Optimal. Leaf size=61 \[ \frac{4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac{4 a (a+b x)^{m+4}}{b (m+4)}+\frac{(a+b x)^{m+5}}{b (m+5)} \]
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Rubi [A] time = 0.0279287, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {627, 43} \[ \frac{4 a^2 (a+b x)^{m+3}}{b (m+3)}-\frac{4 a (a+b x)^{m+4}}{b (m+4)}+\frac{(a+b x)^{m+5}}{b (m+5)} \]
Antiderivative was successfully verified.
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Rule 627
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^m \left (a^2-b^2 x^2\right )^2 \, dx &=\int (a-b x)^2 (a+b x)^{2+m} \, dx\\ &=\int \left (4 a^2 (a+b x)^{2+m}-4 a (a+b x)^{3+m}+(a+b x)^{4+m}\right ) \, dx\\ &=\frac{4 a^2 (a+b x)^{3+m}}{b (3+m)}-\frac{4 a (a+b x)^{4+m}}{b (4+m)}+\frac{(a+b x)^{5+m}}{b (5+m)}\\ \end{align*}
Mathematica [A] time = 0.0530235, size = 50, normalized size = 0.82 \[ \frac{(a+b x)^{m+3} \left (\frac{4 a^2}{m+3}-\frac{4 a (a+b x)}{m+4}+\frac{(a+b x)^2}{m+5}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 94, normalized size = 1.5 \begin{align*}{\frac{ \left ( bx+a \right ) ^{3+m} \left ({b}^{2}{m}^{2}{x}^{2}-2\,ab{m}^{2}x+7\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-18\,abmx+12\,{b}^{2}{x}^{2}+11\,m{a}^{2}-36\,abx+32\,{a}^{2} \right ) }{b \left ({m}^{3}+12\,{m}^{2}+47\,m+60 \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 = 1.96081, size = 363, normalized size = 5.95 \begin{align*} \frac{{\left (a^{5} m^{2} + 11 \, a^{5} m +{\left (b^{5} m^{2} + 7 \, b^{5} m + 12 \, b^{5}\right )} x^{5} + 32 \, a^{5} +{\left (a b^{4} m^{2} + 3 \, a b^{4} m\right )} x^{4} - 2 \,{\left (a^{2} b^{3} m^{2} + 11 \, a^{2} b^{3} m + 20 \, a^{2} b^{3}\right )} x^{3} - 2 \,{\left (a^{3} b^{2} m^{2} + 7 \, a^{3} b^{2} m\right )} x^{2} +{\left (a^{4} b m^{2} + 15 \, a^{4} b m + 60 \, a^{4} b\right )} x\right )}{\left (b x + a\right )}^{m}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.48281, size = 945, normalized size = 15.49 \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 = 1.28296, size = 389, normalized size = 6.38 \begin{align*} \frac{{\left (b x + a\right )}^{m} b^{5} m^{2} x^{5} +{\left (b x + a\right )}^{m} a b^{4} m^{2} x^{4} + 7 \,{\left (b x + a\right )}^{m} b^{5} m x^{5} - 2 \,{\left (b x + a\right )}^{m} a^{2} b^{3} m^{2} x^{3} + 3 \,{\left (b x + a\right )}^{m} a b^{4} m x^{4} + 12 \,{\left (b x + a\right )}^{m} b^{5} x^{5} - 2 \,{\left (b x + a\right )}^{m} a^{3} b^{2} m^{2} x^{2} - 22 \,{\left (b x + a\right )}^{m} a^{2} b^{3} m x^{3} +{\left (b x + a\right )}^{m} a^{4} b m^{2} x - 14 \,{\left (b x + a\right )}^{m} a^{3} b^{2} m x^{2} - 40 \,{\left (b x + a\right )}^{m} a^{2} b^{3} x^{3} +{\left (b x + a\right )}^{m} a^{5} m^{2} + 15 \,{\left (b x + a\right )}^{m} a^{4} b m x + 11 \,{\left (b x + a\right )}^{m} a^{5} m + 60 \,{\left (b x + a\right )}^{m} a^{4} b x + 32 \,{\left (b x + a\right )}^{m} a^{5}}{b m^{3} + 12 \, b m^{2} + 47 \, b m + 60 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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