Optimal. Leaf size=196 \[ -\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+455 a^3 b^{12} \log (x)+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2} \]
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Rubi [A] time = 0.11, antiderivative size = 196, 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} \[ -\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+210 a^2 b^{13} \sqrt {x}+455 a^3 b^{12} \log (x)-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {a^{15}}{6 x^6}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2} \]
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^7} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{13}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (105 a^2 b^{13}+\frac {a^{15}}{x^{13}}+\frac {15 a^{14} b}{x^{12}}+\frac {105 a^{13} b^2}{x^{11}}+\frac {455 a^{12} b^3}{x^{10}}+\frac {1365 a^{11} b^4}{x^9}+\frac {3003 a^{10} b^5}{x^8}+\frac {5005 a^9 b^6}{x^7}+\frac {6435 a^8 b^7}{x^6}+\frac {6435 a^7 b^8}{x^5}+\frac {5005 a^6 b^9}{x^4}+\frac {3003 a^5 b^{10}}{x^3}+\frac {1365 a^4 b^{11}}{x^2}+\frac {455 a^3 b^{12}}{x}+15 a b^{14} x+b^{15} x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2}+455 a^3 b^{12} \log (x)\\ \end {align*}
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Mathematica [A] time = 0.16, size = 196, normalized size = 1.00 \[ -\frac {a^{15}}{6 x^6}-\frac {30 a^{14} b}{11 x^{11/2}}-\frac {21 a^{13} b^2}{x^5}-\frac {910 a^{12} b^3}{9 x^{9/2}}-\frac {1365 a^{11} b^4}{4 x^4}-\frac {858 a^{10} b^5}{x^{7/2}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {2574 a^8 b^7}{x^{5/2}}-\frac {6435 a^7 b^8}{2 x^2}-\frac {10010 a^6 b^9}{3 x^{3/2}}-\frac {3003 a^5 b^{10}}{x}-\frac {2730 a^4 b^{11}}{\sqrt {x}}+455 a^3 b^{12} \log (x)+210 a^2 b^{13} \sqrt {x}+15 a b^{14} x+\frac {2}{3} b^{15} x^{3/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 172, normalized size = 0.88 \[ \frac {5940 \, a b^{14} x^{7} + 360360 \, a^{3} b^{12} x^{6} \log \left (\sqrt {x}\right ) - 1189188 \, a^{5} b^{10} x^{5} - 1274130 \, a^{7} b^{8} x^{4} - 660660 \, a^{9} b^{6} x^{3} - 135135 \, a^{11} b^{4} x^{2} - 8316 \, a^{13} b^{2} x - 66 \, a^{15} + 8 \, {\left (33 \, b^{15} x^{7} + 10395 \, a^{2} b^{13} x^{6} - 135135 \, a^{4} b^{11} x^{5} - 165165 \, a^{6} b^{9} x^{4} - 127413 \, a^{8} b^{7} x^{3} - 42471 \, a^{10} b^{5} x^{2} - 5005 \, a^{12} b^{3} x - 135 \, a^{14} b\right )} \sqrt {x}}{396 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 166, normalized size = 0.85 \[ \frac {2}{3} \, b^{15} x^{\frac {3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \left ({\left | x \right |}\right ) + 210 \, a^{2} b^{13} \sqrt {x} - \frac {1081080 \, a^{4} b^{11} x^{\frac {11}{2}} + 1189188 \, a^{5} b^{10} x^{5} + 1321320 \, a^{6} b^{9} x^{\frac {9}{2}} + 1274130 \, a^{7} b^{8} x^{4} + 1019304 \, a^{8} b^{7} x^{\frac {7}{2}} + 660660 \, a^{9} b^{6} x^{3} + 339768 \, a^{10} b^{5} x^{\frac {5}{2}} + 135135 \, a^{11} b^{4} x^{2} + 40040 \, a^{12} b^{3} x^{\frac {3}{2}} + 8316 \, a^{13} b^{2} x + 1080 \, a^{14} b \sqrt {x} + 66 \, a^{15}}{396 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 165, normalized size = 0.84 \[ \frac {2 b^{15} x^{\frac {3}{2}}}{3}+455 a^{3} b^{12} \ln \relax (x )+15 a \,b^{14} x +210 a^{2} b^{13} \sqrt {x}-\frac {2730 a^{4} b^{11}}{\sqrt {x}}-\frac {3003 a^{5} b^{10}}{x}-\frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}}-\frac {6435 a^{7} b^{8}}{2 x^{2}}-\frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}}-\frac {5005 a^{9} b^{6}}{3 x^{3}}-\frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}}-\frac {1365 a^{11} b^{4}}{4 x^{4}}-\frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}}-\frac {21 a^{13} b^{2}}{x^{5}}-\frac {30 a^{14} b}{11 x^{\frac {11}{2}}}-\frac {a^{15}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 165, normalized size = 0.84 \[ \frac {2}{3} \, b^{15} x^{\frac {3}{2}} + 15 \, a b^{14} x + 455 \, a^{3} b^{12} \log \relax (x) + 210 \, a^{2} b^{13} \sqrt {x} - \frac {1081080 \, a^{4} b^{11} x^{\frac {11}{2}} + 1189188 \, a^{5} b^{10} x^{5} + 1321320 \, a^{6} b^{9} x^{\frac {9}{2}} + 1274130 \, a^{7} b^{8} x^{4} + 1019304 \, a^{8} b^{7} x^{\frac {7}{2}} + 660660 \, a^{9} b^{6} x^{3} + 339768 \, a^{10} b^{5} x^{\frac {5}{2}} + 135135 \, a^{11} b^{4} x^{2} + 40040 \, a^{12} b^{3} x^{\frac {3}{2}} + 8316 \, a^{13} b^{2} x + 1080 \, a^{14} b \sqrt {x} + 66 \, a^{15}}{396 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 167, normalized size = 0.85 \[ \frac {2\,b^{15}\,x^{3/2}}{3}-\frac {\frac {a^{15}}{6}+21\,a^{13}\,b^2\,x+\frac {30\,a^{14}\,b\,\sqrt {x}}{11}+\frac {1365\,a^{11}\,b^4\,x^2}{4}+\frac {5005\,a^9\,b^6\,x^3}{3}+\frac {6435\,a^7\,b^8\,x^4}{2}+3003\,a^5\,b^{10}\,x^5+\frac {910\,a^{12}\,b^3\,x^{3/2}}{9}+858\,a^{10}\,b^5\,x^{5/2}+2574\,a^8\,b^7\,x^{7/2}+\frac {10010\,a^6\,b^9\,x^{9/2}}{3}+2730\,a^4\,b^{11}\,x^{11/2}}{x^6}+910\,a^3\,b^{12}\,\ln \left (\sqrt {x}\right )+210\,a^2\,b^{13}\,\sqrt {x}+15\,a\,b^{14}\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.04, size = 201, normalized size = 1.03 \[ - \frac {a^{15}}{6 x^{6}} - \frac {30 a^{14} b}{11 x^{\frac {11}{2}}} - \frac {21 a^{13} b^{2}}{x^{5}} - \frac {910 a^{12} b^{3}}{9 x^{\frac {9}{2}}} - \frac {1365 a^{11} b^{4}}{4 x^{4}} - \frac {858 a^{10} b^{5}}{x^{\frac {7}{2}}} - \frac {5005 a^{9} b^{6}}{3 x^{3}} - \frac {2574 a^{8} b^{7}}{x^{\frac {5}{2}}} - \frac {6435 a^{7} b^{8}}{2 x^{2}} - \frac {10010 a^{6} b^{9}}{3 x^{\frac {3}{2}}} - \frac {3003 a^{5} b^{10}}{x} - \frac {2730 a^{4} b^{11}}{\sqrt {x}} + 455 a^{3} b^{12} \log {\relax (x )} + 210 a^{2} b^{13} \sqrt {x} + 15 a b^{14} x + \frac {2 b^{15} x^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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