Optimal. Leaf size=198 \[ -\frac {a^{15}}{7 x^7}-\frac {30 a^{14} b}{13 x^{13/2}}-\frac {35 a^{13} b^2}{2 x^6}-\frac {910 a^{12} b^3}{11 x^{11/2}}-\frac {273 a^{11} b^4}{x^5}-\frac {2002 a^{10} b^5}{3 x^{9/2}}-\frac {5005 a^9 b^6}{4 x^4}-\frac {12870 a^8 b^7}{7 x^{7/2}}-\frac {2145 a^7 b^8}{x^3}-\frac {2002 a^6 b^9}{x^{5/2}}-\frac {3003 a^5 b^{10}}{2 x^2}-\frac {910 a^4 b^{11}}{x^{3/2}}-\frac {455 a^3 b^{12}}{x}-\frac {210 a^2 b^{13}}{\sqrt {x}}+15 a b^{14} \log (x)+2 b^{15} \sqrt {x} \]
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Rubi [A] time = 0.11, antiderivative size = 198, 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} \[ -\frac {35 a^{13} b^2}{2 x^6}-\frac {910 a^{12} b^3}{11 x^{11/2}}-\frac {273 a^{11} b^4}{x^5}-\frac {2002 a^{10} b^5}{3 x^{9/2}}-\frac {5005 a^9 b^6}{4 x^4}-\frac {12870 a^8 b^7}{7 x^{7/2}}-\frac {2145 a^7 b^8}{x^3}-\frac {2002 a^6 b^9}{x^{5/2}}-\frac {3003 a^5 b^{10}}{2 x^2}-\frac {910 a^4 b^{11}}{x^{3/2}}-\frac {455 a^3 b^{12}}{x}-\frac {210 a^2 b^{13}}{\sqrt {x}}-\frac {30 a^{14} b}{13 x^{13/2}}-\frac {a^{15}}{7 x^7}+15 a b^{14} \log (x)+2 b^{15} \sqrt {x} \]
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^8} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{15}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (b^{15}+\frac {a^{15}}{x^{15}}+\frac {15 a^{14} b}{x^{14}}+\frac {105 a^{13} b^2}{x^{13}}+\frac {455 a^{12} b^3}{x^{12}}+\frac {1365 a^{11} b^4}{x^{11}}+\frac {3003 a^{10} b^5}{x^{10}}+\frac {5005 a^9 b^6}{x^9}+\frac {6435 a^8 b^7}{x^8}+\frac {6435 a^7 b^8}{x^7}+\frac {5005 a^6 b^9}{x^6}+\frac {3003 a^5 b^{10}}{x^5}+\frac {1365 a^4 b^{11}}{x^4}+\frac {455 a^3 b^{12}}{x^3}+\frac {105 a^2 b^{13}}{x^2}+\frac {15 a b^{14}}{x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{7 x^7}-\frac {30 a^{14} b}{13 x^{13/2}}-\frac {35 a^{13} b^2}{2 x^6}-\frac {910 a^{12} b^3}{11 x^{11/2}}-\frac {273 a^{11} b^4}{x^5}-\frac {2002 a^{10} b^5}{3 x^{9/2}}-\frac {5005 a^9 b^6}{4 x^4}-\frac {12870 a^8 b^7}{7 x^{7/2}}-\frac {2145 a^7 b^8}{x^3}-\frac {2002 a^6 b^9}{x^{5/2}}-\frac {3003 a^5 b^{10}}{2 x^2}-\frac {910 a^4 b^{11}}{x^{3/2}}-\frac {455 a^3 b^{12}}{x}-\frac {210 a^2 b^{13}}{\sqrt {x}}+2 b^{15} \sqrt {x}+15 a b^{14} \log (x)\\ \end {align*}
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Mathematica [A] time = 0.18, size = 198, normalized size = 1.00 \[ -\frac {a^{15}}{7 x^7}-\frac {30 a^{14} b}{13 x^{13/2}}-\frac {35 a^{13} b^2}{2 x^6}-\frac {910 a^{12} b^3}{11 x^{11/2}}-\frac {273 a^{11} b^4}{x^5}-\frac {2002 a^{10} b^5}{3 x^{9/2}}-\frac {5005 a^9 b^6}{4 x^4}-\frac {12870 a^8 b^7}{7 x^{7/2}}-\frac {2145 a^7 b^8}{x^3}-\frac {2002 a^6 b^9}{x^{5/2}}-\frac {3003 a^5 b^{10}}{2 x^2}-\frac {910 a^4 b^{11}}{x^{3/2}}-\frac {455 a^3 b^{12}}{x}-\frac {210 a^2 b^{13}}{\sqrt {x}}+15 a b^{14} \log (x)+2 b^{15} \sqrt {x} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 172, normalized size = 0.87 \[ \frac {360360 \, a b^{14} x^{7} \log \left (\sqrt {x}\right ) - 5465460 \, a^{3} b^{12} x^{6} - 18036018 \, a^{5} b^{10} x^{5} - 25765740 \, a^{7} b^{8} x^{4} - 15030015 \, a^{9} b^{6} x^{3} - 3279276 \, a^{11} b^{4} x^{2} - 210210 \, a^{13} b^{2} x - 1716 \, a^{15} + 8 \, {\left (3003 \, b^{15} x^{7} - 315315 \, a^{2} b^{13} x^{6} - 1366365 \, a^{4} b^{11} x^{5} - 3006003 \, a^{6} b^{9} x^{4} - 2760615 \, a^{8} b^{7} x^{3} - 1002001 \, a^{10} b^{5} x^{2} - 124215 \, a^{12} b^{3} x - 