Optimal. Leaf size=80 \[ -\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4} \]
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Rubi [A] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}-\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^{15} x \, dx &=2 \operatorname {Subst}\left (\int x^3 (a+b x)^{15} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^{15}}{b^3}+\frac {3 a^2 (a+b x)^{16}}{b^3}-\frac {3 a (a+b x)^{17}}{b^3}+\frac {(a+b x)^{18}}{b^3}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.62 \[ -\frac {\left (a+b \sqrt {x}\right )^{16} \left (a^3-16 a^2 b \sqrt {x}+136 a b^2 x-816 b^3 x^{3/2}\right )}{7752 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.14, size = 173, normalized size = 2.16 \[ \frac {5}{3} \, a b^{14} x^{9} + \frac {455}{8} \, a^{3} b^{12} x^{8} + 429 \, a^{5} b^{10} x^{7} + \frac {2145}{2} \, a^{7} b^{8} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac {1365}{4} \, a^{11} b^{4} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {1}{2} \, a^{15} x^{2} + \frac {2}{969} \, {\left (51 \, b^{15} x^{9} + 5985 \, a^{2} b^{13} x^{8} + 88179 \, a^{4} b^{11} x^{7} + 373065 \, a^{6} b^{9} x^{6} + 566865 \, a^{8} b^{7} x^{5} + 323323 \, a^{10} b^{5} x^{4} + 62985 \, a^{12} b^{3} x^{3} + 2907 \, a^{14} b x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 167, normalized size = 2.09 \[ \frac {2}{19} \, b^{15} x^{\frac {19}{2}} + \frac {5}{3} \, a b^{14} x^{9} + \frac {210}{17} \, a^{2} b^{13} x^{\frac {17}{2}} + \frac {455}{8} \, a^{3} b^{12} x^{8} + 182 \, a^{4} b^{11} x^{\frac {15}{2}} + 429 \, a^{5} b^{10} x^{7} + 770 \, a^{6} b^{9} x^{\frac {13}{2}} + \frac {2145}{2} \, a^{7} b^{8} x^{6} + 1170 \, a^{8} b^{7} x^{\frac {11}{2}} + 1001 \, a^{9} b^{6} x^{5} + \frac {2002}{3} \, a^{10} b^{5} x^{\frac {9}{2}} + \frac {1365}{4} \, a^{11} b^{4} x^{4} + 130 \, a^{12} b^{3} x^{\frac {7}{2}} + 35 \, a^{13} b^{2} x^{3} + 6 \, a^{14} b x^{\frac {5}{2}} + \frac {1}{2} \, a^{15} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 168, normalized size = 2.10 \[ \frac {2 b^{15} x^{\frac {19}{2}}}{19}+\frac {5 a \,b^{14} x^{9}}{3}+\frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17}+\frac {455 a^{3} b^{12} x^{8}}{8}+182 a^{4} b^{11} x^{\frac {15}{2}}+429 a^{5} b^{10} x^{7}+770 a^{6} b^{9} x^{\frac {13}{2}}+\frac {2145 a^{7} b^{8} x^{6}}{2}+1170 a^{8} b^{7} x^{\frac {11}{2}}+1001 a^{9} b^{6} x^{5}+\frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{4}}{4}+130 a^{12} b^{3} x^{\frac {7}{2}}+35 a^{13} b^{2} x^{3}+6 a^{14} b \,x^{\frac {5}{2}}+\frac {a^{15} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 64, normalized size = 0.80 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{19}}{19 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{18} a}{3 \, b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{17} a^{2}}{17 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{3}}{8 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 167, normalized size = 2.09 \[ \frac {a^{15}\,x^2}{2}+\frac {2\,b^{15}\,x^{19/2}}{19}+6\,a^{14}\,b\,x^{5/2}+\frac {5\,a\,b^{14}\,x^9}{3}+35\,a^{13}\,b^2\,x^3+\frac {1365\,a^{11}\,b^4\,x^4}{4}+1001\,a^9\,b^6\,x^5+\frac {2145\,a^7\,b^8\,x^6}{2}+429\,a^5\,b^{10}\,x^7+\frac {455\,a^3\,b^{12}\,x^8}{8}+130\,a^{12}\,b^3\,x^{7/2}+\frac {2002\,a^{10}\,b^5\,x^{9/2}}{3}+1170\,a^8\,b^7\,x^{11/2}+770\,a^6\,b^9\,x^{13/2}+182\,a^4\,b^{11}\,x^{15/2}+\frac {210\,a^2\,b^{13}\,x^{17/2}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.79, size = 204, normalized size = 2.55 \[ \frac {a^{15} x^{2}}{2} + 6 a^{14} b x^{\frac {5}{2}} + 35 a^{13} b^{2} x^{3} + 130 a^{12} b^{3} x^{\frac {7}{2}} + \frac {1365 a^{11} b^{4} x^{4}}{4} + \frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3} + 1001 a^{9} b^{6} x^{5} + 1170 a^{8} b^{7} x^{\frac {11}{2}} + \frac {2145 a^{7} b^{8} x^{6}}{2} + 770 a^{6} b^{9} x^{\frac {13}{2}} + 429 a^{5} b^{10} x^{7} + 182 a^{4} b^{11} x^{\frac {15}{2}} + \frac {455 a^{3} b^{12} x^{8}}{8} + \frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17} + \frac {5 a b^{14} x^{9}}{3} + \frac {2 b^{15} x^{\frac {19}{2}}}{19} \]
Verification of antiderivative is not currently implemented for this CAS.
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