Optimal. Leaf size=73 \[ \frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3} \]
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Rubi [A] time = 0.0548598, antiderivative size = 73, 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 = {446, 77} \[ \frac{2 \sqrt{a+b x^3} (A b-2 a B)}{3 b^3}+\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^5 \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (A+B x)}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a (-A b+a B)}{b^2 (a+b x)^{3/2}}+\frac{A b-2 a B}{b^2 \sqrt{a+b x}}+\frac{B \sqrt{a+b x}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{2 a (A b-a B)}{3 b^3 \sqrt{a+b x^3}}+\frac{2 (A b-2 a B) \sqrt{a+b x^3}}{3 b^3}+\frac{2 B \left (a+b x^3\right )^{3/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.033604, size = 55, normalized size = 0.75 \[ \frac{2 \left (-8 a^2 B+a \left (6 A b-4 b B x^3\right )+b^2 x^3 \left (3 A+B x^3\right )\right )}{9 b^3 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}B{x}^{6}+6\,A{x}^{3}{b}^{2}-8\,B{x}^{3}ab+12\,abA-16\,{a}^{2}B}{9\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{3}+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936323, size = 109, normalized size = 1.49 \begin{align*} \frac{2}{9} \, B{\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}}}{b^{3}} - \frac{6 \, \sqrt{b x^{3} + a} a}{b^{3}} - \frac{3 \, a^{2}}{\sqrt{b x^{3} + a} b^{3}}\right )} + \frac{2}{3} \, A{\left (\frac{\sqrt{b x^{3} + a}}{b^{2}} + \frac{a}{\sqrt{b x^{3} + a} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73956, size = 131, normalized size = 1.79 \begin{align*} \frac{2 \,{\left (B b^{2} x^{6} -{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{3} - 8 \, B a^{2} + 6 \, A a b\right )} \sqrt{b x^{3} + a}}{9 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.8081, size = 124, normalized size = 1.7 \begin{align*} \begin{cases} \frac{4 A a}{3 b^{2} \sqrt{a + b x^{3}}} + \frac{2 A x^{3}}{3 b \sqrt{a + b x^{3}}} - \frac{16 B a^{2}}{9 b^{3} \sqrt{a + b x^{3}}} - \frac{8 B a x^{3}}{9 b^{2} \sqrt{a + b x^{3}}} + \frac{2 B x^{6}}{9 b \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{6}}{6} + \frac{B x^{9}}{9}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12547, size = 88, normalized size = 1.21 \begin{align*} \frac{2 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} B - 6 \, \sqrt{b x^{3} + a} B a + 3 \, \sqrt{b x^{3} + a} A b - \frac{3 \,{\left (B a^{2} - A a b\right )}}{\sqrt{b x^{3} + a}}\right )}}{9 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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