Optimal. Leaf size=30 \[ \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1598, 272, 45}
\begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int x \left (a x+b x^3\right )^2 \, dx &=\int x^3 \left (a+b x^2\right )^2 \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x (a+b x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^4}{4}+\frac {1}{3} a b x^6+\frac {b^2 x^8}{8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 25, normalized size = 0.83
method | result | size |
default | \(\frac {1}{4} a^{2} x^{4}+\frac {1}{3} a b \,x^{6}+\frac {1}{8} b^{2} x^{8}\) | \(25\) |
norman | \(\frac {1}{4} a^{2} x^{4}+\frac {1}{3} a b \,x^{6}+\frac {1}{8} b^{2} x^{8}\) | \(25\) |
risch | \(\frac {1}{4} a^{2} x^{4}+\frac {1}{3} a b \,x^{6}+\frac {1}{8} b^{2} x^{8}\) | \(25\) |
gosper | \(\frac {x^{4} \left (3 b^{2} x^{4}+8 a b \,x^{2}+6 a^{2}\right )}{24}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.17, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^{2} x^{4}}{4} + \frac {a b x^{6}}{3} + \frac {b^{2} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, b^{2} x^{8} + \frac {1}{3} \, a b x^{6} + \frac {1}{4} \, a^{2} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^4}{4}+\frac {a\,b\,x^6}{3}+\frac {b^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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