Optimal. Leaf size=80 \[ a^2 d^2 x+\frac {1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac {1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^3}{3 c}+\frac {1}{7} c^2 e^2 x^7 \]
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Rubi [A] time = 0.05, antiderivative size = 80, 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^2 x+\frac {1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac {1}{3} a x^3 \left (a e^2+2 c d^2\right )+\frac {d e \left (a+c x^2\right )^3}{3 c}+\frac {1}{7} c^2 e^2 x^7 \end {gather*}
Antiderivative was successfully verified.
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Rule 696
Rule 1810
Rubi steps
\begin {align*} \int (d+e x)^2 \left (a+c x^2\right )^2 \, dx &=\frac {d e \left (a+c x^2\right )^3}{3 c}+\int \left (a+c x^2\right )^2 \left (-2 d e x+(d+e x)^2\right ) \, dx\\ &=\frac {d e \left (a+c x^2\right )^3}{3 c}+\int \left (a^2 d^2+a \left (2 c d^2+a e^2\right ) x^2+c \left (c d^2+2 a e^2\right ) x^4+c^2 e^2 x^6\right ) \, dx\\ &=a^2 d^2 x+\frac {1}{3} a \left (2 c d^2+a e^2\right ) x^3+\frac {1}{5} c \left (c d^2+2 a e^2\right ) x^5+\frac {1}{7} c^2 e^2 x^7+\frac {d e \left (a+c x^2\right )^3}{3 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 91, normalized size = 1.14 \begin {gather*} a^2 d^2 x+a^2 d e x^2+\frac {1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac {1}{3} a x^3 \left (a e^2+2 c d^2\right )+a c d e x^4+\frac {1}{3} c^2 d e x^6+\frac {1}{7} c^2 e^2 x^7 \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)^2 \left (a+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.36, size = 89, normalized size = 1.11 \begin {gather*} \frac {1}{7} x^{7} e^{2} c^{2} + \frac {1}{3} x^{6} e d c^{2} + \frac {1}{5} x^{5} d^{2} c^{2} + \frac {2}{5} x^{5} e^{2} c a + x^{4} e d c a + \frac {2}{3} x^{3} d^{2} c a + \frac {1}{3} x^{3} e^{2} a^{2} + x^{2} e d a^{2} + x d^{2} a^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 89, normalized size = 1.11 \begin {gather*} \frac {1}{7} \, c^{2} x^{7} e^{2} + \frac {1}{3} \, c^{2} d x^{6} e + \frac {1}{5} \, c^{2} d^{2} x^{5} + \frac {2}{5} \, a c x^{5} e^{2} + a c d x^{4} e + \frac {2}{3} \, a c d^{2} x^{3} + \frac {1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 1.10 \begin {gather*} \frac {c^{2} e^{2} x^{7}}{7}+\frac {c^{2} d e \,x^{6}}{3}+a c d e \,x^{4}+a^{2} d e \,x^{2}+a^{2} d^{2} x +\frac {\left (2 e^{2} a c +c^{2} d^{2}\right ) x^{5}}{5}+\frac {\left (a^{2} e^{2}+2 d^{2} a c \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 87, normalized size = 1.09 \begin {gather*} \frac {1}{7} \, c^{2} e^{2} x^{7} + \frac {1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac {1}{5} \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac {1}{3} \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 87, normalized size = 1.09 \begin {gather*} x^3\,\left (\frac {a^2\,e^2}{3}+\frac {2\,c\,a\,d^2}{3}\right )+x^5\,\left (\frac {c^2\,d^2}{5}+\frac {2\,a\,c\,e^2}{5}\right )+a^2\,d^2\,x+\frac {c^2\,e^2\,x^7}{7}+a^2\,d\,e\,x^2+\frac {c^2\,d\,e\,x^6}{3}+a\,c\,d\,e\,x^4 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 95, normalized size = 1.19 \begin {gather*} a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac {c^{2} d e x^{6}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + x^{5} \left (\frac {2 a c e^{2}}{5} + \frac {c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac {a^{2} e^{2}}{3} + \frac {2 a c d^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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