Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{11/2}}{11 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \begin {gather*} \frac {2 a^2 (a+b x)^{11/2}}{11 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int x^2 (a+b x)^{9/2} \, dx &=\int \left (\frac {a^2 (a+b x)^{9/2}}{b^2}-\frac {2 a (a+b x)^{11/2}}{b^2}+\frac {(a+b x)^{13/2}}{b^2}\right ) \, dx\\ &=\frac {2 a^2 (a+b x)^{11/2}}{11 b^3}-\frac {4 a (a+b x)^{13/2}}{13 b^3}+\frac {2 (a+b x)^{15/2}}{15 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (8 a^2-44 a b x+143 b^2 x^2\right )}{2145 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 39, normalized size = 0.74 \begin {gather*} \frac {2 (a+b x)^{11/2} \left (195 a^2-330 a (a+b x)+143 (a+b x)^2\right )}{2145 b^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 86, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (143 \, b^{7} x^{7} + 671 \, a b^{6} x^{6} + 1218 \, a^{2} b^{5} x^{5} + 1030 \, a^{3} b^{4} x^{4} + 355 \, a^{4} b^{3} x^{3} + 3 \, a^{5} b^{2} x^{2} - 4 \, a^{6} b x + 8 \, a^{7}\right )} \sqrt {b x + a}}{2145 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 421, normalized size = 7.94 \begin {gather*} \frac {2 \, {\left (\frac {3003 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{5}}{b^{2}} + \frac {6435 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{4}}{b^{2}} + \frac {1430 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a^{3}}{b^{2}} + \frac {650 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} a^{2}}{b^{2}} + \frac {75 \, {\left (231 \, {\left (b x + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b x + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b x + a} a^{6}\right )} a}{b^{2}} + \frac {7 \, {\left (429 \, {\left (b x + a\right )}^{\frac {15}{2}} - 3465 \, {\left (b x + a\right )}^{\frac {13}{2}} a + 12285 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2} - 25025 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{3} + 32175 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 27027 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + 15015 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} - 6435 \, \sqrt {b x + a} a^{7}\right )}}{b^{2}}\right )}}{45045 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.60 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {11}{2}} \left (143 b^{2} x^{2}-44 a b x +8 a^{2}\right )}{2145 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {15}{2}}}{15 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {13}{2}} a}{13 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}} a^{2}}{11 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 36, normalized size = 0.68 \begin {gather*} \frac {\frac {2\,{\left (a+b\,x\right )}^{15/2}}{15}-\frac {4\,a\,{\left (a+b\,x\right )}^{13/2}}{13}+\frac {2\,a^2\,{\left (a+b\,x\right )}^{11/2}}{11}}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.86, size = 168, normalized size = 3.17 \begin {gather*} \begin {cases} \frac {16 a^{7} \sqrt {a + b x}}{2145 b^{3}} - \frac {8 a^{6} x \sqrt {a + b x}}{2145 b^{2}} + \frac {2 a^{5} x^{2} \sqrt {a + b x}}{715 b} + \frac {142 a^{4} x^{3} \sqrt {a + b x}}{429} + \frac {412 a^{3} b x^{4} \sqrt {a + b x}}{429} + \frac {812 a^{2} b^{2} x^{5} \sqrt {a + b x}}{715} + \frac {122 a b^{3} x^{6} \sqrt {a + b x}}{195} + \frac {2 b^{4} x^{7} \sqrt {a + b x}}{15} & \text {for}\: b \neq 0 \\\frac {a^{\frac {9}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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