\(\int \frac {(a+b \sqrt [3]{x})^{15}}{x^4} \, dx\) [2348]

Optimal result

Integrand size = 15, antiderivative size = 200 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=-\frac {a^{15}}{3 x^3}-\frac {45 a^{14} b}{8 x^{8/3}}-\frac {45 a^{13} b^2}{x^{7/3}}-\frac {455 a^{12} b^3}{2 x^2}-\frac {819 a^{11} b^4}{x^{5/3}}-\frac {9009 a^{10} b^5}{4 x^{4/3}}-\frac {5005 a^9 b^6}{x}-\frac {19305 a^8 b^7}{2 x^{2/3}}-\frac {19305 a^7 b^8}{\sqrt [3]{x}}+9009 a^5 b^{10} \sqrt [3]{x}+\frac {4095}{2} a^4 b^{11} x^{2/3}+455 a^3 b^{12} x+\frac {315}{4} a^2 b^{13} x^{4/3}+9 a b^{14} x^{5/3}+\frac {b^{15} x^2}{2}+5005 a^6 b^9 \log (x) \]

[Out]

-1/3*a^15/x^3-45/8*a^14*b/x^(8/3)-45*a^13*b^2/x^(7/3)-455/2*a^12*b^3/x^2-819*a^11*b^4/x^(5/3)-9009/4*a^10*b^5/
x^(4/3)-5005*a^9*b^6/x-19305/2*a^8*b^7/x^(2/3)-19305*a^7*b^8/x^(1/3)+9009*a^5*b^10*x^(1/3)+4095/2*a^4*b^11*x^(
2/3)+455*a^3*b^12*x+315/4*a^2*b^13*x^(4/3)+9*a*b^14*x^(5/3)+1/2*b^15*x^2+5005*a^6*b^9*ln(x)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=-\frac {a^{15}}{3 x^3}-\frac {45 a^{14} b}{8 x^{8/3}}-\frac {45 a^{13} b^2}{x^{7/3}}-\frac {455 a^{12} b^3}{2 x^2}-\frac {819 a^{11} b^4}{x^{5/3}}-\frac {9009 a^{10} b^5}{4 x^{4/3}}-\frac {5005 a^9 b^6}{x}-\frac {19305 a^8 b^7}{2 x^{2/3}}-\frac {19305 a^7 b^8}{\sqrt [3]{x}}+5005 a^6 b^9 \log (x)+9009 a^5 b^{10} \sqrt [3]{x}+\frac {4095}{2} a^4 b^{11} x^{2/3}+455 a^3 b^{12} x+\frac {315}{4} a^2 b^{13} x^{4/3}+9 a b^{14} x^{5/3}+\frac {b^{15} x^2}{2} \]

[In]

Int[(a + b*x^(1/3))^15/x^4,x]

[Out]

