Optimal. Leaf size=205 \[ \frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0758189, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1112, 270} \[ \frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+3}}{d^3 (m+3) \left (a+b x^2\right )}+\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+5}}{d^5 (m+5) \left (a+b x^2\right )}+\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^2\right )}+\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} (d x)^{m+1}}{d (m+1) \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 270
Rubi steps
\begin{align*} \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int (d x)^m \left (a b+b^2 x^2\right )^3 \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (a^3 b^3 (d x)^m+\frac{3 a^2 b^4 (d x)^{2+m}}{d^2}+\frac{3 a b^5 (d x)^{4+m}}{d^4}+\frac{b^6 (d x)^{6+m}}{d^6}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{a^3 (d x)^{1+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac{3 a^2 b (d x)^{3+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac{3 a b^2 (d x)^{5+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac{b^3 (d x)^{7+m} \sqrt{a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0693267, size = 131, normalized size = 0.64 \[ \frac{x \sqrt{\left (a+b x^2\right )^2} (d x)^m \left (3 a^2 b \left (m^3+13 m^2+47 m+35\right ) x^2+a^3 \left (m^3+15 m^2+71 m+105\right )+3 a b^2 \left (m^3+11 m^2+31 m+21\right ) x^4+b^3 \left (m^3+9 m^2+23 m+15\right ) x^6\right )}{(m+1) (m+3) (m+5) (m+7) \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 199, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{3}{m}^{3}{x}^{6}+9\,{b}^{3}{m}^{2}{x}^{6}+3\,a{b}^{2}{m}^{3}{x}^{4}+23\,{b}^{3}m{x}^{6}+33\,a{b}^{2}{m}^{2}{x}^{4}+15\,{b}^{3}{x}^{6}+3\,{a}^{2}b{m}^{3}{x}^{2}+93\,a{b}^{2}m{x}^{4}+39\,{a}^{2}b{m}^{2}{x}^{2}+63\,a{x}^{4}{b}^{2}+{a}^{3}{m}^{3}+141\,{a}^{2}bm{x}^{2}+15\,{a}^{3}{m}^{2}+105\,{a}^{2}b{x}^{2}+71\,{a}^{3}m+105\,{a}^{3} \right ) x \left ( dx \right ) ^{m}}{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984213, size = 161, normalized size = 0.79 \begin{align*} \frac{{\left ({\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} b^{3} d^{m} x^{7} + 3 \,{\left (m^{3} + 11 \, m^{2} + 31 \, m + 21\right )} a b^{2} d^{m} x^{5} + 3 \,{\left (m^{3} + 13 \, m^{2} + 47 \, m + 35\right )} a^{2} b d^{m} x^{3} +{\left (m^{3} + 15 \, m^{2} + 71 \, m + 105\right )} a^{3} d^{m} 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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Fricas [A] time = 1.55453, size = 352, normalized size = 1.72 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 9 \, b^{3} m^{2} + 23 \, b^{3} m + 15 \, b^{3}\right )} x^{7} + 3 \,{\left (a b^{2} m^{3} + 11 \, a b^{2} m^{2} + 31 \, a b^{2} m + 21 \, a b^{2}\right )} x^{5} + 3 \,{\left (a^{2} b m^{3} + 13 \, a^{2} b m^{2} + 47 \, a^{2} b m + 35 \, a^{2} b\right )} x^{3} +{\left (a^{3} m^{3} + 15 \, a^{3} m^{2} + 71 \, a^{3} m + 105 \, a^{3}\right )} x\right )} \left (d x\right )^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27073, size = 518, normalized size = 2.53 \begin{align*} \frac{\left (d x\right )^{m} b^{3} m^{3} x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + 9 \, \left (d x\right )^{m} b^{3} m^{2} x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + 3 \, \left (d x\right )^{m} a b^{2} m^{3} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + 23 \, \left (d x\right )^{m} b^{3} m x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + 33 \, \left (d x\right )^{m} a b^{2} m^{2} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + 15 \, \left (d x\right )^{m} b^{3} x^{7} \mathrm{sgn}\left (b x^{2} + a\right ) + 3 \, \left (d x\right )^{m} a^{2} b m^{3} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 93 \, \left (d x\right )^{m} a b^{2} m x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + 39 \, \left (d x\right )^{m} a^{2} b m^{2} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 63 \, \left (d x\right )^{m} a b^{2} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + \left (d x\right )^{m} a^{3} m^{3} x \mathrm{sgn}\left (b x^{2} + a\right ) + 141 \, \left (d x\right )^{m} a^{2} b m x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 15 \, \left (d x\right )^{m} a^{3} m^{2} x \mathrm{sgn}\left (b x^{2} + a\right ) + 105 \, \left (d x\right )^{m} a^{2} b x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) + 71 \, \left (d x\right )^{m} a^{3} m x \mathrm{sgn}\left (b x^{2} + a\right ) + 105 \, \left (d x\right )^{m} a^{3} x \mathrm{sgn}\left (b x^{2} + a\right )}{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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