Optimal. Leaf size=43 \[ \frac {a^3 x^8}{8}+\frac {3}{10} a^2 b x^{10}+\frac {1}{4} a b^2 x^{12}+\frac {b^3 x^{14}}{14} \]
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Rubi [A]
time = 0.02, 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 = {272, 45}
\begin {gather*} \frac {a^3 x^8}{8}+\frac {3}{10} a^2 b x^{10}+\frac {1}{4} a b^2 x^{12}+\frac {b^3 x^{14}}{14} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^7 \left (a+b x^2\right )^3 \, dx &=\frac {1}{2} \text {Subst}\left (\int x^3 (a+b x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^2\right )\\ &=\frac {a^3 x^8}{8}+\frac {3}{10} a^2 b x^{10}+\frac {1}{4} a b^2 x^{12}+\frac {b^3 x^{14}}{14}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {a^3 x^8}{8}+\frac {3}{10} a^2 b x^{10}+\frac {1}{4} a b^2 x^{12}+\frac {b^3 x^{14}}{14} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 36, normalized size = 0.84
method | result | size |
gosper | \(\frac {1}{8} a^{3} x^{8}+\frac {3}{10} a^{2} b \,x^{10}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{14} b^{3} x^{14}\) | \(36\) |
default | \(\frac {1}{8} a^{3} x^{8}+\frac {3}{10} a^{2} b \,x^{10}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{14} b^{3} x^{14}\) | \(36\) |
norman | \(\frac {1}{8} a^{3} x^{8}+\frac {3}{10} a^{2} b \,x^{10}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{14} b^{3} x^{14}\) | \(36\) |
risch | \(\frac {1}{8} a^{3} x^{8}+\frac {3}{10} a^{2} b \,x^{10}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{14} b^{3} x^{14}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{14} \, b^{3} x^{14} + \frac {1}{4} \, a b^{2} x^{12} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{8} \, a^{3} x^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.19, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{14} \, b^{3} x^{14} + \frac {1}{4} \, a b^{2} x^{12} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{8} \, a^{3} x^{8} \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^{8}}{8} + \frac {3 a^{2} b x^{10}}{10} + \frac {a b^{2} x^{12}}{4} + \frac {b^{3} x^{14}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.44, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{14} \, b^{3} x^{14} + \frac {1}{4} \, a b^{2} x^{12} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{8} \, a^{3} x^{8} \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^8}{8}+\frac {3\,a^2\,b\,x^{10}}{10}+\frac {a\,b^2\,x^{12}}{4}+\frac {b^3\,x^{14}}{14} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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