Optimal. Leaf size=194 \[ -\frac {a^{15}}{5 x^5}-\frac {10 a^{14} b}{3 x^{9/2}}-\frac {105 a^{13} b^2}{4 x^4}-\frac {130 a^{12} b^3}{x^{7/2}}-\frac {455 a^{11} b^4}{x^3}-\frac {6006 a^{10} b^5}{5 x^{5/2}}-\frac {5005 a^9 b^6}{2 x^2}-\frac {4290 a^8 b^7}{x^{3/2}}-\frac {6435 a^7 b^8}{x}-\frac {10010 a^6 b^9}{\sqrt {x}}+3003 a^5 b^{10} \log (x)+2730 a^4 b^{11} \sqrt {x}+455 a^3 b^{12} x+70 a^2 b^{13} x^{3/2}+\frac {15}{2} a b^{14} x^2+\frac {2}{5} b^{15} x^{5/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 194, 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} \[ -\frac {105 a^{13} b^2}{4 x^4}-\frac {130 a^{12} b^3}{x^{7/2}}-\frac {455 a^{11} b^4}{x^3}-\frac {6006 a^{10} b^5}{5 x^{5/2}}-\frac {5005 a^9 b^6}{2 x^2}-\frac {4290 a^8 b^7}{x^{3/2}}+70 a^2 b^{13} x^{3/2}-\frac {6435 a^7 b^8}{x}-\frac {10010 a^6 b^9}{\sqrt {x}}+2730 a^4 b^{11} \sqrt {x}+455 a^3 b^{12} x+3003 a^5 b^{10} \log (x)-\frac {10 a^{14} b}{3 x^{9/2}}-\frac {a^{15}}{5 x^5}+\frac {15}{2} a b^{14} x^2+\frac {2}{5} b^{15} x^{5/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^{15}}{x^6} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{11}} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (1365 a^4 b^{11}+\frac {a^{15}}{x^{11}}+\frac {15 a^{14} b}{x^{10}}+\frac {105 a^{13} b^2}{x^9}+\frac {455 a^{12} b^3}{x^8}+\frac {1365 a^{11} b^4}{x^7}+\frac {3003 a^{10} b^5}{x^6}+\frac {5005 a^9 b^6}{x^5}+\frac {6435 a^8 b^7}{x^4}+\frac {6435 a^7 b^8}{x^3}+\frac {5005 a^6 b^9}{x^2}+\frac {3003 a^5 b^{10}}{x}+455 a^3 b^{12} x+105 a^2 b^{13} x^2+15 a b^{14} x^3+b^{15} x^4\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^{15}}{5 x^5}-\frac {10 a^{14} b}{3 x^{9/2}}-\frac {105 a^{13} b^2}{4 x^4}-\frac {130 a^{12} b^3}{x^{7/2}}-\frac {455 a^{11} b^4}{x^3}-\frac {6006 a^{10} b^5}{5 x^{5/2}}-\frac {5005 a^9 b^6}{2 x^2}-\frac {4290 a^8 b^7}{x^{3/2}}-\frac {6435 a^7 b^8}{x}-\frac {10010 a^6 b^9}{\sqrt {x}}+2730 a^4 b^{11} \sqrt {x}+455 a^3 b^{12} x+70 a^2 b^{13} x^{3/2}+\frac {15}{2} a b^{14} x^2+\frac {2}{5} b^{15} x^{5/2}+3003 a^5 b^{10} \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 194, normalized size = 1.00 \[ -\frac {a^{15}}{5 x^5}-\frac {10 a^{14} b}{3 x^{9/2}}-\frac {105 a^{13} b^2}{4 x^4}-\frac {130 a^{12} b^3}{x^{7/2}}-\frac {455 a^{11} b^4}{x^3}-\frac {6006 a^{10} b^5}{5 x^{5/2}}-\frac {5005 a^9 b^6}{2 x^2}-\frac {4290 a^8 b^7}{x^{3/2}}-\frac {6435 a^7 b^8}{x}-\frac {10010 a^6 b^9}{\sqrt {x}}+3003 a^5 b^{10} \log (x)+2730 a^4 b^{11} \sqrt {x}+455 a^3 b^{12} x+70 a^2 b^{13} x^{3/2}+\frac {15}{2} a b^{14} x^2+\frac {2}{5} b^{15} x^{5/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.92, size = 172, normalized size = 0.89 \[ \frac {450 \, a b^{14} x^{7} + 27300 \, a^{3} b^{12} x^{6} + 360360 \, a^{5} b^{10} x^{5} \log \left (\sqrt {x}\right ) - 386100 \, a^{7} b^{8} x^{4} - 150150 \, a^{9} b^{6} x^{3} - 27300 \, a^{11} b^{4} x^{2} - 1575 \, a^{13} b^{2} x - 12 \, a^{15} + 8 \, {\left (3 \, b^{15} x^{7} + 525 \, a^{2} b^{13} x^{6} + 20475 \, a^{4} b^{11} x^{5} - 75075 \, a^{6} b^{9} x^{4} - 32175 \, a^{8} b^{7} x^{3} - 9009 \, a^{10} b^{5} x^{2} - 975 \, a^{12} b^{3} x - 25 \, a^{14} b\right )} \sqrt {x}}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 166, normalized size = 0.86 \[ \frac {2}{5} \, b^{15} x^{\frac {5}{2}} + \frac {15}{2} \, a b^{14} x^{2} + 70 \, a^{2} b^{13} x^{\frac {3}{2}} + 455 \, a^{3} b^{12} x + 3003 \, a^{5} b^{10} \log \left ({\left | x \right |}\right ) + 2730 \, a^{4} b^{11} \sqrt {x} - \frac {600600 \, a^{6} b^{9} x^{\frac {9}{2}} + 386100 \, a^{7} b^{8} x^{4} + 257400 \, a^{8} b^{7} x^{\frac {7}{2}} + 150150 \, a^{9} b^{6} x^{3} + 72072 \, a^{10} b^{5} x^{\frac {5}{2}} + 27300 \, a^{11} b^{4} x^{2} + 7800 \, a^{12} b^{3} x^{\frac {3}{2}} + 1575 \, a^{13} b^{2} x + 200 \, a^{14} b \sqrt {x} + 12 \, a^{15}}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 165, normalized size = 0.85 \[ \frac {2 b^{15} x^{\frac {5}{2}}}{5}+\frac {15 a \,b^{14} x^{2}}{2}+70 a^{2} b^{13} x^{\frac {3}{2}}+3003 a^{5} b^{10} \ln \relax (x )+455 a^{3} b^{12} x +2730 a^{4} b^{11} \sqrt {x}-\frac {10010 a^{6} b^{9}}{\sqrt {x}}-\frac {6435 a^{7} b^{8}}{x}-\frac {4290 a^{8} b^{7}}{x^{\frac {3}{2}}}-\frac {5005 a^{9} b^{6}}{2 x^{2}}-\frac {6006 a^{10} b^{5}}{5 x^{\frac {5}{2}}}-\frac {455 a^{11} b^{4}}{x^{3}}-\frac {130 a^{12} b^{3}}{x^{\frac {7}{2}}}-\frac {105 a^{13} b^{2}}{4 x^{4}}-\frac {10 a^{14} b}{3 x^{\frac {9}{2}}}-\frac {a^{15}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.91, size = 165, normalized size = 0.85 \[ \frac {2}{5} \, b^{15} x^{\frac {5}{2}} + \frac {15}{2} \, a b^{14} x^{2} + 70 \, a^{2} b^{13} x^{\frac {3}{2}} + 455 \, a^{3} b^{12} x + 3003 \, a^{5} b^{10} \log \relax (x) + 2730 \, a^{4} b^{11} \sqrt {x} - \frac {600600 \, a^{6} b^{9} x^{\frac {9}{2}} + 386100 \, a^{7} b^{8} x^{4} + 257400 \, a^{8} b^{7} x^{\frac {7}{2}} + 150150 \, a^{9} b^{6} x^{3} + 72072 \, a^{10} b^{5} x^{\frac {5}{2}} + 27300 \, a^{11} b^{4} x^{2} + 7800 \, a^{12} b^{3} x^{\frac {3}{2}} + 1575 \, a^{13} b^{2} x + 200 \, a^{14} b \sqrt {x} + 12 \, a^{15}}{60 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.12, size = 167, normalized size = 0.86 \[ \frac {2\,b^{15}\,x^{5/2}}{5}-\frac {\frac {a^{15}}{5}+\frac {105\,a^{13}\,b^2\,x}{4}+\frac {10\,a^{14}\,b\,\sqrt {x}}{3}+455\,a^{11}\,b^4\,x^2+\frac {5005\,a^9\,b^6\,x^3}{2}+6435\,a^7\,b^8\,x^4+130\,a^{12}\,b^3\,x^{3/2}+\frac {6006\,a^{10}\,b^5\,x^{5/2}}{5}+4290\,a^8\,b^7\,x^{7/2}+10010\,a^6\,b^9\,x^{9/2}}{x^5}+6006\,a^5\,b^{10}\,\ln \left (\sqrt {x}\right )+455\,a^3\,b^{12}\,x+\frac {15\,a\,b^{14}\,x^2}{2}+2730\,a^4\,b^{11}\,\sqrt {x}+70\,a^2\,b^{13}\,x^{3/2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.05, size = 199, normalized size = 1.03 \[ - \frac {a^{15}}{5 x^{5}} - \frac {10 a^{14} b}{3 x^{\frac {9}{2}}} - \frac {105 a^{13} b^{2}}{4 x^{4}} - \frac {130 a^{12} b^{3}}{x^{\frac {7}{2}}} - \frac {455 a^{11} b^{4}}{x^{3}} - \frac {6006 a^{10} b^{5}}{5 x^{\frac {5}{2}}} - \frac {5005 a^{9} b^{6}}{2 x^{2}} - \frac {4290 a^{8} b^{7}}{x^{\frac {3}{2}}} - \frac {6435 a^{7} b^{8}}{x} - \frac {10010 a^{6} b^{9}}{\sqrt {x}} + 3003 a^{5} b^{10} \log {\relax (x )} + 2730 a^{4} b^{11} \sqrt {x} + 455 a^{3} b^{12} x + 70 a^{2} b^{13} x^{\frac {3}{2}} + \frac {15 a b^{14} x^{2}}{2} + \frac {2 b^{15} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________