Optimal. Leaf size=35 \[ a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200}
\begin {gather*} a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rubi steps
\begin {align*} \int \left (a+c x^2\right )^3 \, dx &=\int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} a^3 x+a^2 c x^3+\frac {3}{5} a c^2 x^5+\frac {c^3 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 32, normalized size = 0.91
method | result | size |
gosper | \(a^{3} x +a^{2} c \,x^{3}+\frac {3}{5} a \,c^{2} x^{5}+\frac {1}{7} c^{3} x^{7}\) | \(32\) |
default | \(a^{3} x +a^{2} c \,x^{3}+\frac {3}{5} a \,c^{2} x^{5}+\frac {1}{7} c^{3} x^{7}\) | \(32\) |
norman | \(a^{3} x +a^{2} c \,x^{3}+\frac {3}{5} a \,c^{2} x^{5}+\frac {1}{7} c^{3} x^{7}\) | \(32\) |
risch | \(a^{3} x +a^{2} c \,x^{3}+\frac {3}{5} a \,c^{2} x^{5}+\frac {1}{7} c^{3} x^{7}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, c^{3} x^{7} + \frac {3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.48, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, c^{3} x^{7} + \frac {3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 32, normalized size = 0.91 \begin {gather*} a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.00, size = 31, normalized size = 0.89 \begin {gather*} \frac {1}{7} \, c^{3} x^{7} + \frac {3}{5} \, a c^{2} x^{5} + a^{2} c x^{3} + a^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 31, normalized size = 0.89 \begin {gather*} a^3\,x+a^2\,c\,x^3+\frac {3\,a\,c^2\,x^5}{5}+\frac {c^3\,x^7}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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