Optimal. Leaf size=103 \[ \frac {1}{7} e x^7 \left (e (a e+3 b d)+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 e (a e+b d)+c d^2\right )+\frac {1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac {1}{9} e^2 x^9 (b e+3 c d)+\frac {1}{11} c e^3 x^{11} \]
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Rubi [A] time = 0.10, antiderivative size = 103, 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} \[ \frac {1}{7} e x^7 \left (e (a e+3 b d)+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 e (a e+b d)+c d^2\right )+\frac {1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac {1}{9} e^2 x^9 (b e+3 c d)+\frac {1}{11} c e^3 x^{11} \]
Antiderivative was successfully verified.
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Rule 1153
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a d^3+d^2 (b d+3 a e) x^2+d \left (c d^2+3 e (b d+a e)\right ) x^4+e \left (3 c d^2+e (3 b d+a e)\right ) x^6+e^2 (3 c d+b e) x^8+c e^3 x^{10}\right ) \, dx\\ &=a d^3 x+\frac {1}{3} d^2 (b d+3 a e) x^3+\frac {1}{5} d \left (c d^2+3 e (b d+a e)\right ) x^5+\frac {1}{7} e \left (3 c d^2+e (3 b d+a e)\right ) x^7+\frac {1}{9} e^2 (3 c d+b e) x^9+\frac {1}{11} c e^3 x^{11}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 104, normalized size = 1.01 \[ \frac {1}{7} e x^7 \left (a e^2+3 b d e+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 a e^2+3 b d e+c d^2\right )+\frac {1}{3} d^2 x^3 (3 a e+b d)+a d^3 x+\frac {1}{9} e^2 x^9 (b e+3 c d)+\frac {1}{11} c e^3 x^{11} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 111, normalized size = 1.08 \[ \frac {1}{11} x^{11} e^{3} c + \frac {1}{3} x^{9} e^{2} d c + \frac {1}{9} x^{9} e^{3} b + \frac {3}{7} x^{7} e d^{2} c + \frac {3}{7} x^{7} e^{2} d b + \frac {1}{7} x^{7} e^{3} a + \frac {1}{5} x^{5} d^{3} c + \frac {3}{5} x^{5} e d^{2} b + \frac {3}{5} x^{5} e^{2} d a + \frac {1}{3} x^{3} d^{3} b + x^{3} e d^{2} a + x d^{3} a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 108, normalized size = 1.05 \[ \frac {1}{11} \, c x^{11} e^{3} + \frac {1}{3} \, c d x^{9} e^{2} + \frac {1}{9} \, b x^{9} e^{3} + \frac {3}{7} \, c d^{2} x^{7} e + \frac {3}{7} \, b d x^{7} e^{2} + \frac {1}{5} \, c d^{3} x^{5} + \frac {1}{7} \, a x^{7} e^{3} + \frac {3}{5} \, b d^{2} x^{5} e + \frac {3}{5} \, a d x^{5} e^{2} + \frac {1}{3} \, b d^{3} x^{3} + a d^{2} x^{3} e + a d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 103, normalized size = 1.00 \[ \frac {c \,e^{3} x^{11}}{11}+\frac {\left (e^{3} b +3 d \,e^{2} c \right ) x^{9}}{9}+\frac {\left (a \,e^{3}+3 d \,e^{2} b +3 c \,d^{2} e \right ) x^{7}}{7}+a \,d^{3} x +\frac {\left (3 d \,e^{2} a +3 d^{2} e b +d^{3} c \right ) x^{5}}{5}+\frac {\left (3 d^{2} e a +d^{3} b \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 102, normalized size = 0.99 \[ \frac {1}{11} \, c e^{3} x^{11} + \frac {1}{9} \, {\left (3 \, c d e^{2} + b e^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{7} + \frac {1}{5} \, {\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x + \frac {1}{3} \, {\left (b d^{3} + 3 \, a d^{2} e\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.63, size = 101, normalized size = 0.98 \[ x^3\,\left (\frac {b\,d^3}{3}+a\,e\,d^2\right )+x^9\,\left (\frac {b\,e^3}{9}+\frac {c\,d\,e^2}{3}\right )+x^5\,\left (\frac {c\,d^3}{5}+\frac {3\,b\,d^2\,e}{5}+\frac {3\,a\,d\,e^2}{5}\right )+x^7\,\left (\frac {3\,c\,d^2\,e}{7}+\frac {3\,b\,d\,e^2}{7}+\frac {a\,e^3}{7}\right )+\frac {c\,e^3\,x^{11}}{11}+a\,d^3\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 112, normalized size = 1.09 \[ a d^{3} x + \frac {c e^{3} x^{11}}{11} + x^{9} \left (\frac {b e^{3}}{9} + \frac {c d e^{2}}{3}\right ) + x^{7} \left (\frac {a e^{3}}{7} + \frac {3 b d e^{2}}{7} + \frac {3 c d^{2} e}{7}\right ) + x^{5} \left (\frac {3 a d e^{2}}{5} + \frac {3 b d^{2} e}{5} + \frac {c d^{3}}{5}\right ) + x^{3} \left (a d^{2} e + \frac {b d^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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