Optimal. Leaf size=120 \[ -\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{38760 a^5 x^8}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{285 a^3 x^9}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}} \]
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Rubi [A] time = 0.05, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ -\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{38760 a^5 x^8}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{285 a^3 x^9}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^{11}} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt {x}\right )}{5 a}\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt {x}\right )}{95 a^2}\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{285 a^3 x^9}-\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt {x}\right )}{285 a^3}\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{285 a^3 x^9}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{4845 a^4 x^{17/2}}+\frac {\left (2 b^4\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt {x}\right )}{4845 a^4}\\ &=-\frac {\left (a+b \sqrt {x}\right )^{16}}{10 a x^{10}}+\frac {2 b \left (a+b \sqrt {x}\right )^{16}}{95 a^2 x^{19/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^{16}}{285 a^3 x^9}+\frac {2 b^3 \left (a+b \sqrt {x}\right )^{16}}{4845 a^4 x^{17/2}}-\frac {b^4 \left (a+b \sqrt {x}\right )^{16}}{38760 a^5 x^8}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.54 \[ -\frac {\left (a+b \sqrt {x}\right )^{16} \left (3876 a^4-816 a^3 b \sqrt {x}+136 a^2 b^2 x-16 a b^3 x^{3/2}+b^4 x^2\right )}{38760 a^5 x^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 168, normalized size = 1.40 \[ -\frac {193800 \, a b^{14} x^{7} + 4408950 \, a^{3} b^{12} x^{6} + 23279256 \, a^{5} b^{10} x^{5} + 41570100 \, a^{7} b^{8} x^{4} + 27713400 \, a^{9} b^{6} x^{3} + 6613425 \, a^{11} b^{4} x^{2} + 452200 \, a^{13} b^{2} x + 3876 \, a^{15} + 16 \, {\left (969 \, b^{15} x^{7} + 72675 \, a^{2} b^{13} x^{6} + 734825 \, a^{4} b^{11} x^{5} + 2204475 \, a^{6} b^{9} x^{4} + 2398275 \, a^{8} b^{7} x^{3} + 969969 \, a^{10} b^{5} x^{2} + 129675 \, a^{12} b^{3} x + 3825 \, a^{14} b\right )} \sqrt {x}}{38760 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 167, normalized size = 1.39 \[ -\frac {15504 \, b^{15} x^{\frac {15}{2}} + 193800 \, a b^{14} x^{7} + 1162800 \, a^{2} b^{13} x^{\frac {13}{2}} + 4408950 \, a^{3} b^{12} x^{6} + 11757200 \, a^{4} b^{11} x^{\frac {11}{2}} + 23279256 \, a^{5} b^{10} x^{5} + 35271600 \, a^{6} b^{9} x^{\frac {9}{2}} + 41570100 \, a^{7} b^{8} x^{4} + 38372400 \, a^{8} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{9} b^{6} x^{3} + 15519504 \, a^{10} b^{5} x^{\frac {5}{2}} + 6613425 \, a^{11} b^{4} x^{2} + 2074800 \, a^{12} b^{3} x^{\frac {3}{2}} + 452200 \, a^{13} b^{2} x + 61200 \, a^{14} b \sqrt {x} + 3876 \, a^{15}}{38760 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 168, normalized size = 1.40 \[ -\frac {2 b^{15}}{5 x^{\frac {5}{2}}}-\frac {5 a \,b^{14}}{x^{3}}-\frac {30 a^{2} b^{13}}{x^{\frac {7}{2}}}-\frac {455 a^{3} b^{12}}{4 x^{4}}-\frac {910 a^{4} b^{11}}{3 x^{\frac {9}{2}}}-\frac {3003 a^{5} b^{10}}{5 x^{5}}-\frac {910 a^{6} b^{9}}{x^{\frac {11}{2}}}-\frac {2145 a^{7} b^{8}}{2 x^{6}}-\frac {990 a^{8} b^{7}}{x^{\frac {13}{2}}}-\frac {715 a^{9} b^{6}}{x^{7}}-\frac {2002 a^{10} b^{5}}{5 x^{\frac {15}{2}}}-\frac {1365 a^{11} b^{4}}{8 x^{8}}-\frac {910 a^{12} b^{3}}{17 x^{\frac {17}{2}}}-\frac {35 a^{13} b^{2}}{3 x^{9}}-\frac {30 a^{14} b}{19 x^{\frac {19}{2}}}-\frac {a^{15}}{10 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 167, normalized size = 1.39 \[ -\frac {15504 \, b^{15} x^{\frac {15}{2}} + 193800 \, a b^{14} x^{7} + 1162800 \, a^{2} b^{13} x^{\frac {13}{2}} + 4408950 \, a^{3} b^{12} x^{6} + 11757200 \, a^{4} b^{11} x^{\frac {11}{2}} + 23279256 \, a^{5} b^{10} x^{5} + 35271600 \, a^{6} b^{9} x^{\frac {9}{2}} + 41570100 \, a^{7} b^{8} x^{4} + 38372400 \, a^{8} b^{7} x^{\frac {7}{2}} + 27713400 \, a^{9} b^{6} x^{3} + 15519504 \, a^{10} b^{5} x^{\frac {5}{2}} + 6613425 \, a^{11} b^{4} x^{2} + 2074800 \, a^{12} b^{3} x^{\frac {3}{2}} + 452200 \, a^{13} b^{2} x + 61200 \, a^{14} b \sqrt {x} + 3876 \, a^{15}}{38760 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 167, normalized size = 1.39 \[ -\frac {\frac {a^{15}}{10}+\frac {2\,b^{15}\,x^{15/2}}{5}+\frac {35\,a^{13}\,b^2\,x}{3}+\frac {30\,a^{14}\,b\,\sqrt {x}}{19}+5\,a\,b^{14}\,x^7+\frac {1365\,a^{11}\,b^4\,x^2}{8}+715\,a^9\,b^6\,x^3+\frac {2145\,a^7\,b^8\,x^4}{2}+\frac {3003\,a^5\,b^{10}\,x^5}{5}+\frac {910\,a^{12}\,b^3\,x^{3/2}}{17}+\frac {455\,a^3\,b^{12}\,x^6}{4}+\frac {2002\,a^{10}\,b^5\,x^{5/2}}{5}+990\,a^8\,b^7\,x^{7/2}+910\,a^6\,b^9\,x^{9/2}+\frac {910\,a^4\,b^{11}\,x^{11/2}}{3}+30\,a^2\,b^{13}\,x^{13/2}}{x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.03, size = 211, normalized size = 1.76 \[ - \frac {a^{15}}{10 x^{10}} - \frac {30 a^{14} b}{19 x^{\frac {19}{2}}} - \frac {35 a^{13} b^{2}}{3 x^{9}} - \frac {910 a^{12} b^{3}}{17 x^{\frac {17}{2}}} - \frac {1365 a^{11} b^{4}}{8 x^{8}} - \frac {2002 a^{10} b^{5}}{5 x^{\frac {15}{2}}} - \frac {715 a^{9} b^{6}}{x^{7}} - \frac {990 a^{8} b^{7}}{x^{\frac {13}{2}}} - \frac {2145 a^{7} b^{8}}{2 x^{6}} - \frac {910 a^{6} b^{9}}{x^{\frac {11}{2}}} - \frac {3003 a^{5} b^{10}}{5 x^{5}} - \frac {910 a^{4} b^{11}}{3 x^{\frac {9}{2}}} - \frac {455 a^{3} b^{12}}{4 x^{4}} - \frac {30 a^{2} b^{13}}{x^{\frac {7}{2}}} - \frac {5 a b^{14}}{x^{3}} - \frac {2 b^{15}}{5 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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