3.2347 \(\int \frac {(a+b \sqrt [3]{x})^{15}}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 200, 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} \[ -\frac {315 a^{13} b^2}{4 x^{4/3}}-\frac {4095 a^{11} b^4}{2 x^{2/3}}+\frac {19305}{2} a^7 b^8 x^{2/3}+\frac {9009}{4} a^5 b^{10} x^{4/3}+819 a^4 b^{11} x^{5/3}+\frac {455}{2} a^3 b^{12} x^2+45 a^2 b^{13} x^{7/3}-\frac {455 a^{12} b^3}{x}-\frac {9009 a^{10} b^5}{\sqrt [3]{x}}+19305 a^8 b^7 \sqrt [3]{x}+5005 a^6 b^9 x+5005 a^9 b^6 \log (x)-\frac {9 a^{14} b}{x^{5/3}}-\frac {a^{15}}{2 x^2}+\frac {45}{8} a b^{14} x^{8/3}+\frac {b^{15} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^3,x]

[Out]

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

Rule 43

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 266

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

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Mathematica [A]  time = 0.11, size = 200, normalized size = 1.00 \[ -\frac {a^{15}}{2 x^2}-\frac {9 a^{14} b}{x^{5/3}}-\frac {315 a^{13} b^2}{4 x^{4/3}}-\frac {455 a^{12} b^3}{x}-\frac {4095 a^{11} b^4}{2 x^{2/3}}-\frac {9009 a^{10} b^5}{\sqrt [3]{x}}+5005 a^9 b^6 \log (x)+19305 a^8 b^7 \sqrt [3]{x}+\frac {19305}{2} a^7 b^8 x^{2/3}+5005 a^6 b^9 x+\frac {9009}{4} a^5 b^{10} x^{4/3}+819 a^4 b^{11} x^{5/3}+\frac {455}{2} a^3 b^{12} x^2+45 a^2 b^{13} x^{7/3}+\frac {45}{8} a b^{14} x^{8/3}+\frac {b^{15} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^3,x]

[Out]

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

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fricas [A]  time = 1.06, size = 173, normalized size = 0.86 \[ \frac {8 \, b^{15} x^{5} + 5460 \, a^{3} b^{12} x^{4} + 120120 \, a^{6} b^{9} x^{3} + 360360 \, a^{9} b^{6} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 10920 \, a^{12} b^{3} x - 12 \, a^{15} + 27 \, {\left (5 \, a b^{14} x^{4} + 728 \, a^{4} b^{11} x^{3} + 8580 \, a^{7} b^{8} x^{2} - 8008 \, a^{10} b^{5} x - 70 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 54 \, {\left (20 \, a^{2} b^{13} x^{4} + 1001 \, a^{5} b^{10} x^{3} + 8580 \, a^{8} b^{7} x^{2} - 910 \, a^{11} b^{4} x - 4 \, a^{14} b\right )} x^{\frac {1}{3}}}{24 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 166, normalized size = 0.83 \[ \frac {1}{3} \, b^{15} x^{3} + \frac {45}{8} \, a b^{14} x^{\frac {8}{3}} + 45 \, a^{2} b^{13} x^{\frac {7}{3}} + \frac {455}{2} \, a^{3} b^{12} x^{2} + 819 \, a^{4} b^{11} x^{\frac {5}{3}} + \frac {9009}{4} \, a^{5} b^{10} x^{\frac {4}{3}} + 5005 \, a^{6} b^{9} x + 5005 \, a^{9} b^{6} \log \left ({\left | x \right |}\right ) + \frac {19305}{2} \, a^{7} b^{8} x^{\frac {2}{3}} + 19305 \, a^{8} b^{7} x^{\frac {1}{3}} - \frac {36036 \, a^{10} b^{5} x^{\frac {5}{3}} + 8190 \, a^{11} b^{4} x^{\frac {4}{3}} + 1820 \, a^{12} b^{3} x + 315 \, a^{13} b^{2} x^{\frac {2}{3}} + 36 \, a^{14} b x^{\frac {1}{3}} + 2 \, a^{15}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^15*x^3 + 45/8*a*b^14*x^(8/3) + 45*a^2*b^13*x^(7/3) + 455/2*a^3*b^12*x^2 + 819*a^4*b^11*x^(5/3) + 9009/4*
a^5*b^10*x^(4/3) + 5005*a^6*b^9*x + 5005*a^9*b^6*log(abs(x)) + 19305/2*a^7*b^8*x^(2/3) + 19305*a^8*b^7*x^(1/3)
- 1/4*(36036*a^10*b^5*x^(5/3) + 8190*a^11*b^4*x^(4/3) + 1820*a^12*b^3*x + 315*a^13*b^2*x^(2/3) + 36*a^14*b*x^
(1/3) + 2*a^15)/x^2

