Optimal. Leaf size=46 \[ \frac {2 \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^2}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^2} \]
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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} \[ \frac {2 \left (a+b x^3\right )^{5/2} (A b-a B)}{15 b^2}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int x^2 \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int (a+b x)^{3/2} (A+B x) \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {(A b-a B) (a+b x)^{3/2}}{b}+\frac {B (a+b x)^{5/2}}{b}\right ) \, dx,x,x^3\right )\\ &=\frac {2 (A b-a B) \left (a+b x^3\right )^{5/2}}{15 b^2}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.74 \[ \frac {2 \left (a+b x^3\right )^{5/2} \left (-2 a B+7 A b+5 b B x^3\right )}{105 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 73, normalized size = 1.59 \[ \frac {2 \, {\left (5 \, B b^{3} x^{9} + {\left (8 \, B a b^{2} + 7 \, A b^{3}\right )} x^{6} - 2 \, B a^{3} + 7 \, A a^{2} b + {\left (B a^{2} b + 14 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{105 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 44, normalized size = 0.96 \[ \frac {2 \, {\left (5 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} B - 7 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B a + 7 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} A b\right )}}{105 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 0.67 \[ \frac {2 \left (b \,x^{3}+a \right )^{\frac {5}{2}} \left (5 B b \,x^{3}+7 A b -2 B a \right )}{105 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 49, normalized size = 1.07 \[ \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} A}{15 \, b} + \frac {2}{105} \, {\left (\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{b^{2}} - \frac {7 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{b^{2}}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.35, size = 150, normalized size = 3.26 \[ \frac {\left (2\,A\,a^2-\frac {2\,a\,\left (2\,B\,a^2+4\,A\,a\,b-\frac {4\,a\,\left (2\,A\,b^2+\frac {16\,B\,a\,b}{7}\right )}{5\,b}\right )}{3\,b}\right )\,\sqrt {b\,x^3+a}}{3\,b}+\frac {x^3\,\sqrt {b\,x^3+a}\,\left (2\,B\,a^2+4\,A\,a\,b-\frac {4\,a\,\left (2\,A\,b^2+\frac {16\,B\,a\,b}{7}\right )}{5\,b}\right )}{9\,b}+\frac {2\,B\,b\,x^9\,\sqrt {b\,x^3+a}}{21}+\frac {x^6\,\left (2\,A\,b^2+\frac {16\,B\,a\,b}{7}\right )\,\sqrt {b\,x^3+a}}{15\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.52, size = 165, normalized size = 3.59 \[ \begin {cases} \frac {2 A a^{2} \sqrt {a + b x^{3}}}{15 b} + \frac {4 A a x^{3} \sqrt {a + b x^{3}}}{15} + \frac {2 A b x^{6} \sqrt {a + b x^{3}}}{15} - \frac {4 B a^{3} \sqrt {a + b x^{3}}}{105 b^{2}} + \frac {2 B a^{2} x^{3} \sqrt {a + b x^{3}}}{105 b} + \frac {16 B a x^{6} \sqrt {a + b x^{3}}}{105} + \frac {2 B b x^{9} \sqrt {a + b x^{3}}}{21} & \text {for}\: b \neq 0 \\a^{\frac {3}{2}} \left (\frac {A x^{3}}{3} + \frac {B x^{6}}{6}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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