Optimal. Leaf size=202 \[ -\frac {a^{15}}{x}-\frac {45 a^{14} b}{2 x^{2/3}}-\frac {315 a^{13} b^2}{\sqrt [3]{x}}+4095 a^{11} b^4 \sqrt [3]{x}+\frac {9009}{2} a^{10} b^5 x^{2/3}+5005 a^9 b^6 x+\frac {19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac {5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac {4095}{8} a^4 b^{11} x^{8/3}+\frac {455}{3} a^3 b^{12} x^3+\frac {63}{2} a^2 b^{13} x^{10/3}+\frac {45}{11} a b^{14} x^{11/3}+\frac {b^{15} x^4}{4}+455 a^{12} b^3 \log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 202, 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 = {272, 45}
\begin {gather*} -\frac {a^{15}}{x}-\frac {45 a^{14} b}{2 x^{2/3}}-\frac {315 a^{13} b^2}{\sqrt [3]{x}}+455 a^{12} b^3 \log (x)+4095 a^{11} b^4 \sqrt [3]{x}+\frac {9009}{2} a^{10} b^5 x^{2/3}+5005 a^9 b^6 x+\frac {19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac {5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac {4095}{8} a^4 b^{11} x^{8/3}+\frac {455}{3} a^3 b^{12} x^3+\frac {63}{2} a^2 b^{13} x^{10/3}+\frac {45}{11} a b^{14} x^{11/3}+\frac {b^{15} x^4}{4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^2} \, dx &=3 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (1365 a^{11} b^4+\frac {a^{15}}{x^4}+\frac {15 a^{14} b}{x^3}+\frac {105 a^{13} b^2}{x^2}+\frac {455 a^{12} b^3}{x}+3003 a^{10} b^5 x+5005 a^9 b^6 x^2+6435 a^8 b^7 x^3+6435 a^7 b^8 x^4+5005 a^6 b^9 x^5+3003 a^5 b^{10} x^6+1365 a^4 b^{11} x^7+455 a^3 b^{12} x^8+105 a^2 b^{13} x^9+15 a b^{14} x^{10}+b^{15} x^{11}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^{15}}{x}-\frac {45 a^{14} b}{2 x^{2/3}}-\frac {315 a^{13} b^2}{\sqrt [3]{x}}+4095 a^{11} b^4 \sqrt [3]{x}+\frac {9009}{2} a^{10} b^5 x^{2/3}+5005 a^9 b^6 x+\frac {19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac {5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac {4095}{8} a^4 b^{11} x^{8/3}+\frac {455}{3} a^3 b^{12} x^3+\frac {63}{2} a^2 b^{13} x^{10/3}+\frac {45}{11} a b^{14} x^{11/3}+\frac {b^{15} x^4}{4}+455 a^{12} b^3 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 195, normalized size = 0.97 \begin {gather*} \frac {-264 a^{15}-5940 a^{14} b \sqrt [3]{x}-83160 a^{13} b^2 x^{2/3}+1081080 a^{11} b^4 x^{4/3}+1189188 a^{10} b^5 x^{5/3}+1321320 a^9 b^6 x^2+1274130 a^8 b^7 x^{7/3}+1019304 a^7 b^8 x^{8/3}+660660 a^6 b^9 x^3+339768 a^5 b^{10} x^{10/3}+135135 a^4 b^{11} x^{11/3}+40040 a^3 b^{12} x^4+8316 a^2 b^{13} x^{13/3}+1080 a b^{14} x^{14/3}+66 b^{15} x^5}{264 x}+1365 a^{12} b^3 \log \left (\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 165, normalized size = 0.82
method | result | size |