-1/3*a^15/x^3 - (45*a^14*b)/(8*x^(8/3)) - (45*a^13*b^2)/x^(7/3) - (455*a^12*b^3)/(2*x^2) - (819*a^11*b^4)/x^(5
/3) - (9009*a^10*b^5)/(4*x^(4/3)) - (5005*a^9*b^6)/x - (19305*a^8*b^7)/(2*x^(2/3)) - (19305*a^7*b^8)/x^(1/3) +
9009*a^5*b^10*x^(1/3) + (4095*a^4*b^11*x^(2/3))/2 + 455*a^3*b^12*x + (315*a^2*b^13*x^(4/3))/4 + 9*a*b^14*x^(5
/3) + (b^15*x^2)/2 + 5005*a^6*b^9*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{10}} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (3003 a^5 b^{10}+\frac {a^{15}}{x^{10}}+\frac {15 a^{14} b}{x^9}+\frac {105 a^{13} b^2}{x^8}+\frac {455 a^{12} b^3}{x^7}+\frac {1365 a^{11} b^4}{x^6}+\frac {3003 a^{10} b^5}{x^5}+\frac {5005 a^9 b^6}{x^4}+\frac {6435 a^8 b^7}{x^3}+\frac {6435 a^7 b^8}{x^2}+\frac {5005 a^6 b^9}{x}+1365 a^4 b^{11} x+455 a^3 b^{12} x^2+105 a^2 b^{13} x^3+15 a b^{14} x^4+b^{15} x^5\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {a^{15}}{3 x^3}-\frac {45 a^{14} b}{8 x^{8/3}}-\frac {45 a^{13} b^2}{x^{7/3}}-\frac {455 a^{12} b^3}{2 x^2}-\frac {819 a^{11} b^4}{x^{5/3}}-\frac {9009 a^{10} b^5}{4 x^{4/3}}-\frac {5005 a^9 b^6}{x}-\frac {19305 a^8 b^7}{2 x^{2/3}}-\frac {19305 a^7 b^8}{\sqrt [3]{x}}+9009 a^5 b^{10} \sqrt [3]{x}+\frac {4095}{2} a^4 b^{11} x^{2/3}+455 a^3 b^{12} x+\frac {315}{4} a^2 b^{13} x^{4/3}+9 a b^{14} x^{5/3}+\frac {b^{15} x^2}{2}+5005 a^6 b^9 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=\frac {-8 a^{15}-135 a^{14} b \sqrt [3]{x}-1080 a^{13} b^2 x^{2/3}-5460 a^{12} b^3 x-19656 a^{11} b^4 x^{4/3}-54054 a^{10} b^5 x^{5/3}-120120 a^9 b^6 x^2-231660 a^8 b^7 x^{7/3}-463320 a^7 b^8 x^{8/3}+216216 a^5 b^{10} x^{10/3}+49140 a^4 b^{11} x^{11/3}+10920 a^3 b^{12} x^4+1890 a^2 b^{13} x^{13/3}+216 a b^{14} x^{14/3}+12 b^{15} x^5}{24 x^3}+15015 a^6 b^9 \log \left (\sqrt [3]{x}\right ) \]

[In]

Integrate[(a + b*x^(1/3))^15/x^4,x]

[Out]

(-8*a^15 - 135*a^14*b*x^(1/3) - 1080*a^13*b^2*x^(2/3) - 5460*a^12*b^3*x - 19656*a^11*b^4*x^(4/3) - 54054*a^10*
b^5*x^(5/3) - 120120*a^9*b^6*x^2 - 231660*a^8*b^7*x^(7/3) - 463320*a^7*b^8*x^(8/3) + 216216*a^5*b^10*x^(10/3)
+ 49140*a^4*b^11*x^(11/3) + 10920*a^3*b^12*x^4 + 1890*a^2*b^13*x^(13/3) + 216*a*b^14*x^(14/3) + 12*b^15*x^5)/(
24*x^3) + 15015*a^6*b^9*Log[x^(1/3)]

Maple [A] (verified)

