Optimal. Leaf size=79 \[ \frac {1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^3 x+a d^2 e x^3+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11} \]
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Rubi [A] time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1154} \begin {gather*} \frac {1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^2 e x^3+a d^3 x+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11} \end {gather*}
Antiderivative was successfully verified.
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Rule 1154
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx &=\int \left (a d^3+3 a d^2 e x^2+d \left (c d^2+3 a e^2\right ) x^4+e \left (3 c d^2+a e^2\right ) x^6+3 c d e^2 x^8+c e^3 x^{10}\right ) \, dx\\ &=a d^3 x+a d^2 e x^3+\frac {1}{5} d \left (c d^2+3 a e^2\right ) x^5+\frac {1}{7} e \left (3 c d^2+a e^2\right ) x^7+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 79, normalized size = 1.00 \begin {gather*} \frac {1}{7} e x^7 \left (a e^2+3 c d^2\right )+\frac {1}{5} d x^5 \left (3 a e^2+c d^2\right )+a d^3 x+a d^2 e x^3+\frac {1}{3} c d e^2 x^9+\frac {1}{11} c e^3 x^{11} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d+e x^2\right )^3 \left (a+c x^4\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 73, normalized size = 0.92 \begin {gather*} \frac {1}{11} x^{11} e^{3} c + \frac {1}{3} x^{9} e^{2} d c + \frac {3}{7} x^{7} e d^{2} c + \frac {1}{7} x^{7} e^{3} a + \frac {1}{5} x^{5} d^{3} c + \frac {3}{5} x^{5} e^{2} d a + x^{3} e d^{2} a + x d^{3} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 71, normalized size = 0.90 \begin {gather*} \frac {1}{11} \, c x^{11} e^{3} + \frac {1}{3} \, c d x^{9} e^{2} + \frac {3}{7} \, c d^{2} x^{7} e + \frac {1}{5} \, c d^{3} x^{5} + \frac {1}{7} \, a x^{7} e^{3} + \frac {3}{5} \, a d x^{5} e^{2} + a d^{2} x^{3} e + a d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 72, normalized size = 0.91 \begin {gather*} \frac {c \,e^{3} x^{11}}{11}+\frac {c d \,e^{2} x^{9}}{3}+a \,d^{2} e \,x^{3}+\frac {\left (a \,e^{3}+3 c \,d^{2} e \right ) x^{7}}{7}+a \,d^{3} x +\frac {\left (3 d \,e^{2} a +d^{3} c \right ) x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 71, normalized size = 0.90 \begin {gather*} \frac {1}{11} \, c e^{3} x^{11} + \frac {1}{3} \, c d e^{2} x^{9} + \frac {1}{7} \, {\left (3 \, c d^{2} e + a e^{3}\right )} x^{7} + a d^{2} e x^{3} + \frac {1}{5} \, {\left (c d^{3} + 3 \, a d e^{2}\right )} x^{5} + a d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 71, normalized size = 0.90 \begin {gather*} x^5\,\left (\frac {c\,d^3}{5}+\frac {3\,a\,d\,e^2}{5}\right )+x^7\,\left (\frac {3\,c\,d^2\,e}{7}+\frac {a\,e^3}{7}\right )+\frac {c\,e^3\,x^{11}}{11}+a\,d^3\,x+a\,d^2\,e\,x^3+\frac {c\,d\,e^2\,x^9}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 78, normalized size = 0.99 \begin {gather*} a d^{3} x + a d^{2} e x^{3} + \frac {c d e^{2} x^{9}}{3} + \frac {c e^{3} x^{11}}{11} + x^{7} \left (\frac {a e^{3}}{7} + \frac {3 c d^{2} e}{7}\right ) + x^{5} \left (\frac {3 a d e^{2}}{5} + \frac {c d^{3}}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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