Optimal. Leaf size=38 \[ \frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2}-\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2}-\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \left (a+b \sqrt {x}\right )^{15} \, dx &=2 \operatorname {Subst}\left (\int x (a+b x)^{15} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^{15}}{b}+\frac {(a+b x)^{16}}{b}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2}+\frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 28, normalized size = 0.74 \[ -\frac {\left (a-16 b \sqrt {x}\right ) \left (a+b \sqrt {x}\right )^{16}}{136 b^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 167, normalized size = 4.39 \[ \frac {15}{8} \, a b^{14} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac {1001}{2} \, a^{5} b^{10} x^{6} + 1287 \, a^{7} b^{8} x^{5} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 455 \, a^{11} b^{4} x^{3} + \frac {105}{2} \, a^{13} b^{2} x^{2} + a^{15} x + \frac {2}{17} \, {\left (b^{15} x^{8} + 119 \, a^{2} b^{13} x^{7} + 1785 \, a^{4} b^{11} x^{6} + 7735 \, a^{6} b^{9} x^{5} + 12155 \, a^{8} b^{7} x^{4} + 7293 \, a^{10} b^{5} x^{3} + 1547 \, a^{12} b^{3} x^{2} + 85 \, a^{14} b x\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 164, normalized size = 4.32 \[ \frac {2}{17} \, b^{15} x^{\frac {17}{2}} + \frac {15}{8} \, a b^{14} x^{8} + 14 \, a^{2} b^{13} x^{\frac {15}{2}} + 65 \, a^{3} b^{12} x^{7} + 210 \, a^{4} b^{11} x^{\frac {13}{2}} + \frac {1001}{2} \, a^{5} b^{10} x^{6} + 910 \, a^{6} b^{9} x^{\frac {11}{2}} + 1287 \, a^{7} b^{8} x^{5} + 1430 \, a^{8} b^{7} x^{\frac {9}{2}} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 858 \, a^{10} b^{5} x^{\frac {7}{2}} + 455 \, a^{11} b^{4} x^{3} + 182 \, a^{12} b^{3} x^{\frac {5}{2}} + \frac {105}{2} \, a^{13} b^{2} x^{2} + 10 \, a^{14} b x^{\frac {3}{2}} + a^{15} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 165, normalized size = 4.34 \[ \frac {2 b^{15} x^{\frac {17}{2}}}{17}+\frac {15 a \,b^{14} x^{8}}{8}+14 a^{2} b^{13} x^{\frac {15}{2}}+65 a^{3} b^{12} x^{7}+210 a^{4} b^{11} x^{\frac {13}{2}}+\frac {1001 a^{5} b^{10} x^{6}}{2}+910 a^{6} b^{9} x^{\frac {11}{2}}+1287 a^{7} b^{8} x^{5}+1430 a^{8} b^{7} x^{\frac {9}{2}}+\frac {5005 a^{9} b^{6} x^{4}}{4}+858 a^{10} b^{5} x^{\frac {7}{2}}+455 a^{11} b^{4} x^{3}+182 a^{12} b^{3} x^{\frac {5}{2}}+\frac {105 a^{13} b^{2} x^{2}}{2}+10 a^{14} b \,x^{\frac {3}{2}}+a^{15} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 30, normalized size = 0.79 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{17}}{17 \, b^{2}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a}{8 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 164, normalized size = 4.32 \[ a^{15}\,x+\frac {2\,b^{15}\,x^{17/2}}{17}+10\,a^{14}\,b\,x^{3/2}+\frac {15\,a\,b^{14}\,x^8}{8}+\frac {105\,a^{13}\,b^2\,x^2}{2}+455\,a^{11}\,b^4\,x^3+\frac {5005\,a^9\,b^6\,x^4}{4}+1287\,a^7\,b^8\,x^5+\frac {1001\,a^5\,b^{10}\,x^6}{2}+65\,a^3\,b^{12}\,x^7+182\,a^{12}\,b^3\,x^{5/2}+858\,a^{10}\,b^5\,x^{7/2}+1430\,a^8\,b^7\,x^{9/2}+910\,a^6\,b^9\,x^{11/2}+210\,a^4\,b^{11}\,x^{13/2}+14\,a^2\,b^{13}\,x^{15/2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.43, size = 197, normalized size = 5.18 \[ a^{15} x + 10 a^{14} b x^{\frac {3}{2}} + \frac {105 a^{13} b^{2} x^{2}}{2} + 182 a^{12} b^{3} x^{\frac {5}{2}} + 455 a^{11} b^{4} x^{3} + 858 a^{10} b^{5} x^{\frac {7}{2}} + \frac {5005 a^{9} b^{6} x^{4}}{4} + 1430 a^{8} b^{7} x^{\frac {9}{2}} + 1287 a^{7} b^{8} x^{5} + 910 a^{6} b^{9} x^{\frac {11}{2}} + \frac {1001 a^{5} b^{10} x^{6}}{2} + 210 a^{4} b^{11} x^{\frac {13}{2}} + 65 a^{3} b^{12} x^{7} + 14 a^{2} b^{13} x^{\frac {15}{2}} + \frac {15 a b^{14} x^{8}}{8} + \frac {2 b^{15} x^{\frac {17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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