Optimal. Leaf size=217 \[ -\frac {a^{15}}{11 x^{11}}-\frac {45 a^{14} b}{32 x^{32/3}}-\frac {315 a^{13} b^2}{31 x^{31/3}}-\frac {91 a^{12} b^3}{2 x^{10}}-\frac {4095 a^{11} b^4}{29 x^{29/3}}-\frac {1287 a^{10} b^5}{4 x^{28/3}}-\frac {5005 a^9 b^6}{9 x^9}-\frac {1485 a^8 b^7}{2 x^{26/3}}-\frac {3861 a^7 b^8}{5 x^{25/3}}-\frac {5005 a^6 b^9}{8 x^8}-\frac {9009 a^5 b^{10}}{23 x^{23/3}}-\frac {4095 a^4 b^{11}}{22 x^{22/3}}-\frac {65 a^3 b^{12}}{x^7}-\frac {63 a^2 b^{13}}{4 x^{20/3}}-\frac {45 a b^{14}}{19 x^{19/3}}-\frac {b^{15}}{6 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 217, 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}}{11 x^{11}}-\frac {45 a^{14} b}{32 x^{32/3}}-\frac {315 a^{13} b^2}{31 x^{31/3}}-\frac {91 a^{12} b^3}{2 x^{10}}-\frac {4095 a^{11} b^4}{29 x^{29/3}}-\frac {1287 a^{10} b^5}{4 x^{28/3}}-\frac {5005 a^9 b^6}{9 x^9}-\frac {1485 a^8 b^7}{2 x^{26/3}}-\frac {3861 a^7 b^8}{5 x^{25/3}}-\frac {5005 a^6 b^9}{8 x^8}-\frac {9009 a^5 b^{10}}{23 x^{23/3}}-\frac {4095 a^4 b^{11}}{22 x^{22/3}}-\frac {65 a^3 b^{12}}{x^7}-\frac {63 a^2 b^{13}}{4 x^{20/3}}-\frac {45 a b^{14}}{19 x^{19/3}}-\frac {b^{15}}{6 x^6} \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^{12}} \, dx &=3 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{34}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {a^{15}}{x^{34}}+\frac {15 a^{14} b}{x^{33}}+\frac {105 a^{13} b^2}{x^{32}}+\frac {455 a^{12} b^3}{x^{31}}+\frac {1365 a^{11} b^4}{x^{30}}+\frac {3003 a^{10} b^5}{x^{29}}+\frac {5005 a^9 b^6}{x^{28}}+\frac {6435 a^8 b^7}{x^{27}}+\frac {6435 a^7 b^8}{x^{26}}+\frac {5005 a^6 b^9}{x^{25}}+\frac {3003 a^5 b^{10}}{x^{24}}+\frac {1365 a^4 b^{11}}{x^{23}}+\frac {455 a^3 b^{12}}{x^{22}}+\frac {105 a^2 b^{13}}{x^{21}}+\frac {15 a b^{14}}{x^{20}}+\frac {b^{15}}{x^{19}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^{15}}{11 x^{11}}-\frac {45 a^{14} b}{32 x^{32/3}}-\frac {315 a^{13} b^2}{31 x^{31/3}}-\frac {91 a^{12} b^3}{2 x^{10}}-\frac {4095 a^{11} b^4}{29 x^{29/3}}-\frac {1287 a^{10} b^5}{4 x^{28/3}}-\frac {5005 a^9 b^6}{9 x^9}-\frac {1485 a^8 b^7}{2 x^{26/3}}-\frac {3861 a^7 b^8}{5 x^{25/3}}-\frac {5005 a^6 b^9}{8 x^8}-\frac {9009 a^5 b^{10}}{23 x^{23/3}}-\frac {4095 a^4 b^{11}}{22 x^{22/3}}-\frac {65 a^3 b^{12}}{x^7}-\frac {63 a^2 b^{13}}{4 x^{20/3}}-\frac {45 a b^{14}}{19 x^{19/3}}-\frac {b^{15}}{6 x^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 189, normalized size = 0.87 \begin {gather*} \frac {-565722720 a^{15}-8751023325 a^{14} b \sqrt [3]{x}-63233200800 a^{13} b^2 x^{2/3}-283144221360 a^{12} b^3 x-878723445600 a^{11} b^4 x^{4/3}-2002234136760 a^{10} b^5 x^{5/3}-3460651594400 a^9 b^6 x^2-4620540315600 a^8 b^7 x^{7/3}-4805361928224 a^7 b^8 x^{8/3}-3893233043700 a^6 b^9 x^3-2437502427360 a^5 b^{10} x^{10/3}-1158317269200 a^4 b^{11} x^{11/3}-404491744800 a^3 b^{12} x^4-98011461240 a^2 b^{13} x^{13/3}-14738565600 a b^{14} x^{14/3}-1037158320 b^{15} x^5}{6222949920 x^{11}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 168, normalized size = 0.77
method | result | size |
derivativedivides | \(-\frac {a^{15}}{11 x^{11}}-\frac {45 a^{14} b}{32 x^{\frac {32}{3}}}-\frac {315 a^{13} b^{2}}{31 x^{\frac {31}{3}}}-\frac {91 a^{12} b^{3}}{2 x^{10}}-\frac {4095 a^{11} b^{4}}{29 x^{\frac {29}{3}}}-\frac {1287 a^{10} b^{5}}{4 x^{\frac {28}{3}}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}-\frac {1485 a^{8} b^{7}}{2 x^{\frac {26}{3}}}-\frac {3861 a^{7} b^{8}}{5 x^{\frac {25}{3}}}-\frac {5005 a^{6} b^{9}}{8 x^{8}}-\frac {9009 a^{5} b^{10}}{23 x^{\frac {23}{3}}}-\frac {4095 a^{4} b^{11}}{22 x^{\frac {22}{3}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {63 a^{2} b^{13}}{4 x^{\frac {20}{3}}}-\frac {45 a \,b^{14}}{19 x^{\frac {19}{3}}}-\frac {b^{15}}{6 x^{6}}\) | \(168\) |
default | \(-\frac {a^{15}}{11 x^{11}}-\frac {45 a^{14} b}{32 x^{\frac {32}{3}}}-\frac {315 a^{13} b^{2}}{31 x^{\frac {31}{3}}}-\frac {91 a^{12} b^{3}}{2 x^{10}}-\frac {4095 a^{11} b^{4}}{29 x^{\frac {29}{3}}}-\frac {1287 a^{10} b^{5}}{4 x^{\frac {28}{3}}}-\frac {5005 a^{9} b^{6}}{9 x^{9}}-\frac {1485 a^{8} b^{7}}{2 x^{\frac {26}{3}}}-\frac {3861 a^{7} b^{8}}{5 x^{\frac {25}{3}}}-\frac {5005 a^{6} b^{9}}{8 x^{8}}-\frac {9009 a^{5} b^{10}}{23 x^{\frac {23}{3}}}-\frac {4095 a^{4} b^{11}}{22 x^{\frac {22}{3}}}-\frac {65 a^{3} b^{12}}{x^{7}}-\frac {63 a^{2} b^{13}}{4 x^{\frac {20}{3}}}-\frac {45 a \,b^{14}}{19 x^{\frac {19}{3}}}-\frac {b^{15}}{6 x^{6}}\) | \(168\) |
trager | \(\frac {\left (x -1\right ) \left (72 a^{15} x^{10}+36036 a^{12} b^{3} x^{10}+440440 a^{9} b^{6} x^{10}+495495 a^{6} b^{9} x^{10}+51480 a^{3} b^{12} x^{10}+132 b^{15} x^{10}+72 a^{15} x^{9}+36036 a^{12} b^{3} x^{9}+440440 a^{9} b^{6} x^{9}+495495 a^{6} b^{9} x^{9}+51480 a^{3} b^{12} x^{9}+132 b^{15} x^{9}+72 a^{15} x^{8}+36036 a^{12} b^{3} x^{8}+440440 a^{9} b^{6} x^{8}+495495 a^{6} b^{9} x^{8}+51480 a^{3} b^{12} x^{8}+132 