {gather*} x^7\,\left (\frac {B\,b^2}{7}+\frac {2\,A\,c\,b}{7}\right )+x^8\,\left (\frac {A\,c^2}{8}+\frac {B\,b\,c}{4}\right )+\frac {A\,b^2\,x^6}{6}+\frac {B\,c^2\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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