derivativedivides | \(-\frac {a^{15}}{x}-\frac {45 a^{14} b}{2 x^{\frac {2}{3}}}-\frac {315 a^{13} b^{2}}{x^{\frac {1}{3}}}+4095 a^{11} b^{4} x^{\frac {1}{3}}+\frac {9009 a^{10} b^{5} x^{\frac {2}{3}}}{2}+5005 a^{9} b^{6} x +\frac {19305 a^{8} b^{7} x^{\frac {4}{3}}}{4}+3861 a^{7} b^{8} x^{\frac {5}{3}}+\frac {5005 a^{6} b^{9} x^{2}}{2}+1287 a^{5} b^{10} x^{\frac {7}{3}}+\frac {4095 a^{4} b^{11} x^{\frac {8}{3}}}{8}+\frac {455 a^{3} b^{12} x^{3}}{3}+\frac {63 a^{2} b^{13} x^{\frac {10}{3}}}{2}+\frac {45 a \,b^{14} x^{\frac {11}{3}}}{11}+\frac {b^{15} x^{4}}{4}+455 a^{12} b^{3} \ln \left (x \right )\) | \(165\) |
default | \(-\frac {a^{15}}{x}-\frac {45 a^{14} b}{2 x^{\frac {2}{3}}}-\frac {315 a^{13} b^{2}}{x^{\frac {1}{3}}}+4095 a^{11} b^{4} x^{\frac {1}{3}}+\frac {9009 a^{10} b^{5} x^{\frac {2}{3}}}{2}+5005 a^{9} b^{6} x +\frac {19305 a^{8} b^{7} x^{\frac {4}{3}}}{4}+3861 a^{7} b^{8} x^{\frac {5}{3}}+\frac {5005 a^{6} b^{9} x^{2}}{2}+1287 a^{5} b^{10} x^{\frac {7}{3}}+\frac {4095 a^{4} b^{11} x^{\frac {8}{3}}}{8}+\frac {455 a^{3} b^{12} x^{3}}{3}+\frac {63 a^{2} b^{13} x^{\frac {10}{3}}}{2}+\frac {45 a \,b^{14} x^{\frac {11}{3}}}{11}+\frac {b^{15} x^{4}}{4}+455 a^{12} b^{3} \ln \left (x \right )\) | \(165\) |
trager | \(\frac {\left (x -1\right ) \left (3 b^{15} x^{4}+1820 a^{3} b^{12} x^{3}+3 b^{15} x^{3}+30030 a^{6} b^{9} x^{2}+1820 a^{3} b^{12} x^{2}+3 b^{15} x^{2}+60060 a^{9} b^{6} x +30030 a^{6} b^{9} x +1820 a^{3} b^{12} x +3 b^{15} x +12 a^{15}\right )}{12 x}-\frac {9 \left (-14 b^{12} x^{4}-572 a^{3} b^{9} x^{3}-2145 a^{6} b^{6} x^{2}-1820 a^{9} b^{3} x +10 a^{12}\right ) a^{2} b}{4 x^{\frac {2}{3}}}-\frac {9 \left (-40 b^{12} x^{4}-5005 a^{3} b^{9} x^{3}-37752 a^{6} b^{6} x^{2}-44044 a^{9} b^{3} x +3080 a^{12}\right ) a \,b^{2}}{88 x^{\frac {1}{3}}}+455 a^{12} b^{3} \ln \left (x \right )\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 167, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, b^{15} x^{4} + \frac {45}{11} \, a b^{14} x^{\frac {11}{3}} + \frac {63}{2} \, a^{2} b^{13} x^{\frac {10}{3}} + \frac {455}{3} \, a^{3} b^{12} x^{3} + \frac {4095}{8} \, a^{4} b^{11} x^{\frac {8}{3}} + 1287 \, a^{5} b^{10} x^{\frac {7}{3}} + \frac {5005}{2} \, a^{6} b^{9} x^{2} + 3861 \, a^{7} b^{8} x^{\frac {5}{3}} + \frac {19305}{4} \, a^{8} b^{7} x^{\frac {4}{3}} + 5005 \, a^{9} b^{6} x + 455 \, a^{12} b^{3} \log \left (x\right ) + \frac {9009}{2} \, a^{10} b^{5} x^{\frac {2}{3}} + 4095 \, a^{11} b^{4} x^{\frac {1}{3}} - \frac {630 \, a^{13} b^{2} x^{\frac {2}{3}} + 45 \, a^{14} b x^{\frac {1}{3}} + 2 \, a^{15}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 173, normalized size = 0.86 \begin {gather*} \frac {66 \, b^{15} x^{5} + 