Time = 3.65 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(-\frac {a^{15}}{3 x^{3}}-\frac {45 a^{14} b}{8 x^{\frac {8}{3}}}-\frac {45 a^{13} b^{2}}{x^{\frac {7}{3}}}-\frac {455 a^{12} b^{3}}{2 x^{2}}-\frac {819 a^{11} b^{4}}{x^{\frac {5}{3}}}-\frac {9009 a^{10} b^{5}}{4 x^{\frac {4}{3}}}-\frac {5005 a^{9} b^{6}}{x}-\frac {19305 a^{8} b^{7}}{2 x^{\frac {2}{3}}}-\frac {19305 a^{7} b^{8}}{x^{\frac {1}{3}}}+9009 a^{5} b^{10} x^{\frac {1}{3}}+\frac {4095 a^{4} b^{11} x^{\frac {2}{3}}}{2}+455 a^{3} b^{12} x +\frac {315 a^{2} b^{13} x^{\frac {4}{3}}}{4}+9 a \,b^{14} x^{\frac {5}{3}}+\frac {b^{15} x^{2}}{2}+5005 a^{6} b^{9} \ln \left (x \right )\)	\(165\)
default	\(-\frac {a^{15}}{3 x^{3}}-\frac {45 a^{14} b}{8 x^{\frac {8}{3}}}-\frac {45 a^{13} b^{2}}{x^{\frac {7}{3}}}-\frac {455 a^{12} b^{3}}{2 x^{2}}-\frac {819 a^{11} b^{4}}{x^{\frac {5}{3}}}-\frac {9009 a^{10} b^{5}}{4 x^{\frac {4}{3}}}-\frac {5005 a^{9} b^{6}}{x}-\frac {19305 a^{8} b^{7}}{2 x^{\frac {2}{3}}}-\frac {19305 a^{7} b^{8}}{x^{\frac {1}{3}}}+9009 a^{5} b^{10} x^{\frac {1}{3}}+\frac {4095 a^{4} b^{11} x^{\frac {2}{3}}}{2}+455 a^{3} b^{12} x +\frac {315 a^{2} b^{13} x^{\frac {4}{3}}}{4}+9 a \,b^{14} x^{\frac {5}{3}}+\frac {b^{15} x^{2}}{2}+5005 a^{6} b^{9} \ln \left (x \right )\)	\(165\)
trager	\(\frac {\left (-1+x \right ) \left (3 b^{15} x^{4}+2730 a^{3} b^{12} x^{3}+3 b^{15} x^{3}+2 x^{2} a^{15}+1365 a^{12} b^{3} x^{2}+30030 a^{9} b^{6} x^{2}+2 x \,a^{15}+1365 a^{12} b^{3} x +2 a^{15}\right )}{6 x^{3}}-\frac {9 \left (-70 b^{12} x^{4}-8008 a^{3} b^{9} x^{3}+8580 a^{6} b^{6} x^{2}+728 a^{9} b^{3} x +5 a^{12}\right ) a^{2} b}{8 x^{\frac {8}{3}}}-\frac {9 \left (-4 b^{12} x^{4}-910 a^{3} b^{9} x^{3}+8580 a^{6} b^{6} x^{2}+1001 a^{9} b^{3} x +20 a^{12}\right ) a \,b^{2}}{4 x^{\frac {7}{3}}}+5005 a^{6} b^{9} \ln \left (x \right )\)	\(206\)

[In]

int((a+b*x^(1/3))^15/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^15/x^3-45/8*a^14*b/x^(8/3)-45*a^13*b^2/x^(7/3)-455/2*a^12*b^3/x^2-819*a^11*b^4/x^(5/3)-9009/4*a^10*b^5/
x^(4/3)-5005*a^9*b^6/x-19305/2*a^8*b^7/x^(2/3)-19305*a^7*b^8/x^(1/3)+9009*a^5*b^10*x^(1/3)+4095/2*a^4*b^11*x^(
2/3)+455*a^3*b^12*x+315/4*a^2*b^13*x^(4/3)+9*a*b^14*x^(5/3)+1/2*b^15*x^2+5005*a^6*b^9*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=\frac {12 \, b^{15} x^{5} + 10920 \, a^{3} b^{12} x^{4} + 360360 \, a^{6} b^{9} x^{3} \log \left (x^{\frac {1}{3}}\right ) - 120120 \, a^{9} b^{6} x^{2} - 5460 \, a^{12} b^{3} x - 8 \, a^{15} + 54 \, {\left (4 \, a b^{14} x^{4} + 910 \, a^{4} b^{11} x^{3} - 8580 \, a^{7} b^{8} x^{2} - 1001 \, a^{10} b^{5} x - 20 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 27 \, {\left (70 \, a^{2} b^{13} x^{4} + 8008 \, a^{5} b^{10} x^{3} - 8580 \, a^{8} b^{7} x^{2} - 728 \, a^{11} b^{4} x - 5 \, a^{14} b\right )} x^{\frac {1}{3}}}{24 \, x^{3}} \]

[In]

integrate((a+b*x^(1/3))^15/x^4,x, algorithm="fricas")

[Out]

