Optimal. Leaf size=96 \[ a^2 d x+\frac {1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{3} a x^3 (a e+2 b d)+\frac {1}{9} c x^9 (2 b e+c d)+\frac {1}{11} c^2 e x^{11} \]
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Rubi [A] time = 0.07, antiderivative size = 96, 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 = {1153} \[ a^2 d x+\frac {1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{3} a x^3 (a e+2 b d)+\frac {1}{9} c x^9 (2 b e+c d)+\frac {1}{11} c^2 e x^{11} \]
Antiderivative was successfully verified.
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Rule 1153
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^2 \, dx &=\int \left (a^2 d+a (2 b d+a e) x^2+\left (b^2 d+2 a c d+2 a b e\right ) x^4+\left (2 b c d+b^2 e+2 a c e\right ) x^6+c (c d+2 b e) x^8+c^2 e x^{10}\right ) \, dx\\ &=a^2 d x+\frac {1}{3} a (2 b d+a e) x^3+\frac {1}{5} \left (b^2 d+2 a c d+2 a b e\right ) x^5+\frac {1}{7} \left (2 b c d+b^2 e+2 a c e\right ) x^7+\frac {1}{9} c (c d+2 b e) x^9+\frac {1}{11} c^2 e x^{11}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 96, normalized size = 1.00 \[ a^2 d x+\frac {1}{7} x^7 \left (2 a c e+b^2 e+2 b c d\right )+\frac {1}{5} x^5 \left (2 a b e+2 a c d+b^2 d\right )+\frac {1}{3} a x^3 (a e+2 b d)+\frac {1}{9} c x^9 (2 b e+c d)+\frac {1}{11} c^2 e x^{11} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 100, normalized size = 1.04 \[ \frac {1}{11} x^{11} e c^{2} + \frac {1}{9} x^{9} d c^{2} + \frac {2}{9} x^{9} e c b + \frac {2}{7} x^{7} d c b + \frac {1}{7} x^{7} e b^{2} + \frac {2}{7} x^{7} e c a + \frac {1}{5} x^{5} d b^{2} + \frac {2}{5} x^{5} d c a + \frac {2}{5} x^{5} e b a + \frac {2}{3} x^{3} d b a + \frac {1}{3} x^{3} e a^{2} + x d a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 106, normalized size = 1.10 \[ \frac {1}{11} \, c^{2} x^{11} e + \frac {1}{9} \, c^{2} d x^{9} + \frac {2}{9} \, b c x^{9} e + \frac {2}{7} \, b c d x^{7} + \frac {1}{7} \, b^{2} x^{7} e + \frac {2}{7} \, a c x^{7} e + \frac {1}{5} \, b^{2} d x^{5} + \frac {2}{5} \, a c d x^{5} + \frac {2}{5} \, a b x^{5} e + \frac {2}{3} \, a b d x^{3} + \frac {1}{3} \, a^{2} x^{3} e + a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 91, normalized size = 0.95 \[ \frac {c^{2} e \,x^{11}}{11}+\frac {\left (2 e b c +d \,c^{2}\right ) x^{9}}{9}+\frac {\left (2 b c d +\left (2 a c +b^{2}\right ) e \right ) x^{7}}{7}+\frac {\left (2 a b e +\left (2 a c +b^{2}\right ) d \right ) x^{5}}{5}+a^{2} d x +\frac {\left (e \,a^{2}+2 d a b \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 90, normalized size = 0.94 \[ \frac {1}{11} \, c^{2} e x^{11} + \frac {1}{9} \, {\left (c^{2} d + 2 \, b c e\right )} x^{9} + \frac {1}{7} \, {\left (2 \, b c d + {\left (b^{2} + 2 \, a c\right )} e\right )} x^{7} + \frac {1}{5} \, {\left (2 \, a b e + {\left (b^{2} + 2 \, a c\right )} d\right )} x^{5} + a^{2} d x + \frac {1}{3} \, {\left (2 \, a b d + a^{2} e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 90, normalized size = 0.94 \[ x^5\,\left (\frac {d\,b^2}{5}+\frac {2\,a\,e\,b}{5}+\frac {2\,a\,c\,d}{5}\right )+x^7\,\left (\frac {e\,b^2}{7}+\frac {2\,c\,d\,b}{7}+\frac {2\,a\,c\,e}{7}\right )+x^3\,\left (\frac {e\,a^2}{3}+\frac {2\,b\,d\,a}{3}\right )+x^9\,\left (\frac {d\,c^2}{9}+\frac {2\,b\,e\,c}{9}\right )+\frac {c^2\,e\,x^{11}}{11}+a^2\,d\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 107, normalized size = 1.11 \[ a^{2} d x + \frac {c^{2} e x^{11}}{11} + x^{9} \left (\frac {2 b c e}{9} + \frac {c^{2} d}{9}\right ) + x^{7} \left (\frac {2 a c e}{7} + \frac {b^{2} e}{7} + \frac {2 b c d}{7}\right ) + x^{5} \left (\frac {2 a b e}{5} + \frac {2 a c d}{5} + \frac {b^{2} d}{5}\right ) + x^{3} \left (\frac {a^{2} e}{3} + \frac {2 a b d}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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