Optimal. Leaf size=73 \[ a d^2 x+\frac {1}{3} d (b d+2 a e) x^3+\frac {1}{5} \left (c d^2+e (2 b d+a e)\right ) x^5+\frac {1}{7} e (2 c d+b e) x^7+\frac {1}{9} c e^2 x^9 \]
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Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1167}
\begin {gather*} \frac {1}{5} x^5 \left (e (a e+2 b d)+c d^2\right )+\frac {1}{3} d x^3 (2 a e+b d)+a d^2 x+\frac {1}{7} e x^7 (b e+2 c d)+\frac {1}{9} c e^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 1167
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d^2+d (b d+2 a e) x^2+\left (c d^2+e (2 b d+a e)\right ) x^4+e (2 c d+b e) x^6+c e^2 x^8\right ) \, dx\\ &=a d^2 x+\frac {1}{3} d (b d+2 a e) x^3+\frac {1}{5} \left (c d^2+e (2 b d+a e)\right ) x^5+\frac {1}{7} e (2 c d+b e) x^7+\frac {1}{9} c e^2 x^9\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 73, normalized size = 1.00 \begin {gather*} a d^2 x+\frac {1}{3} d (b d+2 a e) x^3+\frac {1}{5} \left (c d^2+2 b d e+a e^2\right ) x^5+\frac {1}{7} e (2 c d+b e) x^7+\frac {1}{9} c e^2 x^9 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 70, normalized size = 0.96
method | result | size |
default | \(\frac {c \,e^{2} x^{9}}{9}+\frac {\left (e^{2} b +2 c d e \right ) x^{7}}{7}+\frac {\left (a \,e^{2}+2 d e b +c \,d^{2}\right ) x^{5}}{5}+\frac {\left (2 a d e +d^{2} b \right ) x^{3}}{3}+a \,d^{2} x\) | \(70\) |
norman | \(\frac {c \,e^{2} x^{9}}{9}+\left (\frac {1}{7} e^{2} b +\frac {2}{7} c d e \right ) x^{7}+\left (\frac {1}{5} a \,e^{2}+\frac {2}{5} d e b +\frac {1}{5} c \,d^{2}\right ) x^{5}+\left (\frac {2}{3} a d e +\frac {1}{3} d^{2} b \right ) x^{3}+a \,d^{2} x\) | \(71\) |
gosper | \(\frac {1}{9} c \,e^{2} x^{9}+\frac {1}{7} x^{7} e^{2} b +\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {2}{5} x^{5} d e b +\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+\frac {1}{3} x^{3} d^{2} b +a \,d^{2} x\) | \(77\) |
risch | \(\frac {1}{9} c \,e^{2} x^{9}+\frac {1}{7} x^{7} e^{2} b +\frac {2}{7} c d e \,x^{7}+\frac {1}{5} x^{5} a \,e^{2}+\frac {2}{5} x^{5} d e b +\frac {1}{5} x^{5} c \,d^{2}+\frac {2}{3} a d e \,x^{3}+\frac {1}{3} x^{3} d^{2} b +a \,d^{2} x\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 69, normalized size = 0.95 \begin {gather*} \frac {1}{9} \, c x^{9} e^{2} + \frac {1}{7} \, {\left (2 \, c d e + b e^{2}\right )} x^{7} + \frac {1}{5} \, {\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{5} + a d^{2} x + \frac {1}{3} \, {\left (b d^{2} + 2 \, a d e\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 74, normalized size = 1.01 \begin {gather*} \frac {1}{5} \, c d^{2} x^{5} + \frac {1}{3} \, b d^{2} x^{3} + a d^{2} x + \frac {1}{315} \, {\left (35 \, c x^{9} + 45 \, b x^{7} + 63 \, a x^{5}\right )} e^{2} + \frac {2}{105} \, {\left (15 \, c d x^{7} + 21 \, b d x^{5} + 35 \, a d 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 = 78, normalized size = 1.07 \begin {gather*} a d^{2} x + \frac {c e^{2} x^{9}}{9} + x^{7} \left (\frac {b e^{2}}{7} + \frac {2 c d e}{7}\right ) + x^{5} \left (\frac {a e^{2}}{5} + \frac {2 b d e}{5} + \frac {c d^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a d e}{3} + \frac {b d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.91, size = 76, normalized size = 1.04 \begin {gather*} \frac {1}{9} \, c x^{9} e^{2} + \frac {2}{7} \, c d x^{7} e + \frac {1}{7} \, b x^{7} e^{2} + \frac {1}{5} \, c d^{2} x^{5} + \frac {2}{5} \, b d x^{5} e + \frac {1}{5} \, a x^{5} e^{2} + \frac {1}{3} \, b d^{2} x^{3} + \frac {2}{3} \, a d x^{3} e + a d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 70, normalized size = 0.96 \begin {gather*} x^5\,\left (\frac {c\,d^2}{5}+\frac {2\,b\,d\,e}{5}+\frac {a\,e^2}{5}\right )+x^3\,\left (\frac {b\,d^2}{3}+\frac {2\,a\,e\,d}{3}\right )+x^7\,\left (\frac {b\,e^2}{7}+\frac {2\,c\,d\,e}{7}\right )+\frac {c\,e^2\,x^9}{9}+a\,d^2\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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