Optimal. Leaf size=122 \[ -\frac {2 a^5 \left (a+b \sqrt {x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{12}}{6 b^6}-\frac {20 a^3 \left (a+b \sqrt {x}\right )^{13}}{13 b^6}+\frac {10 a^2 \left (a+b \sqrt {x}\right )^{14}}{7 b^6}+\frac {\left (a+b \sqrt {x}\right )^{16}}{8 b^6}-\frac {2 a \left (a+b \sqrt {x}\right )^{15}}{3 b^6} \]
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Rubi [A] time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {10 a^2 \left (a+b \sqrt {x}\right )^{14}}{7 b^6}-\frac {20 a^3 \left (a+b \sqrt {x}\right )^{13}}{13 b^6}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{12}}{6 b^6}-\frac {2 a^5 \left (a+b \sqrt {x}\right )^{11}}{11 b^6}+\frac {\left (a+b \sqrt {x}\right )^{16}}{8 b^6}-\frac {2 a \left (a+b \sqrt {x}\right )^{15}}{3 b^6} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^{10} x^2 \, dx &=2 \operatorname {Subst}\left (\int x^5 (a+b x)^{10} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^5 (a+b x)^{10}}{b^5}+\frac {5 a^4 (a+b x)^{11}}{b^5}-\frac {10 a^3 (a+b x)^{12}}{b^5}+\frac {10 a^2 (a+b x)^{13}}{b^5}-\frac {5 a (a+b x)^{14}}{b^5}+\frac {(a+b x)^{15}}{b^5}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {2 a^5 \left (a+b \sqrt {x}\right )^{11}}{11 b^6}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{12}}{6 b^6}-\frac {20 a^3 \left (a+b \sqrt {x}\right )^{13}}{13 b^6}+\frac {10 a^2 \left (a+b \sqrt {x}\right )^{14}}{7 b^6}-\frac {2 a \left (a+b \sqrt {x}\right )^{15}}{3 b^6}+\frac {\left (a+b \sqrt {x}\right )^{16}}{8 b^6}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 74, normalized size = 0.61 \[ -\frac {\left (a+b \sqrt {x}\right )^{11} \left (a^5-11 a^4 b \sqrt {x}+66 a^3 b^2 x-286 a^2 b^3 x^{3/2}+1001 a b^4 x^2-3003 b^5 x^{5/2}\right )}{24024 b^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 118, normalized size = 0.97 \[ \frac {1}{8} \, b^{10} x^{8} + \frac {45}{7} \, a^{2} b^{8} x^{7} + 35 \, a^{4} b^{6} x^{6} + 42 \, a^{6} b^{4} x^{5} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {1}{3} \, a^{10} x^{3} + \frac {4}{3003} \, {\left (1001 \, a b^{9} x^{7} + 13860 \, a^{3} b^{7} x^{6} + 34398 \, a^{5} b^{5} x^{5} + 20020 \, a^{7} b^{3} x^{4} + 2145 \, a^{9} b x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 112, normalized size = 0.92 \[ \frac {1}{8} \, b^{10} x^{8} + \frac {4}{3} \, a b^{9} x^{\frac {15}{2}} + \frac {45}{7} \, a^{2} b^{8} x^{7} + \frac {240}{13} \, a^{3} b^{7} x^{\frac {13}{2}} + 35 \, a^{4} b^{6} x^{6} + \frac {504}{11} \, a^{5} b^{5} x^{\frac {11}{2}} + 42 \, a^{6} b^{4} x^{5} + \frac {80}{3} \, a^{7} b^{3} x^{\frac {9}{2}} + \frac {45}{4} \, a^{8} b^{2} x^{4} + \frac {20}{7} \, a^{9} b x^{\frac {7}{2}} + \frac {1}{3} \, a^{10} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 113, normalized size = 0.93 \[ \frac {b^{10} x^{8}}{8}+\frac {4 a \,b^{9} x^{\frac {15}{2}}}{3}+\frac {45 a^{2} b^{8} x^{7}}{7}+\frac {240 a^{3} b^{7} x^{\frac {13}{2}}}{13}+35 a^{4} b^{6} x^{6}+\frac {504 a^{5} b^{5} x^{\frac {11}{2}}}{11}+42 a^{6} b^{4} x^{5}+\frac {80 a^{7} b^{3} x^{\frac {9}{2}}}{3}+\frac {45 a^{8} b^{2} x^{4}}{4}+\frac {20 a^{9} b \,x^{\frac {7}{2}}}{7}+\frac {a^{10} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 98, normalized size = 0.80 \[ \frac {{\left (b \sqrt {x} + a\right )}^{16}}{8 \, b^{6}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{15} a}{3 \, b^{6}} + \frac {10 \, {\left (b \sqrt {x} + a\right )}^{14} a^{2}}{7 \, b^{6}} - \frac {20 \, {\left (b \sqrt {x} + a\right )}^{13} a^{3}}{13 \, b^{6}} + \frac {5 \, {\left (b \sqrt {x} + a\right )}^{12} a^{4}}{6 \, b^{6}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{11} a^{5}}{11 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 112, normalized size = 0.92 \[ \frac {a^{10}\,x^3}{3}+\frac {b^{10}\,x^8}{8}+\frac {20\,a^9\,b\,x^{7/2}}{7}+\frac {4\,a\,b^9\,x^{15/2}}{3}+\frac {45\,a^8\,b^2\,x^4}{4}+42\,a^6\,b^4\,x^5+35\,a^4\,b^6\,x^6+\frac {45\,a^2\,b^8\,x^7}{7}+\frac {80\,a^7\,b^3\,x^{9/2}}{3}+\frac {504\,a^5\,b^5\,x^{11/2}}{11}+\frac {240\,a^3\,b^7\,x^{13/2}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.01, size = 139, normalized size = 1.14 \[ \frac {a^{10} x^{3}}{3} + \frac {20 a^{9} b x^{\frac {7}{2}}}{7} + \frac {45 a^{8} b^{2} x^{4}}{4} + \frac {80 a^{7} b^{3} x^{\frac {9}{2}}}{3} + 42 a^{6} b^{4} x^{5} + \frac {504 a^{5} b^{5} x^{\frac {11}{2}}}{11} + 35 a^{4} b^{6} x^{6} + \frac {240 a^{3} b^{7} x^{\frac {13}{2}}}{13} + \frac {45 a^{2} b^{8} x^{7}}{7} + \frac {4 a b^{9} x^{\frac {15}{2}}}{3} + \frac {b^{10} x^{8}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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