Optimal. Leaf size=161 \[ a^3 d^3 x+\frac {1}{4} a^3 e^3 x^4+a^2 d x^3 \left (a e^2+c d^2\right )+\frac {1}{2} a^2 c e^3 x^6+\frac {1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac {3}{8} a c^2 e^3 x^8+\frac {3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac {1}{3} c^3 d e^2 x^9+\frac {1}{10} c^3 e^3 x^{10} \]
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Rubi [A] time = 0.14, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {696, 1810} \begin {gather*} a^2 d x^3 \left (a e^2+c d^2\right )+\frac {1}{2} a^2 c e^3 x^6+a^3 d^3 x+\frac {1}{4} a^3 e^3 x^4+\frac {1}{7} c^2 d x^7 \left (9 a e^2+c d^2\right )+\frac {3}{8} a c^2 e^3 x^8+\frac {3}{5} a c d x^5 \left (3 a e^2+c d^2\right )+\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}+\frac {1}{3} c^3 d e^2 x^9+\frac {1}{10} c^3 e^3 x^{10} \end {gather*}
Antiderivative was successfully verified.
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Rule 696
Rule 1810
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx &=\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}+\int \left (a+c x^2\right )^3 \left (-3 d^2 e x+(d+e x)^3\right ) \, dx\\ &=\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}+\int \left (a^3 d^3+3 a^2 d \left (c d^2+a e^2\right ) x^2+a^3 e^3 x^3+3 a c d \left (c d^2+3 a e^2\right ) x^4+3 a^2 c e^3 x^5+c^2 d \left (c d^2+9 a e^2\right ) x^6+3 a c^2 e^3 x^7+3 c^3 d e^2 x^8+c^3 e^3 x^9\right ) \, dx\\ &=a^3 d^3 x+a^2 d \left (c d^2+a e^2\right ) x^3+\frac {1}{4} a^3 e^3 x^4+\frac {3}{5} a c d \left (c d^2+3 a e^2\right ) x^5+\frac {1}{2} a^2 c e^3 x^6+\frac {1}{7} c^2 d \left (c d^2+9 a e^2\right ) x^7+\frac {3}{8} a c^2 e^3 x^8+\frac {1}{3} c^3 d e^2 x^9+\frac {1}{10} c^3 e^3 x^{10}+\frac {3 d^2 e \left (a+c x^2\right )^4}{8 c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 155, normalized size = 0.96 \begin {gather*} \frac {1}{840} x \left (210 a^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+42 a^2 c x^2 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+9 a c^2 x^4 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right )+c^3 x^6 \left (120 d^3+315 d^2 e x+280 d e^2 x^2+84 e^3 x^3\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^3 \left (a+c x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 188, normalized size = 1.17 \begin {gather*} \frac {1}{10} x^{10} e^{3} c^{3} + \frac {1}{3} x^{9} e^{2} d c^{3} + \frac {3}{8} x^{8} e d^{2} c^{3} + \frac {3}{8} x^{8} e^{3} c^{2} a + \frac {1}{7} x^{7} d^{3} c^{3} + \frac {9}{7} x^{7} e^{2} d c^{2} a + \frac {3}{2} x^{6} e d^{2} c^{2} a + \frac {1}{2} x^{6} e^{3} c a^{2} + \frac {3}{5} x^{5} d^{3} c^{2} a + \frac {9}{5} x^{5} e^{2} d c a^{2} + \frac {9}{4} x^{4} e d^{2} c a^{2} + \frac {1}{4} x^{4} e^{3} a^{3} + x^{3} d^{3} c a^{2} + x^{3} e^{2} d a^{3} + \frac {3}{2} x^{2} e d^{2} a^{3} + x d^{3} a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 184, normalized size = 1.14 \begin {gather*} \frac {1}{10} \, c^{3} x^{10} e^{3} + \frac {1}{3} \, c^{3} d x^{9} e^{2} + \frac {3}{8} \, c^{3} d^{2} x^{8} e + \frac {1}{7} \, c^{3} d^{3} x^{7} + \frac {3}{8} \, a c^{2} x^{8} e^{3} + \frac {9}{7} \, a c^{2} d x^{7} e^{2} + \frac {3}{2} \, a c^{2} d^{2} x^{6} e + \frac {3}{5} \, a c^{2} d^{3} x^{5} + \frac {1}{2} \, a^{2} c x^{6} e^{3} + \frac {9}{5} \, a^{2} c d x^{5} e^{2} + \frac {9}{4} \, a^{2} c d^{2} x^{4} e + a^{2} c d^{3} x^{3} + \frac {1}{4} \, a^{3} x^{4} e^{3} + a^{3} d x^{3} e^{2} + \frac {3}{2} \, a^{3} d^{2} x^{2} e + a^{3} d^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 189, normalized size = 1.17 \begin {gather*} \frac {c^{3} e^{3} x^{10}}{10}+\frac {c^{3} d \,e^{2} x^{9}}{3}+\frac {3 a^{3} d^{2} e \,x^{2}}{2}+\frac {\left (3 e^{3} a \,c^{2}+3 d^{2} e \,c^{3}\right ) x^{8}}{8}+a^{3} d^{3} x +\frac {\left (9 d \,e^{2} a \,c^{2}+c^{3} d^{3}\right ) x^{7}}{7}+\frac {\left (3 a^{2} c \,e^{3}+9 d^{2} e a \,c^{2}\right ) x^{6}}{6}+\frac {\left (9 d \,e^{2} a^{2} c +3 d^{3} a \,c^{2}\right ) x^{5}}{5}+\frac {\left (e^{3} a^{3}+9 d^{2} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (3 d \,e^{2} a^{3}+3 d^{3} a^{2} c \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 181, normalized size = 1.12 \begin {gather*} \frac {1}{10} \, c^{3} e^{3} x^{10} + \frac {1}{3} \, c^{3} d e^{2} x^{9} + \frac {3}{8} \, {\left (c^{3} d^{2} e + a c^{2} e^{3}\right )} x^{8} + \frac {3}{2} \, a^{3} d^{2} e x^{2} + \frac {1}{7} \, {\left (c^{3} d^{3} + 9 \, a c^{2} d e^{2}\right )} x^{7} + a^{3} d^{3} x + \frac {1}{2} \, {\left (3 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{6} + \frac {3}{5} \, {\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (9 \, a^{2} c d^{2} e + a^{3} e^{3}\right )} x^{4} + {\left (a^{2} c d^{3} + a^{3} d e^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 174, normalized size = 1.08 \begin {gather*} x^3\,\left (a^3\,d\,e^2+c\,a^2\,d^3\right )+x^4\,\left (\frac {a^3\,e^3}{4}+\frac {9\,c\,a^2\,d^2\,e}{4}\right )+x^7\,\left (\frac {c^3\,d^3}{7}+\frac {9\,a\,c^2\,d\,e^2}{7}\right )+x^8\,\left (\frac {3\,c^3\,d^2\,e}{8}+\frac {3\,a\,c^2\,e^3}{8}\right )+a^3\,d^3\,x+\frac {c^3\,e^3\,x^{10}}{10}+\frac {3\,a^3\,d^2\,e\,x^2}{2}+\frac {c^3\,d\,e^2\,x^9}{3}+\frac {3\,a\,c\,d\,x^5\,\left (c\,d^2+3\,a\,e^2\right )}{5}+\frac {a\,c\,e\,x^6\,\left (3\,c\,d^2+a\,e^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 202, normalized size = 1.25 \begin {gather*} a^{3} d^{3} x + \frac {3 a^{3} d^{2} e x^{2}}{2} + \frac {c^{3} d e^{2} x^{9}}{3} + \frac {c^{3} e^{3} x^{10}}{10} + x^{8} \left (\frac {3 a c^{2} e^{3}}{8} + \frac {3 c^{3} d^{2} e}{8}\right ) + x^{7} \left (\frac {9 a c^{2} d e^{2}}{7} + \frac {c^{3} d^{3}}{7}\right ) + x^{6} \left (\frac {a^{2} c e^{3}}{2} + \frac {3 a c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac {9 a^{2} c d e^{2}}{5} + \frac {3 a c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac {a^{3} e^{3}}{4} + \frac {9 a^{2} c d^{2} e}{4}\right ) + x^{3} \left (a^{3} d e^{2} + a^{2} c d^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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