Optimal. Leaf size=71 \[ \frac{a^2 c x^{m+1}}{m+1}+\frac{a x^{m+3} (a d+2 b c)}{m+3}+\frac{b x^{m+5} (2 a d+b c)}{m+5}+\frac{b^2 d x^{m+7}}{m+7} \]
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Rubi [A] time = 0.0374977, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{a^2 c x^{m+1}}{m+1}+\frac{a x^{m+3} (a d+2 b c)}{m+3}+\frac{b x^{m+5} (2 a d+b c)}{m+5}+\frac{b^2 d x^{m+7}}{m+7} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c x^m+a (2 b c+a d) x^{2+m}+b (b c+2 a d) x^{4+m}+b^2 d x^{6+m}\right ) \, dx\\ &=\frac{a^2 c x^{1+m}}{1+m}+\frac{a (2 b c+a d) x^{3+m}}{3+m}+\frac{b (b c+2 a d) x^{5+m}}{5+m}+\frac{b^2 d x^{7+m}}{7+m}\\ \end{align*}
Mathematica [A] time = 0.0633677, size = 66, normalized size = 0.93 \[ x^{m+1} \left (\frac{a^2 c}{m+1}+\frac{b x^4 (2 a d+b c)}{m+5}+\frac{a x^2 (a d+2 b c)}{m+3}+\frac{b^2 d x^6}{m+7}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 262, normalized size = 3.7 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{2}d{m}^{3}{x}^{6}+9\,{b}^{2}d{m}^{2}{x}^{6}+2\,abd{m}^{3}{x}^{4}+{b}^{2}c{m}^{3}{x}^{4}+23\,{b}^{2}dm{x}^{6}+22\,abd{m}^{2}{x}^{4}+11\,{b}^{2}c{m}^{2}{x}^{4}+15\,{b}^{2}d{x}^{6}+{a}^{2}d{m}^{3}{x}^{2}+2\,abc{m}^{3}{x}^{2}+62\,abdm{x}^{4}+31\,{b}^{2}cm{x}^{4}+13\,{a}^{2}d{m}^{2}{x}^{2}+26\,abc{m}^{2}{x}^{2}+42\,{x}^{4}abd+21\,{b}^{2}c{x}^{4}+{a}^{2}c{m}^{3}+47\,{a}^{2}dm{x}^{2}+94\,abcm{x}^{2}+15\,{a}^{2}c{m}^{2}+35\,{x}^{2}{a}^{2}d+70\,abc{x}^{2}+71\,{a}^{2}cm+105\,{a}^{2}c \right ) }{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.881687, size = 490, normalized size = 6.9 \begin{align*} \frac{{\left ({\left (b^{2} d m^{3} + 9 \, b^{2} d m^{2} + 23 \, b^{2} d m + 15 \, b^{2} d\right )} x^{7} +{\left ({\left (b^{2} c + 2 \, a b d\right )} m^{3} + 21 \, b^{2} c + 42 \, a b d + 11 \,{\left (b^{2} c + 2 \, a b d\right )} m^{2} + 31 \,{\left (b^{2} c + 2 \, a b d\right )} m\right )} x^{5} +{\left ({\left (2 \, a b c + a^{2} d\right )} m^{3} + 70 \, a b c + 35 \, a^{2} d + 13 \,{\left (2 \, a b c + a^{2} d\right )} m^{2} + 47 \,{\left (2 \, a b c + a^{2} d\right )} m\right )} x^{3} +{\left (a^{2} c m^{3} + 15 \, a^{2} c m^{2} + 71 \, a^{2} c m + 105 \, a^{2} c\right )} x\right )} x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.68384, size = 1044, normalized size = 14.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15522, size = 448, normalized size = 6.31 \begin{align*} \frac{b^{2} d m^{3} x^{7} x^{m} + 9 \, b^{2} d m^{2} x^{7} x^{m} + b^{2} c m^{3} x^{5} x^{m} + 2 \, a b d m^{3} x^{5} x^{m} + 23 \, b^{2} d m x^{7} x^{m} + 11 \, b^{2} c m^{2} x^{5} x^{m} + 22 \, a b d m^{2} x^{5} x^{m} + 15 \, b^{2} d x^{7} x^{m} + 2 \, a b c m^{3} x^{3} x^{m} + a^{2} d m^{3} x^{3} x^{m} + 31 \, b^{2} c m x^{5} x^{m} + 62 \, a b d m x^{5} x^{m} + 26 \, a b c m^{2} x^{3} x^{m} + 13 \, a^{2} d m^{2} x^{3} x^{m} + 21 \, b^{2} c x^{5} x^{m} + 42 \, a b d x^{5} x^{m} + a^{2} c m^{3} x x^{m} + 94 \, a b c m x^{3} x^{m} + 47 \, a^{2} d m x^{3} x^{m} + 15 \, a^{2} c m^{2} x x^{m} + 70 \, a b c x^{3} x^{m} + 35 \, a^{2} d x^{3} x^{m} + 71 \, a^{2} c m x x^{m} + 105 \, a^{2} c x x^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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