Optimal. Leaf size=73 \[ \frac {2 \left (a+b x^3\right )^{5/2} (A b-2 a B)}{15 b^3}-\frac {2 a \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^3}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 73, 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 = {446, 77} \begin {gather*} \frac {2 \left (a+b x^3\right )^{5/2} (A b-2 a B)}{15 b^3}-\frac {2 a \left (a+b x^3\right )^{3/2} (A b-a B)}{9 b^3}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int x^5 \sqrt {a+b x^3} \left (A+B x^3\right ) \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x \sqrt {a+b x} (A+B x) \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a (-A b+a B) \sqrt {a+b x}}{b^2}+\frac {(A b-2 a B) (a+b x)^{3/2}}{b^2}+\frac {B (a+b x)^{5/2}}{b^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 a (A b-a B) \left (a+b x^3\right )^{3/2}}{9 b^3}+\frac {2 (A b-2 a B) \left (a+b x^3\right )^{5/2}}{15 b^3}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.78 \begin {gather*} \frac {2 \left (a+b x^3\right )^{3/2} \left (8 a^2 B-2 a b \left (7 A+6 B x^3\right )+3 b^2 x^3 \left (7 A+5 B x^3\right )\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 56, normalized size = 0.77 \begin {gather*} \frac {2 \left (a+b x^3\right )^{3/2} \left (8 a^2 B-14 a A b-12 a b B x^3+21 A b^2 x^3+15 b^2 B x^6\right )}{315 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 75, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (15 \, B b^{3} x^{9} + 3 \, {\left (B a b^{2} + 7 \, A b^{3}\right )} x^{6} + 8 \, B a^{3} - 14 \, A a^{2} b - {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 73, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} B - 42 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B a + 35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a^{2} + 21 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} A b - 35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} A a b\right )}}{315 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 53, normalized size = 0.73 \begin {gather*} -\frac {2 \left (b \,x^{3}+a \right )^{\frac {3}{2}} \left (-15 B \,b^{2} x^{6}-21 A \,b^{2} x^{3}+12 B a b \,x^{3}+14 A a b -8 B \,a^{2}\right )}{315 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 84, normalized size = 1.15 \begin {gather*} \frac {2}{315} \, B {\left (\frac {15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{b^{3}} - \frac {42 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{b^{3}} + \frac {35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{b^{3}}\right )} + \frac {2}{45} \, A {\left (\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 114, normalized size = 1.56 \begin {gather*} \frac {2\,B\,x^9\,\sqrt {b\,x^3+a}}{21}+\frac {x^6\,\sqrt {b\,x^3+a}\,\left (2\,A\,b+\frac {2\,B\,a}{7}\right )}{15\,b}-\frac {2\,a\,\left (2\,A\,a-\frac {4\,a\,\left (2\,A\,b+\frac {2\,B\,a}{7}\right )}{5\,b}\right )\,\sqrt {b\,x^3+a}}{9\,b^2}+\frac {x^3\,\left (2\,A\,a-\frac {4\,a\,\left (2\,A\,b+\frac {2\,B\,a}{7}\right )}{5\,b}\right )\,\sqrt {b\,x^3+a}}{9\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 168, normalized size = 2.30 \begin {gather*} \begin {cases} - \frac {4 A a^{2} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 A a x^{3} \sqrt {a + b x^{3}}}{45 b} + \frac {2 A x^{6} \sqrt {a + b x^{3}}}{15} + \frac {16 B a^{3} \sqrt {a + b x^{3}}}{315 b^{3}} - \frac {8 B a^{2} x^{3} \sqrt {a + b x^{3}}}{315 b^{2}} + \frac {2 B a x^{6} \sqrt {a + b x^{3}}}{105 b} + \frac {2 B x^{9} \sqrt {a + b x^{3}}}{21} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {A x^{6}}{6} + \frac {B x^{9}}{9}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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