Optimal. Leaf size=50 \[ a^2 c x+\frac {1}{5} b x^5 (2 a d+b c)+\frac {1}{3} a x^3 (a d+2 b c)+\frac {1}{7} b^2 d x^7 \]
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Rubi [A] time = 0.03, antiderivative size = 50, 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 = {373} \[ a^2 c x+\frac {1}{5} b x^5 (2 a d+b c)+\frac {1}{3} a x^3 (a d+2 b c)+\frac {1}{7} b^2 d x^7 \]
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c+a (2 b c+a d) x^2+b (b c+2 a d) x^4+b^2 d x^6\right ) \, dx\\ &=a^2 c x+\frac {1}{3} a (2 b c+a d) x^3+\frac {1}{5} b (b c+2 a d) x^5+\frac {1}{7} b^2 d x^7\\ \end {align*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ a^2 c x+\frac {1}{5} b x^5 (2 a d+b c)+\frac {1}{3} a x^3 (a d+2 b c)+\frac {1}{7} b^2 d x^7 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 50, normalized size = 1.00 \[ \frac {1}{7} x^{7} d b^{2} + \frac {1}{5} x^{5} c b^{2} + \frac {2}{5} x^{5} d b a + \frac {2}{3} x^{3} c b a + \frac {1}{3} x^{3} d a^{2} + x c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 50, normalized size = 1.00 \[ \frac {1}{7} \, b^{2} d x^{7} + \frac {1}{5} \, b^{2} c x^{5} + \frac {2}{5} \, a b d x^{5} + \frac {2}{3} \, a b c x^{3} + \frac {1}{3} \, a^{2} d x^{3} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 49, normalized size = 0.98 \[ \frac {b^{2} d \,x^{7}}{7}+\frac {\left (2 a b d +b^{2} c \right ) x^{5}}{5}+a^{2} c x +\frac {\left (a^{2} d +2 a b c \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 48, normalized size = 0.96 \[ \frac {1}{7} \, b^{2} d x^{7} + \frac {1}{5} \, {\left (b^{2} c + 2 \, a b d\right )} x^{5} + a^{2} c x + \frac {1}{3} \, {\left (2 \, a b c + a^{2} d\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 48, normalized size = 0.96 \[ x^3\,\left (\frac {d\,a^2}{3}+\frac {2\,b\,c\,a}{3}\right )+x^5\,\left (\frac {c\,b^2}{5}+\frac {2\,a\,d\,b}{5}\right )+\frac {b^2\,d\,x^7}{7}+a^2\,c\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 53, normalized size = 1.06 \[ a^{2} c x + \frac {b^{2} d x^{7}}{7} + x^{5} \left (\frac {2 a b d}{5} + \frac {b^{2} c}{5}\right ) + x^{3} \left (\frac {a^{2} d}{3} + \frac {2 a b c}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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