3465 \, a^{14} b\right )} \sqrt {x}}{12012 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 168, normalized size = 0.85 \[ 15 \, a b^{14} \log \left ({\left | x \right |}\right ) + 2 \, b^{15} \sqrt {x} - \frac {2522520 \, a^{2} b^{13} x^{\frac {13}{2}} + 5465460 \, a^{3} b^{12} x^{6} + 10930920 \, a^{4} b^{11} x^{\frac {11}{2}} + 18036018 \, a^{5} b^{10} x^{5} + 24048024 \, a^{6} b^{9} x^{\frac {9}{2}} + 25765740 \, a^{7} b^{8} x^{4} + 22084920 \, a^{8} b^{7} x^{\frac {7}{2}} + 15030015 \, a^{9} b^{6} x^{3} + 8016008 \, a^{10} b^{5} x^{\frac {5}{2}} + 3279276 \, a^{11} b^{4} x^{2} + 993720 \, a^{12} b^{3} x^{\frac {3}{2}} + 210210 \, a^{13} b^{2} x + 27720 \, a^{14} b \sqrt {x} + 1716 \, a^{15}}{12012 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 167, normalized size = 0.84 \[ 15 a \,b^{14} \ln \relax (x )+2 b^{15} \sqrt {x}-\frac {210 a^{2} b^{13}}{\sqrt {x}}-\frac {455 a^{3} b^{12}}{x}-\frac {910 a^{4} b^{11}}{x^{\frac {3}{2}}}-\frac {3003 a^{5} b^{10}}{2 x^{2}}-\frac {2002 a^{6} b^{9}}{x^{\frac {5}{2}}}-\frac {2145 a^{7} b^{8}}{x^{3}}-\frac {12870 a^{8} b^{7}}{7 x^{\frac {7}{2}}}-\frac {5005 a^{9} b^{6}}{4 x^{4}}-\frac {2002 a^{10} b^{5}}{3 x^{\frac {9}{2}}}-\frac {273 a^{11} b^{4}}{x^{5}}-\frac {910 a^{12} b^{3}}{11 x^{\frac {11}{2}}}-\frac {35 a^{13} b^{2}}{2 x^{6}}-\frac {30 a^{14} b}{13 x^{\frac {13}{2}}}-\frac {a^{15}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 167, normalized size = 0.84 \[ 15 \, a b^{14} \log \relax (x) + 2 \, b^{15} \sqrt {x} - \frac {2522520 \, a^{2} b^{13} x^{\frac {13}{2}} + 5465460 \, a^{3} b^{12} x^{6} + 10930920 \, a^{4} b^{11} x^{\frac {11}{2}} + 18036018 \, a^{5} b^{10} x^{5} + 24048024 \, a^{6} b^{9} x^{\frac {9}{2}} + 25765740 \, a^{7} b^{8} x^{4} + 22084920 \, a^{8} b^{7} x^{\frac {7}{2}} + 15030015 \, a^{9} b^{6} x^{3} + 8016008 \, a^{10} b^{5} x^{\frac {5}{2}} + 3279276 \, a^{11} b^{4} x^{2} + 993720 \, a^{12} b^{3} x^{\frac {3}{2}} + 210210 \, a^{13} b^{2} x + 27720 \, a^{14} b \sqrt {x} + 1716 \, a^{15}}{12012 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 168, normalized size = 0.85 \[ 2\,b^{15}\,\sqrt {x}-\frac {a^{15}}{7\,x^7}-\frac {30\,a^{14}\,b}{13\,x^{13/2}}-\frac {455\,a^3\,b^{12}}{x}-\frac {3003\,a^5\,b^{10}}{2\,x^2}-\frac {210\,a^2\,b^{13}}{\sqrt {x}}-\frac {2145\,a^7\,b^8}{x^3}-\frac {5005\,a^9\,b^6}{4\,x^4}-\frac {910\,a^4\,b^{11}}{x^{3/2}}-\frac {273\,a^{11}\,b^4}{x^5}-\frac {35\,a^{13}\,b^2}{2\,x^6}-\frac {2002\,a^6\,b^9}{x^{5/2}}-\frac {12870\,a^8\,b^7}{7\,x^{7/2}}-\frac {2002\,a^{10}\,b^5}{3\,x^{9/2}}-\frac {910\,a^{12}\,b^3}{11\,x^{11/2}}+30\,a\,b^{14}\,\ln \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.87, size = 202, normalized size = 1.02 \[ - \frac {a^{15}}{7 x^{7}} - \frac {30 a^{14} b}{13 x^{\frac {13}{2}}} - \frac {35 a^{13} b^{2}}{2 x^{6}} - \frac {910 a^{12} b^{3}}{11 x^{\frac {11}{2}}} - \frac {273 a^{11} b^{4}}{x^{5}} - \frac {2002 a^{10} b^{5}}{3 x^{\frac {9}{2}}} - \frac {5005 a^{9} b^{6}}{4 x^{4}} - \frac {12870 a^{8} b^{7}}{7 x^{\frac {7}{2}}} - \frac {2145 a^{7} b^{8}}{x^{3}} - \frac {2002 a^{6} b^{9}}{x^{\frac {5}{2}}} - \frac {3003 a^{5} b^{10}}{2 x^{2}} - \frac {910 a^{4} b^{11}}{x^{\frac {3}{2}}} - \frac {455 a^{3} b^{12}}{x} - \frac {210 a^{2} b^{13}}{\sqrt {x}} + 15 a b^{14} \log {\relax (x )} + 2 b^{15} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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