b^{15} x^{8}+72 a^{15} x^{7}+36036 a^{12} b^{3} x^{7}+440440 a^{9} b^{6} x^{7}+495495 a^{6} b^{9} x^{7}+51480 a^{3} b^{12} x^{7}+132 b^{15} x^{7}+72 a^{15} x^{6}+36036 a^{12} b^{3} x^{6}+440440 a^{9} b^{6} x^{6}+495495 a^{6} b^{9} x^{6}+51480 a^{3} b^{12} x^{6}+132 b^{15} x^{6}+72 a^{15} x^{5}+36036 a^{12} b^{3} x^{5}+440440 a^{9} b^{6} x^{5}+495495 a^{6} b^{9} x^{5}+51480 a^{3} b^{12} x^{5}+132 b^{15} x^{5}+72 a^{15} x^{4}+36036 a^{12} b^{3} x^{4}+440440 a^{9} b^{6} x^{4}+495495 a^{6} b^{9} x^{4}+51480 a^{3} b^{12} x^{4}+72 a^{15} x^{3}+36036 a^{12} b^{3} x^{3}+440440 a^{9} b^{6} x^{3}+495495 a^{6} b^{9} x^{3}+72 a^{15} x^{2}+36036 a^{12} b^{3} x^{2}+440440 a^{9} b^{6} x^{2}+72 a^{15} x +36036 a^{12} b^{3} x +72 a^{15}\right )}{792 x^{11}}-\frac {9 \left (37352 b^{12} x^{4}+928928 a^{3} b^{9} x^{3}+1760880 a^{6} b^{6} x^{2}+334880 a^{9} b^{3} x +3335 a^{12}\right ) a^{2} b}{21344 x^{\frac {32}{3}}}-\frac {9 \left (34100 b^{12} x^{4}+2679950 a^{3} b^{9} x^{3}+11117964 a^{6} b^{6} x^{2}+4632485 a^{9} b^{3} x +146300 a^{12}\right ) a \,b^{2}}{129580 x^{\frac {31}{3}}}\) | \(622\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 167, normalized size = 0.77 \begin {gather*} -\frac {1037158320 \, b^{15} x^{5} + 14738565600 \, a b^{14} x^{\frac {14}{3}} + 98011461240 \, a^{2} b^{13} x^{\frac {13}{3}} + 404491744800 \, a^{3} b^{12} x^{4} + 1158317269200 \, a^{4} b^{11} x^{\frac {11}{3}} + 2437502427360 \, a^{5} b^{10} x^{\frac {10}{3}} + 3893233043700 \, a^{6} b^{9} x^{3} + 4805361928224 \, a^{7} b^{8} x^{\frac {8}{3}} + 4620540315600 \, a^{8} b^{7} x^{\frac {7}{3}} + 3460651594400 \, a^{9} b^{6} x^{2} + 2002234136760 \, a^{10} b^{5} x^{\frac {5}{3}} + 878723445600 \, a^{11} b^{4} x^{\frac {4}{3}} + 283144221360 \, a^{12} b^{3} x + 63233200800 \, a^{13} b^{2} x^{\frac {2}{3}} + 8751023325 \, a^{14} b x^{\frac {1}{3}} + 565722720 \, a^{15}}{6222949920 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.44, size = 169, normalized size = 0.78 \begin {gather*} -\frac {1037158320 \, b^{15} x^{5} + 404491744800 \, a^{3} b^{12} x^{4} + 3893233043700 \, a^{6} b^{9} x^{3} + 3460651594400 \, a^{9} b^{6} x^{2} + 283144221360 \, a^{12} b^{3} x + 565722720 \, a^{15} + 432216 \, {\left (34100 \, a b^{14} x^{4} + 2679950 \, a^{4} b^{11} x^{3} + 11117964 \, a^{7} b^{8} x^{2} + 4632485 \, a^{10} b^{5} x + 146300 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 2623995 \, {\left (37352 \, a^{2} b^{13} x^{4} + 