Optimal. Leaf size=103 \[ \frac {2 a^2 (A b-a B) \sqrt {a+b x^3}}{3 b^4}-\frac {2 a (2 A b-3 a B) \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 (A b-3 a B) \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^4} \]
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Rubi [A]
time = 0.05, antiderivative size = 103, 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 = {457, 78}
\begin {gather*} \frac {2 a^2 \sqrt {a+b x^3} (A b-a B)}{3 b^4}+\frac {2 \left (a+b x^3\right )^{5/2} (A b-3 a B)}{15 b^4}-\frac {2 a \left (a+b x^3\right )^{3/2} (2 A b-3 a B)}{9 b^4}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^8 \left (A+B x^3\right )}{\sqrt {a+b x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2 (A+B x)}{\sqrt {a+b x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a^2 (-A b+a B)}{b^3 \sqrt {a+b x}}+\frac {a (-2 A b+3 a B) \sqrt {a+b x}}{b^3}+\frac {(A b-3 a B) (a+b x)^{3/2}}{b^3}+\frac {B (a+b x)^{5/2}}{b^3}\right ) \, dx,x,x^3\right )\\ &=\frac {2 a^2 (A b-a B) \sqrt {a+b x^3}}{3 b^4}-\frac {2 a (2 A b-3 a B) \left (a+b x^3\right )^{3/2}}{9 b^4}+\frac {2 (A b-3 a B) \left (a+b x^3\right )^{5/2}}{15 b^4}+\frac {2 B \left (a+b x^3\right )^{7/2}}{21 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 80, normalized size = 0.78 \begin {gather*} \frac {2 \sqrt {a+b x^3} \left (56 a^2 A b-48 a^3 B-28 a A b^2 x^3+24 a^2 b B x^3+21 A b^3 x^6-18 a b^2 B x^6+15 b^3 B x^9\right )}{315 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 132, normalized size = 1.28
method | result | size |
gosper | \(\frac {2 \sqrt {b \,x^{3}+a}\, \left (15 B \,x^{9} b^{3}+21 A \,b^{3} x^{6}-18 B a \,b^{2} x^{6}-28 A a \,b^{2} x^{3}+24 B \,a^{2} b \,x^{3}+56 A \,a^{2} b -48 B \,a^{3}\right )}{315 b^{4}}\) | \(77\) |
trager | \(\frac {2 \sqrt {b \,x^{3}+a}\, \left (15 B \,x^{9} b^{3}+21 A \,b^{3} x^{6}-18 B a \,b^{2} x^{6}-28 A a \,b^{2} x^{3}+24 B \,a^{2} b \,x^{3}+56 A \,a^{2} b -48 B \,a^{3}\right )}{315 b^{4}}\) | \(77\) |
risch | \(\frac {2 \sqrt {b \,x^{3}+a}\, \left (15 B \,x^{9} b^{3}+21 A \,b^{3} x^{6}-18 B a \,b^{2} x^{6}-28 A a \,b^{2} x^{3}+24 B \,a^{2} b \,x^{3}+56 A \,a^{2} b -48 B \,a^{3}\right )}{315 b^{4}}\) | \(77\) |
elliptic | \(\frac {2 B \,x^{9} \sqrt {b \,x^{3}+a}}{21 b}+\frac {2 \left (A -\frac {6 a B}{7 b}\right ) x^{6} \sqrt {b \,x^{3}+a}}{15 b}-\frac {8 a \left (A -\frac {6 a B}{7 b}\right ) x^{3} \sqrt {b \,x^{3}+a}}{45 b^{2}}+\frac {16 a^{2} \left (A -\frac {6 a B}{7 b}\right ) \sqrt {b \,x^{3}+a}}{45 b^{3}}\) | \(99\) |
default | \(B \left (\frac {2 x^{9} \sqrt {b \,x^{3}+a}}{21 b}-\frac {4 a \,x^{6} \sqrt {b \,x^{3}+a}}{35 b^{2}}+\frac {16 a^{2} x^{3} \sqrt {b \,x^{3}+a}}{105 b^{3}}-\frac {32 a^{3} \sqrt {b \,x^{3}+a}}{105 b^{4}}\right )+A \left (\frac {2 x^{6} \sqrt {b \,x^{3}+a}}{15 b}-\frac {8 a \,x^{3} \sqrt {b \,x^{3}+a}}{45 b^{2}}+\frac {16 a^{2} \sqrt {b \,x^{3}+a}}{45 b^{3}}\right )\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 118, normalized size = 1.15 \begin {gather*} \frac {2}{105} \, B {\left (\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}}}{b^{4}} - \frac {21 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} a}{b^{4}} + \frac {35 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a^{2}}{b^{4}} - \frac {35 \, \sqrt {b x^{3} + a} a^{3}}{b^{4}}\right )} + \frac {2}{45} \, A {\left (\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}}}{b^{3}} - \frac {10 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a}{b^{3}} + \frac {15 \, \sqrt {b x^{3} + a} a^{2}}{b^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.85, size = 76, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (15 \, B b^{3} x^{9} - 3 \, {\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} - 48 \, B a^{3} + 56 \, A a^{2} b + 4 \, {\left (6 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {b x^{3} + a}}{315 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 175, normalized size = 1.70 \begin {gather*} \begin {cases} \frac {16 A a^{2} \sqrt {a + b x^{3}}}{45 b^{3}} - \frac {8 A a x^{3} \sqrt {a + b x^{3}}}{45 b^{2}} + \frac {2 A x^{6} \sqrt {a + b x^{3}}}{15 b} - \frac {32 B a^{3} \sqrt {a + b x^{3}}}{105 b^{4}} + \frac {16 B a^{2} x^{3} \sqrt {a + b x^{3}}}{105 b^{3}} - \frac {4 B a x^{6} \sqrt {a + b x^{3}}}{35 b^{2}} + \frac {2 B x^{9} \sqrt {a + b x^{3}}}{21 b} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{9}}{9} + \frac {B x^{12}}{12}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.02, size = 101, normalized size = 0.98 \begin {gather*} -\frac {2 \, {\left (B a^{3} - A a^{2} b\right )} \sqrt {b x^{3} + a}}{3 \, b^{4}} + \frac {2 \, {\left (15 \, {\left (b x^{3} + a\right )}^{\frac {7}{2}} B - 63 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} B a + 105 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} B a^{2} + 21 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} A b - 70 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} A a b\right )}}{315 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.68, size = 104, normalized size = 1.01 \begin {gather*} \frac {8\,a^2\,\sqrt {b\,x^3+a}\,\left (2\,A-\frac {12\,B\,a}{7\,b}\right )}{45\,b^3}+\frac {x^6\,\sqrt {b\,x^3+a}\,\left (2\,A-\frac {12\,B\,a}{7\,b}\right )}{15\,b}+\frac {2\,B\,x^9\,\sqrt {b\,x^3+a}}{21\,b}-\frac {4\,a\,x^3\,\sqrt {b\,x^3+a}\,\left (2\,A-\frac {12\,B\,a}{7\,b}\right )}{45\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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