Optimal. Leaf size=43 \[ \frac {b^3 x^4}{4}+\frac {3}{5} a b^2 x^5+\frac {1}{2} a^2 b x^6+\frac {a^3 x^7}{7} \]
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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45}
\begin {gather*} \frac {a^3 x^7}{7}+\frac {1}{2} a^2 b x^6+\frac {3}{5} a b^2 x^5+\frac {b^3 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^3 x^6 \, dx &=\int x^3 (b+a x)^3 \, dx\\ &=\int \left (b^3 x^3+3 a b^2 x^4+3 a^2 b x^5+a^3 x^6\right ) \, dx\\ &=\frac {b^3 x^4}{4}+\frac {3}{5} a b^2 x^5+\frac {1}{2} a^2 b x^6+\frac {a^3 x^7}{7}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {b^3 x^4}{4}+\frac {3}{5} a b^2 x^5+\frac {1}{2} a^2 b x^6+\frac {a^3 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 36, normalized size = 0.84
method | result | size |
gosper | \(\frac {x^{4} \left (20 a^{3} x^{3}+70 a^{2} b \,x^{2}+84 a \,b^{2} x +35 b^{3}\right )}{140}\) | \(36\) |
default | \(\frac {1}{4} b^{3} x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{7} a^{3} x^{7}\) | \(36\) |
risch | \(\frac {1}{4} b^{3} x^{4}+\frac {3}{5} a \,b^{2} x^{5}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{7} a^{3} x^{7}\) | \(36\) |
norman | \(\frac {\frac {1}{7} a^{3} x^{9}+\frac {1}{4} b^{3} x^{6}+\frac {3}{5} a \,b^{2} x^{7}+\frac {1}{2} a^{2} b \,x^{8}}{x^{2}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{3} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 37, normalized size = 0.86 \begin {gather*} \frac {a^{3} x^{7}}{7} + \frac {a^{2} b x^{6}}{2} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.68, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{7} \, a^{3} x^{7} + \frac {1}{2} \, a^{2} b x^{6} + \frac {3}{5} \, a b^{2} x^{5} + \frac {1}{4} \, b^{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 = 35, normalized size = 0.81 \begin {gather*} \frac {a^3\,x^7}{7}+\frac {a^2\,b\,x^6}{2}+\frac {3\,a\,b^2\,x^5}{5}+\frac {b^3\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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