Optimal. Leaf size=46 \[ -\frac {2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 B}{3 b^2 \sqrt {a+b x^3}} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 = {444, 43} \[ -\frac {2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 B}{3 b^2 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{(a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^{5/2}}+\frac {B}{b (a+b x)^{3/2}}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 B}{3 b^2 \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 33, normalized size = 0.72 \[ -\frac {2 \left (2 a B+A b+3 b B x^3\right )}{9 b^2 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 52, normalized size = 1.13 \[ -\frac {2 \, {\left (3 \, B b x^{3} + 2 \, B a + A b\right )} \sqrt {b x^{3} + a}}{9 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 32, normalized size = 0.70 \[ -\frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )} B - B a + A b\right )}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 30, normalized size = 0.65 \[ -\frac {2 \left (3 B b \,x^{3}+A b +2 B a \right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 49, normalized size = 1.07 \[ -\frac {2}{9} \, B {\left (\frac {3}{\sqrt {b x^{3} + a} b^{2}} - \frac {a}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}}\right )} - \frac {2 \, A}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.68, size = 33, normalized size = 0.72 \[ -\frac {2\,A\,b-2\,B\,a+6\,B\,\left (b\,x^3+a\right )}{9\,b^2\,{\left (b\,x^3+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.34, size = 144, normalized size = 3.13 \[ \begin {cases} - \frac {2 A b}{9 a b^{2} \sqrt {a + b x^{3}} + 9 b^{3} x^{3} \sqrt {a + b x^{3}}} - \frac {4 B a}{9 a b^{2} \sqrt {a + b x^{3}} + 9 b^{3} x^{3} \sqrt {a + b x^{3}}} - \frac {6 B b x^{3}}{9 a b^{2} \sqrt {a + b x^{3}} + 9 b^{3} x^{3} \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{3}}{3} + \frac {B x^{6}}{6}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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