Optimal. Leaf size=55 \[ \frac {1}{5} a^2 A x^5+\frac {1}{6} a (2 A b+a B) x^6+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{8} b^2 B x^8 \]
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Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 77}
\begin {gather*} \frac {1}{5} a^2 A x^5+\frac {1}{7} b x^7 (2 a B+A b)+\frac {1}{6} a x^6 (a B+2 A b)+\frac {1}{8} b^2 B x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int x^4 (A+B x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int x^4 (a+b x)^2 (A+B x) \, dx\\ &=\int \left (a^2 A x^4+a (2 A b+a B) x^5+b (A b+2 a B) x^6+b^2 B x^7\right ) \, dx\\ &=\frac {1}{5} a^2 A x^5+\frac {1}{6} a (2 A b+a B) x^6+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{8} b^2 B x^8\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{5} a^2 A x^5+\frac {1}{6} a (2 A b+a B) x^6+\frac {1}{7} b (A b+2 a B) x^7+\frac {1}{8} b^2 B x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 52, normalized size = 0.95
method | result | size |
gosper | \(\frac {x^{5} \left (105 b^{2} B \,x^{3}+120 b^{2} A \,x^{2}+240 B a b \,x^{2}+280 a b A x +140 a^{2} B x +168 a^{2} A \right )}{840}\) | \(52\) |
default | \(\frac {b^{2} B \,x^{8}}{8}+\frac {\left (b^{2} A +2 a b B \right ) x^{7}}{7}+\frac {\left (2 a b A +a^{2} B \right ) x^{6}}{6}+\frac {a^{2} A \,x^{5}}{5}\) | \(52\) |
norman | \(\frac {b^{2} B \,x^{8}}{8}+\left (\frac {1}{7} b^{2} A +\frac {2}{7} a b B \right ) x^{7}+\left (\frac {1}{3} a b A +\frac {1}{6} a^{2} B \right ) x^{6}+\frac {a^{2} A \,x^{5}}{5}\) | \(52\) |
risch | \(\frac {1}{8} b^{2} B \,x^{8}+\frac {1}{7} x^{7} b^{2} A +\frac {2}{7} x^{7} a b B +\frac {1}{3} x^{6} a b A +\frac {1}{6} x^{6} a^{2} B +\frac {1}{5} a^{2} A \,x^{5}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{8} \, B b^{2} x^{8} + \frac {1}{5} \, A a^{2} x^{5} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{2} + 2 \, A a b\right )} x^{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.43, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{8} \, B b^{2} x^{8} + \frac {1}{5} \, A a^{2} x^{5} + \frac {1}{7} \, {\left (2 \, B a b + A b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B a^{2} + 2 \, A a b\right )} x^{6} \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 a^{2} x^{5}}{5} + \frac {B b^{2} x^{8}}{8} + x^{7} \left (\frac {A b^{2}}{7} + \frac {2 B a b}{7}\right ) + x^{6} \left (\frac {A a b}{3} + \frac {B a^{2}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{8} \, B b^{2} x^{8} + \frac {2}{7} \, B a b x^{7} + \frac {1}{7} \, A b^{2} x^{7} + \frac {1}{6} \, B a^{2} x^{6} + \frac {1}{3} \, A a b x^{6} + \frac {1}{5} \, A a^{2} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 51, normalized size = 0.93 \begin {gather*} x^6\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+x^7\,\left (\frac {A\,b^2}{7}+\frac {2\,B\,a\,b}{7}\right )+\frac {A\,a^2\,x^5}{5}+\frac {B\,b^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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