Optimal. Leaf size=25 \[ b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 200}
\begin {gather*} b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^4} \, dx &=\int \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2+2 b c x^2+c^2 x^4\right ) \, dx\\ &=b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 22, normalized size = 0.88
method | result | size |
default | \(b^{2} x +\frac {2}{3} b c \,x^{3}+\frac {1}{5} c^{2} x^{5}\) | \(22\) |
risch | \(b^{2} x +\frac {2}{3} b c \,x^{3}+\frac {1}{5} c^{2} x^{5}\) | \(22\) |
gosper | \(\frac {x \left (3 c^{2} x^{4}+10 b c \,x^{2}+15 b^{2}\right )}{15}\) | \(25\) |
norman | \(\frac {b^{2} x^{4}+\frac {1}{5} c^{2} x^{8}+\frac {2}{3} b c \,x^{6}}{x^{3}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + b^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + b^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 22, normalized size = 0.88 \begin {gather*} b^{2} x + \frac {2 b c x^{3}}{3} + \frac {c^{2} x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.47, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + b^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 21, normalized size = 0.84 \begin {gather*} b^2\,x+\frac {2\,b\,c\,x^3}{3}+\frac {c^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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