Optimal. Leaf size=162 \[ -\frac {a^7 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{17}}{17 b^8}-\frac {7 a^5 \left (a+b \sqrt {x}\right )^{18}}{3 b^8}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{19}}{19 b^8}-\frac {7 a^3 \left (a+b \sqrt {x}\right )^{20}}{2 b^8}+\frac {2 a^2 \left (a+b \sqrt {x}\right )^{21}}{b^8}+\frac {2 \left (a+b \sqrt {x}\right )^{23}}{23 b^8}-\frac {7 a \left (a+b \sqrt {x}\right )^{22}}{11 b^8} \]
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Rubi [A] time = 0.09, antiderivative size = 162, 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} \[ \frac {2 a^2 \left (a+b \sqrt {x}\right )^{21}}{b^8}-\frac {7 a^3 \left (a+b \sqrt {x}\right )^{20}}{2 b^8}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{19}}{19 b^8}-\frac {7 a^5 \left (a+b \sqrt {x}\right )^{18}}{3 b^8}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{17}}{17 b^8}-\frac {a^7 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}+\frac {2 \left (a+b \sqrt {x}\right )^{23}}{23 b^8}-\frac {7 a \left (a+b \sqrt {x}\right )^{22}}{11 b^8} \]
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^3 \, dx &=2 \operatorname {Subst}\left (\int x^7 (a+b x)^{15} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a^7 (a+b x)^{15}}{b^7}+\frac {7 a^6 (a+b x)^{16}}{b^7}-\frac {21 a^5 (a+b x)^{17}}{b^7}+\frac {35 a^4 (a+b x)^{18}}{b^7}-\frac {35 a^3 (a+b x)^{19}}{b^7}+\frac {21 a^2 (a+b x)^{20}}{b^7}-\frac {7 a (a+b x)^{21}}{b^7}+\frac {(a+b x)^{22}}{b^7}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^7 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}+\frac {14 a^6 \left (a+b \sqrt {x}\right )^{17}}{17 b^8}-\frac {7 a^5 \left (a+b \sqrt {x}\right )^{18}}{3 b^8}+\frac {70 a^4 \left (a+b \sqrt {x}\right )^{19}}{19 b^8}-\frac {7 a^3 \left (a+b \sqrt {x}\right )^{20}}{2 b^8}+\frac {2 a^2 \left (a+b \sqrt {x}\right )^{21}}{b^8}-\frac {7 a \left (a+b \sqrt {x}\right )^{22}}{11 b^8}+\frac {2 \left (a+b \sqrt {x}\right )^{23}}{23 b^8}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 98, normalized size = 0.60 \[ -\frac {\left (a+b \sqrt {x}\right )^{16} \left (a^7-16 a^6 b \sqrt {x}+136 a^5 b^2 x-816 a^4 b^3 x^{3/2}+3876 a^3 b^4 x^2-15504 a^2 b^5 x^{5/2}+54264 a b^6 x^3-170544 b^7 x^{7/2}\right )}{1961256 b^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 173, normalized size = 1.07 \[ \frac {15}{11} \, a b^{14} x^{11} + \frac {91}{2} \, a^{3} b^{12} x^{10} + \frac {1001}{3} \, a^{5} b^{10} x^{9} + \frac {6435}{8} \, a^{7} b^{8} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {455}{2} \, a^{11} b^{4} x^{6} + 21 \, a^{13} b^{2} x^{5} + \frac {1}{4} \, a^{15} x^{4} + \frac {2}{245157} \, {\left (10659 \, b^{15} x^{11} + 1225785 \, a^{2} b^{13} x^{10} + 17612595 \, a^{4} b^{11} x^{9} + 72177105 \, a^{6} b^{9} x^{8} + 105172353 \, a^{8} b^{7} x^{7} + 56631267 \, a^{10} b^{5} x^{6} + 10140585 \, a^{12} b^{3} x^{5} + 408595 \, a^{14} b x^{4}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 167, normalized size = 1.03 \[ \frac {2}{23} \, b^{15} x^{\frac {23}{2}} + \frac {15}{11} \, a b^{14} x^{11} + 10 \, a^{2} b^{13} x^{\frac {21}{2}} + \frac {91}{2} \, a^{3} b^{12} x^{10} + \frac {2730}{19} \, a^{4} b^{11} x^{\frac {19}{2}} + \frac {1001}{3} \, a^{5} b^{10} x^{9} + \frac {10010}{17} \, a^{6} b^{9} x^{\frac {17}{2}} + \frac {6435}{8} \, a^{7} b^{8} x^{8} + 858 \, a^{8} b^{7} x^{\frac {15}{2}} + 715 \, a^{9} b^{6} x^{7} + 462 \, a^{10} b^{5} x^{\frac {13}{2}} + \frac {455}{2} \, a^{11} b^{4} x^{6} + \frac {910}{11} \, a^{12} b^{3} x^{\frac {11}{2}} + 21 \, a^{13} b^{2} x^{5} + \frac {10}{3} \, a^{14} b x^{\frac {9}{2}} + \frac {1}{4} \, a^{15} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 168, normalized size = 1.04 \[ \frac {2 b^{15} x^{\frac {23}{2}}}{23}+\frac {15 a \,b^{14} x^{11}}{11}+10 a^{2} b^{13} x^{\frac {21}{2}}+\frac {91 a^{3} b^{12} x^{10}}{2}+\frac {2730 a^{4} b^{11} x^{\frac {19}{2}}}{19}+\frac {1001 a^{5} b^{10} x^{9}}{3}+\frac {10010 a^{6} b^{9} x^{\frac {17}{2}}}{17}+\frac {6435 a^{7} b^{8} x^{8}}{8}+858 a^{8} b^{7} x^{\frac {15}{2}}+715 a^{9} b^{6} x^{7}+462 a^{10} b^{5} x^{\frac {13}{2}}+\frac {455 a^{11} b^{4} x^{6}}{2}+\frac {910 a^{12} b^{3} x^{\frac {11}{2}}}{11}+21 a^{13} b^{2} x^{5}+\frac {10 a^{14} b \,x^{\frac {9}{2}}}{3}+\frac {a^{15} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 132, normalized size = 0.81 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{23}}{23 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{22} a}{11 \, b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{21} a^{2}}{b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{20} a^{3}}{2 \, b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )}^{19} a^{4}}{19 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{18} a^{5}}{3 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{17} a^{6}}{17 \, b^{8}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{7}}{8 \, b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 167, normalized size = 1.03 \[ \frac {a^{15}\,x^4}{4}+\frac {2\,b^{15}\,x^{23/2}}{23}+\frac {15\,a\,b^{14}\,x^{11}}{11}+\frac {10\,a^{14}\,b\,x^{9/2}}{3}+21\,a^{13}\,b^2\,x^5+\frac {455\,a^{11}\,b^4\,x^6}{2}+715\,a^9\,b^6\,x^7+\frac {6435\,a^7\,b^8\,x^8}{8}+\frac {1001\,a^5\,b^{10}\,x^9}{3}+\frac {91\,a^3\,b^{12}\,x^{10}}{2}+\frac {910\,a^{12}\,b^3\,x^{11/2}}{11}+462\,a^{10}\,b^5\,x^{13/2}+858\,a^8\,b^7\,x^{15/2}+\frac {10010\,a^6\,b^9\,x^{17/2}}{17}+\frac {2730\,a^4\,b^{11}\,x^{19/2}}{19}+10\,a^2\,b^{13}\,x^{21/2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.35, size = 209, normalized size = 1.29 \[ \frac {a^{15} x^{4}}{4} + \frac {10 a^{14} b x^{\frac {9}{2}}}{3} + 21 a^{13} b^{2} x^{5} + \frac {910 a^{12} b^{3} x^{\frac {11}{2}}}{11} + \frac {455 a^{11} b^{4} x^{6}}{2} + 462 a^{10} b^{5} x^{\frac {13}{2}} + 715 a^{9} b^{6} x^{7} + 858 a^{8} b^{7} x^{\frac {15}{2}} + \frac {6435 a^{7} b^{8} x^{8}}{8} + \frac {10010 a^{6} b^{9} x^{\frac {17}{2}}}{17} + \frac {1001 a^{5} b^{10} x^{9}}{3} + \frac {2730 a^{4} b^{11} x^{\frac {19}{2}}}{19} + \frac {91 a^{3} b^{12} x^{10}}{2} + 10 a^{2} b^{13} x^{\frac {21}{2}} + \frac {15 a b^{14} x^{11}}{11} + \frac {2 b^{15} x^{\frac {23}{2}}}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
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