40040 \, a^{3} b^{12} x^{4} + 660660 \, a^{6} b^{9} x^{3} + 1321320 \, a^{9} b^{6} x^{2} + 360360 \, a^{12} b^{3} x \log \left (x^{\frac {1}{3}}\right ) - 264 \, a^{15} + 27 \, {\left (40 \, a b^{14} x^{4} + 5005 \, a^{4} b^{11} x^{3} + 37752 \, a^{7} b^{8} x^{2} + 44044 \, a^{10} b^{5} x - 3080 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 594 \, {\left (14 \, a^{2} b^{13} x^{4} + 572 \, a^{5} b^{10} x^{3} + 2145 \, a^{8} b^{7} x^{2} + 1820 \, a^{11} b^{4} x - 10 \, a^{14} b\right )} x^{\frac {1}{3}}}{264 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.58, size = 204, normalized size = 1.01 \begin {gather*} - \frac {a^{15}}{x} - \frac {45 a^{14} b}{2 x^{\frac {2}{3}}} - \frac {315 a^{13} b^{2}}{\sqrt [3]{x}} + 455 a^{12} b^{3} \log {\left (x \right )} + 4095 a^{11} b^{4} \sqrt [3]{x} + \frac {9009 a^{10} b^{5} x^{\frac {2}{3}}}{2} + 5005 a^{9} b^{6} x + \frac {19305 a^{8} b^{7} x^{\frac {4}{3}}}{4} + 3861 a^{7} b^{8} x^{\frac {5}{3}} + \frac {5005 a^{6} b^{9} x^{2}}{2} + 1287 a^{5} b^{10} x^{\frac {7}{3}} + \frac {4095 a^{4} b^{11} x^{\frac {8}{3}}}{8} + \frac {455 a^{3} b^{12} x^{3}}{3} + \frac {63 a^{2} b^{13} x^{\frac {10}{3}}}{2} + \frac {45 a b^{14} x^{\frac {11}{3}}}{11} + \frac {b^{15} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.01, size = 168, normalized size = 0.83 \begin {gather*} \frac {1}{4} \, b^{15} x^{4} + \frac {45}{11} \, a b^{14} x^{\frac {11}{3}} + \frac {63}{2} \, a^{2} b^{13} x^{\frac {10}{3}} + \frac {455}{3} \, a^{3} b^{12} x^{3} + \frac {4095}{8} \, a^{4} b^{11} x^{\frac {8}{3}} + 1287 \, a^{5} b^{10} x^{\frac {7}{3}} + \frac {5005}{2} \, a^{6} b^{9} x^{2} + 3861 \, a^{7} b^{8} x^{\frac {5}{3}} + \frac {19305}{4} \, a^{8} b^{7} x^{\frac {4}{3}} + 5005 \, a^{9} b^{6} x + 455 \, a^{12} b^{3} \log \left ({\left | x \right |}\right ) + \frac {9009}{2} \, a^{10} b^{5} x^{\frac {2}{3}} + 4095 \, a^{11} b^{4} x^{\frac {1}{3}} - \frac {630 \, a^{13} b^{2} x^{\frac {2}{3}} + 45 \, a^{14} b x^{\frac {1}{3}} + 2 \, a^{15}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.19, size = 167, normalized size = 0.83 \begin {gather*} \frac {b^{15}\,x^4}{4}-\frac {a^{15}+\frac {45\,a^{14}\,b\,x^{1/3}}{2}+315\,a^{13}\,b^2\,x^{2/3}}{x}+1365\,a^{12}\,b^3\,\ln \left (x^{1/3}\right )+5005\,a^9\,b^6\,x+\frac {45\,a\,b^{14}\,x^{11/3}}{11}+\frac {5005\,a^6\,b^9\,x^2}{2}+\frac {455\,a^3\,b^{12}\,x^3}{3}+4095\,a^{11}\,b^4\,x^{1/3}+\frac {9009\,a^{10}\,b^5\,x^{2/3}}{2}+\frac {19305\,a^8\,b^7\,x^{4/3}}{4}+3861\,a^7\,b^8\,x^{5/3}+1287\,a^5\,b^{10}\,x^{7/3}+\frac {4095\,a^4\,b^{11}\,x^{8/3}}{8}+\frac {63\,a^2\,b^{13}\,x^{10/3}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________