Optimal. Leaf size=190 \[ -\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2} \]
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Rubi [A] time = 0.12, antiderivative size = 190, 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} \[ 4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+1365 a^{11} b^4 \log (x)-\frac {10 a^{14} b}{x^{3/2}}-\frac {a^{15}}{2 x^2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^5} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (3003 a^{10} b^5+\frac {a^{15}}{x^5}+\frac {15 a^{14} b}{x^4}+\frac {105 a^{13} b^2}{x^3}+\frac {455 a^{12} b^3}{x^2}+\frac {1365 a^{11} b^4}{x}+5005 a^9 b^6 x+6435 a^8 b^7 x^2+6435 a^7 b^8 x^3+5005 a^6 b^9 x^4+3003 a^5 b^{10} x^5+1365 a^4 b^{11} x^6+455 a^3 b^{12} x^7+105 a^2 b^{13} x^8+15 a b^{14} x^9+b^{15} x^{10}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2}+1365 a^{11} b^4 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.14, size = 190, normalized size = 1.00 \[ -\frac {a^{15}}{2 x^2}-\frac {10 a^{14} b}{x^{3/2}}-\frac {105 a^{13} b^2}{x}-\frac {910 a^{12} b^3}{\sqrt {x}}+1365 a^{11} b^4 \log (x)+6006 a^{10} b^5 \sqrt {x}+5005 a^9 b^6 x+4290 a^8 b^7 x^{3/2}+\frac {6435}{2} a^7 b^8 x^2+2002 a^6 b^9 x^{5/2}+1001 a^5 b^{10} x^3+390 a^4 b^{11} x^{7/2}+\frac {455}{4} a^3 b^{12} x^4+\frac {70}{3} a^2 b^{13} x^{9/2}+3 a b^{14} x^5+\frac {2}{11} b^{15} x^{11/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 172, normalized size = 0.91 \[ \frac {396 \, a b^{14} x^{7} + 15015 \, a^{3} b^{12} x^{6} + 132132 \, a^{5} b^{10} x^{5} + 424710 \, a^{7} b^{8} x^{4} + 660660 \, a^{9} b^{6} x^{3} + 360360 \, a^{11} b^{4} x^{2} \log \left (\sqrt {x}\right ) - 13860 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \, {\left (3 \, b^{15} x^{7} + 385 \, a^{2} b^{13} x^{6} + 6435 \, a^{4} b^{11} x^{5} + 33033 \, a^{6} b^{9} x^{4} + 70785 \, a^{8} b^{7} x^{3} + 99099 \, a^{10} b^{5} x^{2} - 15015 \, a^{12} b^{3} x - 165 \, a^{14} b\right )} \sqrt {x}}{132 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 164, normalized size = 0.86 \[ \frac {2}{11} \, b^{15} x^{\frac {11}{2}} + 3 \, a b^{14} x^{5} + \frac {70}{3} \, a^{2} b^{13} x^{\frac {9}{2}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac {7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac {5}{2}} + \frac {6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac {3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \left ({\left | x \right |}\right ) + 6006 \, a^{10} b^{5} \sqrt {x} - \frac {1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt {x} + a^{15}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 165, normalized size = 0.87 \[ \frac {2 b^{15} x^{\frac {11}{2}}}{11}+3 a \,b^{14} x^{5}+\frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3}+\frac {455 a^{3} b^{12} x^{4}}{4}+390 a^{4} b^{11} x^{\frac {7}{2}}+1001 a^{5} b^{10} x^{3}+2002 a^{6} b^{9} x^{\frac {5}{2}}+\frac {6435 a^{7} b^{8} x^{2}}{2}+4290 a^{8} b^{7} x^{\frac {3}{2}}+1365 a^{11} b^{4} \ln \relax (x )+5005 a^{9} b^{6} x +6006 a^{10} b^{5} \sqrt {x}-\frac {910 a^{12} b^{3}}{\sqrt {x}}-\frac {105 a^{13} b^{2}}{x}-\frac {10 a^{14} b}{x^{\frac {3}{2}}}-\frac {a^{15}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 163, normalized size = 0.86 \[ \frac {2}{11} \, b^{15} x^{\frac {11}{2}} + 3 \, a b^{14} x^{5} + \frac {70}{3} \, a^{2} b^{13} x^{\frac {9}{2}} + \frac {455}{4} \, a^{3} b^{12} x^{4} + 390 \, a^{4} b^{11} x^{\frac {7}{2}} + 1001 \, a^{5} b^{10} x^{3} + 2002 \, a^{6} b^{9} x^{\frac {5}{2}} + \frac {6435}{2} \, a^{7} b^{8} x^{2} + 4290 \, a^{8} b^{7} x^{\frac {3}{2}} + 5005 \, a^{9} b^{6} x + 1365 \, a^{11} b^{4} \log \relax (x) + 6006 \, a^{10} b^{5} \sqrt {x} - \frac {1820 \, a^{12} b^{3} x^{\frac {3}{2}} + 210 \, a^{13} b^{2} x + 20 \, a^{14} b \sqrt {x} + a^{15}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 167, normalized size = 0.88 \[ \frac {2\,b^{15}\,x^{11/2}}{11}-\frac {\frac {a^{15}}{2}+105\,a^{13}\,b^2\,x+10\,a^{14}\,b\,\sqrt {x}+910\,a^{12}\,b^3\,x^{3/2}}{x^2}+2730\,a^{11}\,b^4\,\ln \left (\sqrt {x}\right )+5005\,a^9\,b^6\,x+3\,a\,b^{14}\,x^5+\frac {6435\,a^7\,b^8\,x^2}{2}+1001\,a^5\,b^{10}\,x^3+6006\,a^{10}\,b^5\,\sqrt {x}+\frac {455\,a^3\,b^{12}\,x^4}{4}+4290\,a^8\,b^7\,x^{3/2}+2002\,a^6\,b^9\,x^{5/2}+390\,a^4\,b^{11}\,x^{7/2}+\frac {70\,a^2\,b^{13}\,x^{9/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.63, size = 196, normalized size = 1.03 \[ - \frac {a^{15}}{2 x^{2}} - \frac {10 a^{14} b}{x^{\frac {3}{2}}} - \frac {105 a^{13} b^{2}}{x} - \frac {910 a^{12} b^{3}}{\sqrt {x}} + 1365 a^{11} b^{4} \log {\relax (x )} + 6006 a^{10} b^{5} \sqrt {x} + 5005 a^{9} b^{6} x + 4290 a^{8} b^{7} x^{\frac {3}{2}} + \frac {6435 a^{7} b^{8} x^{2}}{2} + 2002 a^{6} b^{9} x^{\frac {5}{2}} + 1001 a^{5} b^{10} x^{3} + 390 a^{4} b^{11} x^{\frac {7}{2}} + \frac {455 a^{3} b^{12} x^{4}}{4} + \frac {70 a^{2} b^{13} x^{\frac {9}{2}}}{3} + 3 a b^{14} x^{5} + \frac {2 b^{15} x^{\frac {11}{2}}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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