Optimal. Leaf size=47 \[ \frac {a^2 (a+b x)^6}{6 b^3}-\frac {2 a (a+b x)^7}{7 b^3}+\frac {(a+b x)^8}{8 b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {a^2 (a+b x)^6}{6 b^3}+\frac {(a+b x)^8}{8 b^3}-\frac {2 a (a+b x)^7}{7 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x^2 (a+b x)^5 \, dx &=\int \left (\frac {a^2 (a+b x)^5}{b^2}-\frac {2 a (a+b x)^6}{b^2}+\frac {(a+b x)^7}{b^2}\right ) \, dx\\ &=\frac {a^2 (a+b x)^6}{6 b^3}-\frac {2 a (a+b x)^7}{7 b^3}+\frac {(a+b x)^8}{8 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 67, normalized size = 1.43 \begin {gather*} \frac {a^5 x^3}{3}+\frac {5}{4} a^4 b x^4+2 a^3 b^2 x^5+\frac {5}{3} a^2 b^3 x^6+\frac {5}{7} a b^4 x^7+\frac {b^5 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 58, normalized size = 1.23
method | result | size |
gosper | \(\frac {1}{8} b^{5} x^{8}+\frac {5}{7} a \,b^{4} x^{7}+\frac {5}{3} a^{2} b^{3} x^{6}+2 a^{3} b^{2} x^{5}+\frac {5}{4} a^{4} b \,x^{4}+\frac {1}{3} a^{5} x^{3}\) | \(58\) |
default | \(\frac {1}{8} b^{5} x^{8}+\frac {5}{7} a \,b^{4} x^{7}+\frac {5}{3} a^{2} b^{3} x^{6}+2 a^{3} b^{2} x^{5}+\frac {5}{4} a^{4} b \,x^{4}+\frac {1}{3} a^{5} x^{3}\) | \(58\) |
norman | \(\frac {1}{8} b^{5} x^{8}+\frac {5}{7} a \,b^{4} x^{7}+\frac {5}{3} a^{2} b^{3} x^{6}+2 a^{3} b^{2} x^{5}+\frac {5}{4} a^{4} b \,x^{4}+\frac {1}{3} a^{5} x^{3}\) | \(58\) |
risch | \(\frac {1}{8} b^{5} x^{8}+\frac {5}{7} a \,b^{4} x^{7}+\frac {5}{3} a^{2} b^{3} x^{6}+2 a^{3} b^{2} x^{5}+\frac {5}{4} a^{4} b \,x^{4}+\frac {1}{3} a^{5} x^{3}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 57, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} + \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{4} \, a^{4} b x^{4} + \frac {1}{3} \, a^{5} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 57, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} + \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{4} \, a^{4} b x^{4} + \frac {1}{3} \, a^{5} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 65, normalized size = 1.38 \begin {gather*} \frac {a^{5} x^{3}}{3} + \frac {5 a^{4} b x^{4}}{4} + 2 a^{3} b^{2} x^{5} + \frac {5 a^{2} b^{3} x^{6}}{3} + \frac {5 a b^{4} x^{7}}{7} + \frac {b^{5} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.33, size = 57, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, b^{5} x^{8} + \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{3} \, a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac {5}{4} \, a^{4} b x^{4} + \frac {1}{3} \, a^{5} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 57, normalized size = 1.21 \begin {gather*} \frac {a^5\,x^3}{3}+\frac {5\,a^4\,b\,x^4}{4}+2\,a^3\,b^2\,x^5+\frac {5\,a^2\,b^3\,x^6}{3}+\frac {5\,a\,b^4\,x^7}{7}+\frac {b^5\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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