1/24*(12*b^15*x^5 + 10920*a^3*b^12*x^4 + 360360*a^6*b^9*x^3*log(x^(1/3)) - 120120*a^9*b^6*x^2 - 5460*a^12*b^3*
x - 8*a^15 + 54*(4*a*b^14*x^4 + 910*a^4*b^11*x^3 - 8580*a^7*b^8*x^2 - 1001*a^10*b^5*x - 20*a^13*b^2)*x^(2/3) +
27*(70*a^2*b^13*x^4 + 8008*a^5*b^10*x^3 - 8580*a^8*b^7*x^2 - 728*a^11*b^4*x - 5*a^14*b)*x^(1/3))/x^3

Sympy [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=- \frac {a^{15}}{3 x^{3}} - \frac {45 a^{14} b}{8 x^{\frac {8}{3}}} - \frac {45 a^{13} b^{2}}{x^{\frac {7}{3}}} - \frac {455 a^{12} b^{3}}{2 x^{2}} - \frac {819 a^{11} b^{4}}{x^{\frac {5}{3}}} - \frac {9009 a^{10} b^{5}}{4 x^{\frac {4}{3}}} - \frac {5005 a^{9} b^{6}}{x} - \frac {19305 a^{8} b^{7}}{2 x^{\frac {2}{3}}} - \frac {19305 a^{7} b^{8}}{\sqrt [3]{x}} + 5005 a^{6} b^{9} \log {\left (x \right )} + 9009 a^{5} b^{10} \sqrt [3]{x} + \frac {4095 a^{4} b^{11} x^{\frac {2}{3}}}{2} + 455 a^{3} b^{12} x + \frac {315 a^{2} b^{13} x^{\frac {4}{3}}}{4} + 9 a b^{14} x^{\frac {5}{3}} + \frac {b^{15} x^{2}}{2} \]

[In]

integrate((a+b*x**(1/3))**15/x**4,x)

[Out]

-a**15/(3*x**3) - 45*a**14*b/(8*x**(8/3)) - 45*a**13*b**2/x**(7/3) - 455*a**12*b**3/(2*x**2) - 819*a**11*b**4/
x**(5/3) - 9009*a**10*b**5/(4*x**(4/3)) - 5005*a**9*b**6/x - 19305*a**8*b**7/(2*x**(2/3)) - 19305*a**7*b**8/x*
*(1/3) + 5005*a**6*b**9*log(x) + 9009*a**5*b**10*x**(1/3) + 4095*a**4*b**11*x**(2/3)/2 + 455*a**3*b**12*x + 31
5*a**2*b**13*x**(4/3)/4 + 9*a*b**14*x**(5/3) + b**15*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=\frac {1}{2} \, b^{15} x^{2} + 9 \, a b^{14} x^{\frac {5}{3}} + \frac {315}{4} \, a^{2} b^{13} x^{\frac {4}{3}} + 455 \, a^{3} b^{12} x + 5005 \, a^{6} b^{9} \log \left (x\right ) + \frac {4095}{2} \, a^{4} b^{11} x^{\frac {2}{3}} + 9009 \, a^{5} b^{10} x^{\frac {1}{3}} - \frac {463320 \, a^{7} b^{8} x^{\frac {8}{3}} + 231660 \, a^{8} b^{7} x^{\frac {7}{3}} + 120120 \, a^{9} b^{6} x^{2} + 54054 \, a^{10} b^{5} x^{\frac {5}{3}} + 19656 \, a^{11} b^{4} x^{\frac {4}{3}} + 5460 \, a^{12} b^{3} x + 1080 \, a^{13} b^{2} x^{\frac {2}{3}} + 135 \, a^{14} b x^{\frac {1}{3}} + 8 \, a^{15}}{24 \, x^{3}} \]

[In]

integrate((a+b*x^(1/3))^15/x^4,x, algorithm="maxima")

[Out]

