Optimal. Leaf size=32 \[ a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1168}
\begin {gather*} a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 1168
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+c x^4\right ) \, dx &=\int \left (a d+a e x^2+c d x^4+c e x^6\right ) \, dx\\ &=a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 32, normalized size = 1.00 \begin {gather*} a d x+\frac {1}{3} a e x^3+\frac {1}{5} c d x^5+\frac {1}{7} c e x^7 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 27, normalized size = 0.84
method | result | size |
gosper | \(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\) | \(27\) |
default | \(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\) | \(27\) |
norman | \(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\) | \(27\) |
risch | \(a d x +\frac {1}{3} a e \,x^{3}+\frac {1}{5} c d \,x^{5}+\frac {1}{7} c e \,x^{7}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, c x^{7} e + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a x^{3} e + a d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 29, normalized size = 0.91 \begin {gather*} \frac {1}{5} \, c d x^{5} + a d x + \frac {1}{21} \, {\left (3 \, c x^{7} + 7 \, a x^{3}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 29, normalized size = 0.91 \begin {gather*} a d x + \frac {a e x^{3}}{3} + \frac {c d x^{5}}{5} + \frac {c e x^{7}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.91, size = 28, normalized size = 0.88 \begin {gather*} \frac {1}{7} \, c x^{7} e + \frac {1}{5} \, c d x^{5} + \frac {1}{3} \, a x^{3} e + a d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 26, normalized size = 0.81 \begin {gather*} \frac {c\,e\,x^7}{7}+\frac {c\,d\,x^5}{5}+\frac {a\,e\,x^3}{3}+a\,d\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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