Optimal. Leaf size=205 \[ \frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )}+\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0851968, 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 = {1355, 270} \[ \frac{3 a^2 b \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+4}}{d^4 (m+4) \left (a+b x^3\right )}+\frac{3 a b^2 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+7}}{d^7 (m+7) \left (a+b x^3\right )}+\frac{b^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+10}}{d^{10} (m+10) \left (a+b x^3\right )}+\frac{a^3 \sqrt{a^2+2 a b x^3+b^2 x^6} (d x)^{m+1}}{d (m+1) \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 270
Rubi steps
\begin{align*} \int (d x)^m \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int (d x)^m \left (a b+b^2 x^3\right )^3 \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (a^3 b^3 (d x)^m+\frac{3 a^2 b^4 (d x)^{3+m}}{d^3}+\frac{3 a b^5 (d x)^{6+m}}{d^6}+\frac{b^6 (d x)^{9+m}}{d^9}\right ) \, dx}{b^2 \left (a b+b^2 x^3\right )}\\ &=\frac{a^3 (d x)^{1+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d (1+m) \left (a+b x^3\right )}+\frac{3 a^2 b (d x)^{4+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^4 (4+m) \left (a+b x^3\right )}+\frac{3 a b^2 (d x)^{7+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^7 (7+m) \left (a+b x^3\right )}+\frac{b^3 (d x)^{10+m} \sqrt{a^2+2 a b x^3+b^2 x^6}}{d^{10} (10+m) \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.0709297, size = 131, normalized size = 0.64 \[ \frac{x \sqrt{\left (a+b x^3\right )^2} (d x)^m \left (3 a^2 b \left (m^3+18 m^2+87 m+70\right ) x^3+a^3 \left (m^3+21 m^2+138 m+280\right )+3 a b^2 \left (m^3+15 m^2+54 m+40\right ) x^6+b^3 \left (m^3+12 m^2+39 m+28\right ) x^9\right )}{(m+1) (m+4) (m+7) (m+10) \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 199, normalized size = 1. \begin{align*}{\frac{ \left ({b}^{3}{m}^{3}{x}^{9}+12\,{b}^{3}{m}^{2}{x}^{9}+39\,{b}^{3}m{x}^{9}+3\,a{b}^{2}{m}^{3}{x}^{6}+28\,{b}^{3}{x}^{9}+45\,a{b}^{2}{m}^{2}{x}^{6}+162\,a{b}^{2}m{x}^{6}+3\,{a}^{2}b{m}^{3}{x}^{3}+120\,a{b}^{2}{x}^{6}+54\,{a}^{2}b{m}^{2}{x}^{3}+261\,{a}^{2}bm{x}^{3}+{a}^{3}{m}^{3}+210\,{a}^{2}b{x}^{3}+21\,{a}^{3}{m}^{2}+138\,{a}^{3}m+280\,{a}^{3} \right ) x \left ( dx \right ) ^{m}}{ \left ( 10+m \right ) \left ( 7+m \right ) \left ( 4+m \right ) \left ( 1+m \right ) \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+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 = 1.09974, size = 161, normalized size = 0.79 \begin{align*} \frac{{\left ({\left (m^{3} + 12 \, m^{2} + 39 \, m + 28\right )} b^{3} d^{m} x^{10} + 3 \,{\left (m^{3} + 15 \, m^{2} + 54 \, m + 40\right )} a b^{2} d^{m} x^{7} + 3 \,{\left (m^{3} + 18 \, m^{2} + 87 \, m + 70\right )} a^{2} b d^{m} x^{4} +{\left (m^{3} + 21 \, m^{2} + 138 \, m + 280\right )} a^{3} d^{m} x\right )} x^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58441, size = 358, normalized size = 1.75 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 12 \, b^{3} m^{2} + 39 \, b^{3} m + 28 \, b^{3}\right )} x^{10} + 3 \,{\left (a b^{2} m^{3} + 15 \, a b^{2} m^{2} + 54 \, a b^{2} m + 40 \, a b^{2}\right )} x^{7} + 3 \,{\left (a^{2} b m^{3} + 18 \, a^{2} b m^{2} + 87 \, a^{2} b m + 70 \, a^{2} b\right )} x^{4} +{\left (a^{3} m^{3} + 21 \, a^{3} m^{2} + 138 \, a^{3} m + 280 \, a^{3}\right )} x\right )} \left (d x\right )^{m}}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \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^{3}\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 = 1.12653, size = 518, normalized size = 2.53 \begin{align*} \frac{\left (d x\right )^{m} b^{3} m^{3} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + 12 \, \left (d x\right )^{m} b^{3} m^{2} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + 39 \, \left (d x\right )^{m} b^{3} m x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + 3 \, \left (d x\right )^{m} a b^{2} m^{3} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + 28 \, \left (d x\right )^{m} b^{3} x^{10} \mathrm{sgn}\left (b x^{3} + a\right ) + 45 \, \left (d x\right )^{m} a b^{2} m^{2} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + 162 \, \left (d x\right )^{m} a b^{2} m x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + 3 \, \left (d x\right )^{m} a^{2} b m^{3} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 120 \, \left (d x\right )^{m} a b^{2} x^{7} \mathrm{sgn}\left (b x^{3} + a\right ) + 54 \, \left (d x\right )^{m} a^{2} b m^{2} x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 261 \, \left (d x\right )^{m} a^{2} b m x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + \left (d x\right )^{m} a^{3} m^{3} x \mathrm{sgn}\left (b x^{3} + a\right ) + 210 \, \left (d x\right )^{m} a^{2} b x^{4} \mathrm{sgn}\left (b x^{3} + a\right ) + 21 \, \left (d x\right )^{m} a^{3} m^{2} x \mathrm{sgn}\left (b x^{3} + a\right ) + 138 \, \left (d x\right )^{m} a^{3} m x \mathrm{sgn}\left (b x^{3} + a\right ) + 280 \, \left (d x\right )^{m} a^{3} x \mathrm{sgn}\left (b x^{3} + a\right )}{m^{4} + 22 \, m^{3} + 159 \, m^{2} + 418 \, m + 280} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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