928928 \, a^{5} b^{10} x^{3} + 1760880 \, a^{8} b^{7} x^{2} + 334880 \, a^{11} b^{4} x + 3335 \, a^{14} b\right )} x^{\frac {1}{3}}}{6222949920 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 3.61, size = 219, normalized size = 1.01 \begin {gather*} - \frac {a^{15}}{11 x^{11}} - \frac {45 a^{14} b}{32 x^{\frac {32}{3}}} - \frac {315 a^{13} b^{2}}{31 x^{\frac {31}{3}}} - \frac {91 a^{12} b^{3}}{2 x^{10}} - \frac {4095 a^{11} b^{4}}{29 x^{\frac {29}{3}}} - \frac {1287 a^{10} b^{5}}{4 x^{\frac {28}{3}}} - \frac {5005 a^{9} b^{6}}{9 x^{9}} - \frac {1485 a^{8} b^{7}}{2 x^{\frac {26}{3}}} - \frac {3861 a^{7} b^{8}}{5 x^{\frac {25}{3}}} - \frac {5005 a^{6} b^{9}}{8 x^{8}} - \frac {9009 a^{5} b^{10}}{23 x^{\frac {23}{3}}} - \frac {4095 a^{4} b^{11}}{22 x^{\frac {22}{3}}} - \frac {65 a^{3} b^{12}}{x^{7}} - \frac {63 a^{2} b^{13}}{4 x^{\frac {20}{3}}} - \frac {45 a b^{14}}{19 x^{\frac {19}{3}}} - \frac {b^{15}}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.69, size = 167, normalized size = 0.77 \begin {gather*} -\frac {1037158320 \, b^{15} x^{5} + 14738565600 \, a b^{14} x^{\frac {14}{3}} + 98011461240 \, a^{2} b^{13} x^{\frac {13}{3}} + 404491744800 \, a^{3} b^{12} x^{4} + 1158317269200 \, a^{4} b^{11} x^{\frac {11}{3}} + 2437502427360 \, a^{5} b^{10} x^{\frac {10}{3}} + 3893233043700 \, a^{6} b^{9} x^{3} + 4805361928224 \, a^{7} b^{8} x^{\frac {8}{3}} + 4620540315600 \, a^{8} b^{7} x^{\frac {7}{3}} + 3460651594400 \, a^{9} b^{6} x^{2} + 2002234136760 \, a^{10} b^{5} x^{\frac {5}{3}} + 878723445600 \, a^{11} b^{4} x^{\frac {4}{3}} + 283144221360 \, a^{12} b^{3} x + 63233200800 \, a^{13} b^{2} x^{\frac {2}{3}} + 8751023325 \, a^{14} b x^{\frac {1}{3}} + 565722720 \, a^{15}}{6222949920 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.28, size = 167, normalized size = 0.77 \begin {gather*} -\frac {\frac {a^{15}}{11}+\frac {b^{15}\,x^5}{6}+\frac {91\,a^{12}\,b^3\,x}{2}+\frac {45\,a^{14}\,b\,x^{1/3}}{32}+\frac {45\,a\,b^{14}\,x^{14/3}}{19}+\frac {5005\,a^9\,b^6\,x^2}{9}+\frac {5005\,a^6\,b^9\,x^3}{8}+65\,a^3\,b^{12}\,x^4+\frac {315\,a^{13}\,b^2\,x^{2/3}}{31}+\frac {4095\,a^{11}\,b^4\,x^{4/3}}{29}+\frac {1287\,a^{10}\,b^5\,x^{5/3}}{4}+\frac {1485\,a^8\,b^7\,x^{7/3}}{2}+\frac {3861\,a^7\,b^8\,x^{8/3}}{5}+\frac {9009\,a^5\,b^{10}\,x^{10/3}}{23}+\frac {4095\,a^4\,b^{11}\,x^{11/3}}{22}+\frac {63\,a^2\,b^{13}\,x^{13/3}}{4}}{x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________