Optimal. Leaf size=75 \[ \frac {1}{4} a^3 A x^4+\frac {1}{5} a^2 (3 A b+a B) x^5+\frac {1}{2} a b (A b+a B) x^6+\frac {1}{7} b^2 (A b+3 a B) x^7+\frac {1}{8} b^3 B x^8 \]
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Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77}
\begin {gather*} \frac {1}{4} a^3 A x^4+\frac {1}{5} a^2 x^5 (a B+3 A b)+\frac {1}{7} b^2 x^7 (3 a B+A b)+\frac {1}{2} a b x^6 (a B+A b)+\frac {1}{8} b^3 B x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int x^3 (a+b x)^3 (A+B x) \, dx &=\int \left (a^3 A x^3+a^2 (3 A b+a B) x^4+3 a b (A b+a B) x^5+b^2 (A b+3 a B) x^6+b^3 B x^7\right ) \, dx\\ &=\frac {1}{4} a^3 A x^4+\frac {1}{5} a^2 (3 A b+a B) x^5+\frac {1}{2} a b (A b+a B) x^6+\frac {1}{7} b^2 (A b+3 a B) x^7+\frac {1}{8} b^3 B x^8\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{4} a^3 A x^4+\frac {1}{5} a^2 (3 A b+a B) x^5+\frac {1}{2} a b (A b+a B) x^6+\frac {1}{7} b^2 (A b+3 a B) x^7+\frac {1}{8} b^3 B x^8 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 76, normalized size = 1.01
method | result | size |
norman | \(\frac {b^{3} B \,x^{8}}{8}+\left (\frac {1}{7} b^{3} A +\frac {3}{7} a \,b^{2} B \right ) x^{7}+\left (\frac {1}{2} a \,b^{2} A +\frac {1}{2} a^{2} b B \right ) x^{6}+\left (\frac {3}{5} a^{2} b A +\frac {1}{5} a^{3} B \right ) x^{5}+\frac {a^{3} A \,x^{4}}{4}\) | \(75\) |
default | \(\frac {b^{3} B \,x^{8}}{8}+\frac {\left (b^{3} A +3 a \,b^{2} B \right ) x^{7}}{7}+\frac {\left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{6}}{6}+\frac {\left (3 a^{2} b A +a^{3} B \right ) x^{5}}{5}+\frac {a^{3} A \,x^{4}}{4}\) | \(76\) |
gosper | \(\frac {1}{8} b^{3} B \,x^{8}+\frac {1}{7} x^{7} b^{3} A +\frac {3}{7} x^{7} a \,b^{2} B +\frac {1}{2} x^{6} a \,b^{2} A +\frac {1}{2} x^{6} a^{2} b B +\frac {3}{5} x^{5} a^{2} b A +\frac {1}{5} x^{5} a^{3} B +\frac {1}{4} a^{3} A \,x^{4}\) | \(78\) |
risch | \(\frac {1}{8} b^{3} B \,x^{8}+\frac {1}{7} x^{7} b^{3} A +\frac {3}{7} x^{7} a \,b^{2} B +\frac {1}{2} x^{6} a \,b^{2} A +\frac {1}{2} x^{6} a^{2} b B +\frac {3}{5} x^{5} a^{2} b A +\frac {1}{5} x^{5} a^{3} B +\frac {1}{4} a^{3} A \,x^{4}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{8} \, B b^{3} x^{8} + \frac {1}{4} \, A a^{3} x^{4} + \frac {1}{7} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.97, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{8} \, B b^{3} x^{8} + \frac {1}{4} \, A a^{3} x^{4} + \frac {1}{7} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + \frac {1}{2} \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 80, normalized size = 1.07 \begin {gather*} \frac {A a^{3} x^{4}}{4} + \frac {B b^{3} x^{8}}{8} + x^{7} \left (\frac {A b^{3}}{7} + \frac {3 B a b^{2}}{7}\right ) + x^{6} \left (\frac {A a b^{2}}{2} + \frac {B a^{2} b}{2}\right ) + x^{5} \cdot \left (\frac {3 A a^{2} b}{5} + \frac {B a^{3}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{8} \, B b^{3} x^{8} + \frac {3}{7} \, B a b^{2} x^{7} + \frac {1}{7} \, A b^{3} x^{7} + \frac {1}{2} \, B a^{2} b x^{6} + \frac {1}{2} \, A a b^{2} x^{6} + \frac {1}{5} \, B a^{3} x^{5} + \frac {3}{5} \, A a^{2} b x^{5} + \frac {1}{4} \, A a^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 69, normalized size = 0.92 \begin {gather*} x^5\,\left (\frac {B\,a^3}{5}+\frac {3\,A\,b\,a^2}{5}\right )+x^7\,\left (\frac {A\,b^3}{7}+\frac {3\,B\,a\,b^2}{7}\right )+\frac {A\,a^3\,x^4}{4}+\frac {B\,b^3\,x^8}{8}+\frac {a\,b\,x^6\,\left (A\,b+B\,a\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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