1/2*b^15*x^2 + 9*a*b^14*x^(5/3) + 315/4*a^2*b^13*x^(4/3) + 455*a^3*b^12*x + 5005*a^6*b^9*log(x) + 4095/2*a^4*b
^11*x^(2/3) + 9009*a^5*b^10*x^(1/3) - 1/24*(463320*a^7*b^8*x^(8/3) + 231660*a^8*b^7*x^(7/3) + 120120*a^9*b^6*x
^2 + 54054*a^10*b^5*x^(5/3) + 19656*a^11*b^4*x^(4/3) + 5460*a^12*b^3*x + 1080*a^13*b^2*x^(2/3) + 135*a^14*b*x^
(1/3) + 8*a^15)/x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=\frac {1}{2} \, b^{15} x^{2} + 9 \, a b^{14} x^{\frac {5}{3}} + \frac {315}{4} \, a^{2} b^{13} x^{\frac {4}{3}} + 455 \, a^{3} b^{12} x + 5005 \, a^{6} b^{9} \log \left ({\left | x \right |}\right ) + \frac {4095}{2} \, a^{4} b^{11} x^{\frac {2}{3}} + 9009 \, a^{5} b^{10} x^{\frac {1}{3}} - \frac {463320 \, a^{7} b^{8} x^{\frac {8}{3}} + 231660 \, a^{8} b^{7} x^{\frac {7}{3}} + 120120 \, a^{9} b^{6} x^{2} + 54054 \, a^{10} b^{5} x^{\frac {5}{3}} + 19656 \, a^{11} b^{4} x^{\frac {4}{3}} + 5460 \, a^{12} b^{3} x + 1080 \, a^{13} b^{2} x^{\frac {2}{3}} + 135 \, a^{14} b x^{\frac {1}{3}} + 8 \, a^{15}}{24 \, x^{3}} \]

[In]

integrate((a+b*x^(1/3))^15/x^4,x, algorithm="giac")

[Out]

1/2*b^15*x^2 + 9*a*b^14*x^(5/3) + 315/4*a^2*b^13*x^(4/3) + 455*a^3*b^12*x + 5005*a^6*b^9*log(abs(x)) + 4095/2*
a^4*b^11*x^(2/3) + 9009*a^5*b^10*x^(1/3) - 1/24*(463320*a^7*b^8*x^(8/3) + 231660*a^8*b^7*x^(7/3) + 120120*a^9*
b^6*x^2 + 54054*a^10*b^5*x^(5/3) + 19656*a^11*b^4*x^(4/3) + 5460*a^12*b^3*x + 1080*a^13*b^2*x^(2/3) + 135*a^14
*b*x^(1/3) + 8*a^15)/x^3

Mupad [B] (verification not implemented)

Time = 5.83 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx=\frac {b^{15}\,x^2}{2}-\frac {\frac {a^{15}}{3}+\frac {455\,a^{12}\,b^3\,x}{2}+\frac {45\,a^{14}\,b\,x^{1/3}}{8}+5005\,a^9\,b^6\,x^2+45\,a^{13}\,b^2\,x^{2/3}+819\,a^{11}\,b^4\,x^{4/3}+\frac {9009\,a^{10}\,b^5\,x^{5/3}}{4}+\frac {19305\,a^8\,b^7\,x^{7/3}}{2}+19305\,a^7\,b^8\,x^{8/3}}{x^3}+15015\,a^6\,b^9\,\ln \left (x^{1/3}\right )+455\,a^3\,b^{12}\,x+9\,a\,b^{14}\,x^{5/3}+9009\,a^5\,b^{10}\,x^{1/3}+\frac {4095\,a^4\,b^{11}\,x^{2/3}}{2}+\frac {315\,a^2\,b^{13}\,x^{4/3}}{4} \]

[In]

int((a + b*x^(1/3))^15/x^4,x)

[Out]

(b^15*x^2)/2 - (a^15/3 + (455*a^12*b^3*x)/2 + (45*a^14*b*x^(1/3))/8 + 5005*a^9*b^6*x^2 + 45*a^13*b^2*x^(2/3) +
819*a^11*b^4*x^(4/3) + (9009*a^10*b^5*x^(5/3))/4 + (19305*a^8*b^7*x^(7/3))/2 + 19305*a^7*b^8*x^(8/3))/x^3 + 1
5015*a^6*b^9*log(x^(1/3)) + 455*a^3*b^12*x + 9*a*b^14*x^(5/3) + 9009*a^5*b^10*x^(1/3) + (4095*a^4*b^11*x^(2/3)
)/2 + (315*a^2*b^13*x^(4/3))/4