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maple [A]  time = 0.01, size = 165, normalized size = 0.82 \[ \frac {b^{15} x^{3}}{3}+\frac {45 a \,b^{14} x^{\frac {8}{3}}}{8}+45 a^{2} b^{13} x^{\frac {7}{3}}+\frac {455 a^{3} b^{12} x^{2}}{2}+819 a^{4} b^{11} x^{\frac {5}{3}}+\frac {9009 a^{5} b^{10} x^{\frac {4}{3}}}{4}+5005 a^{9} b^{6} \ln \relax (x )+5005 a^{6} b^{9} x +\frac {19305 a^{7} b^{8} x^{\frac {2}{3}}}{2}+19305 a^{8} b^{7} x^{\frac {1}{3}}-\frac {9009 a^{10} b^{5}}{x^{\frac {1}{3}}}-\frac {4095 a^{11} b^{4}}{2 x^{\frac {2}{3}}}-\frac {455 a^{12} b^{3}}{x}-\frac {315 a^{13} b^{2}}{4 x^{\frac {4}{3}}}-\frac {9 a^{14} b}{x^{\frac {5}{3}}}-\frac {a^{15}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^3,x)

[Out]

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

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maxima [A]  time = 0.91, size = 165, normalized size = 0.82 \[ \frac {1}{3} \, b^{15} x^{3} + \frac {45}{8} \, a b^{14} x^{\frac {8}{3}} + 45 \, a^{2} b^{13} x^{\frac {7}{3}} + \frac {455}{2} \, a^{3} b^{12} x^{2} + 819 \, a^{4} b^{11} x^{\frac {5}{3}} + \frac {9009}{4} \, a^{5} b^{10} x^{\frac {4}{3}} + 5005 \, a^{6} b^{9} x + 5005 \, a^{9} b^{6} \log \relax (x) + \frac {19305}{2} \, a^{7} b^{8} x^{\frac {2}{3}} + 19305 \, a^{8} b^{7} x^{\frac {1}{3}} - \frac {36036 \, a^{10} b^{5} x^{\frac {5}{3}} + 8190 \, a^{11} b^{4} x^{\frac {4}{3}} + 1820 \, a^{12} b^{3} x + 315 \, a^{13} b^{2} x^{\frac {2}{3}} + 36 \, a^{14} b x^{\frac {1}{3}} + 2 \, a^{15}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^15*x^3 + 45/8*a*b^14*x^(8/3) + 45*a^2*b^13*x^(7/3) + 455/2*a^3*b^12*x^2 + 819*a^4*b^11*x^(5/3) + 9009/4*
a^5*b^10*x^(4/3) + 5005*a^6*b^9*x + 5005*a^9*b^6*log(x) + 19305/2*a^7*b^8*x^(2/3) + 19305*a^8*b^7*x^(1/3) - 1/
4*(36036*a^10*b^5*x^(5/3) + 8190*a^11*b^4*x^(4/3) + 1820*a^12*b^3*x + 315*a^13*b^2*x^(2/3) + 36*a^14*b*x^(1/3)
+ 2*a^15)/x^2

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mupad [B]  time = 0.08, size = 167, normalized size = 0.84 \[ \frac {b^{15}\,x^3}{3}-\frac {\frac {a^{15}}{2}+455\,a^{12}\,b^3\,x+9\,a^{14}\,b\,x^{1/3}+\frac {315\,a^{13}\,b^2\,x^{2/3}}{4}+\frac {4095\,a^{11}\,b^4\,x^{4/3}}{2}+9009\,a^{10}\,b^5\,x^{5/3}}{x^2}+15015\,a^9\,b^6\,\ln \left (x^{1/3}\right )+5005\,a^6\,b^9\,x+\frac {45\,a\,b^{14}\,x^{8/3}}{8}+\frac {455\,a^3\,b^{12}\,x^2}{2}+19305\,a^8\,b^7\,x^{1/3}+\frac {19305\,a^7\,b^8\,x^{2/3}}{2}+\frac {9009\,a^5\,b^{10}\,x^{4/3}}{4}+819\,a^4\,b^{11}\,x^{5/3}+45\,a^2\,b^{13}\,x^{7/3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^15/x^3,x)

[Out]

(b^15*x^3)/3 - (a^15/2 + 455*a^12*b^3*x + 9*a^14*b*x^(1/3) + (315*a^13*b^2*x^(2/3))/4 + (4095*a^11*b^4*x^(4/3)
)/2 + 9009*a^10*b^5*x^(5/3))/x^2 + 15015*a^9*b^6*log(x^(1/3)) + 5005*a^6*b^9*x + (45*a*b^14*x^(8/3))/8 + (455*
a^3*b^12*x^2)/2 + 19305*a^8*b^7*x^(1/3) + (19305*a^7*b^8*x^(2/3))/2 + (9009*a^5*b^10*x^(4/3))/4 + 819*a^4*b^11
*x^(5/3) + 45*a^2*b^13*x^(7/3)

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sympy [A]  time = 4.63, size = 202, normalized size = 1.01 \[ - \frac {a^{15}}{2 x^{2}} - \frac {9 a^{14} b}{x^{\frac {5}{3}}} - \frac {315 a^{13} b^{2}}{4 x^{\frac {4}{3}}} - \frac {455 a^{12} b^{3}}{x} - \frac {4095 a^{11} b^{4}}{2 x^{\frac {2}{3}}} - \frac {9009 a^{10} b^{5}}{\sqrt [3]{x}} + 5005 a^{9} b^{6} \log {\relax (x )} + 19305 a^{8} b^{7} \sqrt [3]{x} + \frac {19305 a^{7} b^{8} x^{\frac {2}{3}}}{2} + 5005 a^{6} b^{9} x + \frac {9009 a^{5} b^{10} x^{\frac {4}{3}}}{4} + 819 a^{4} b^{11} x^{\frac {5}{3}} + \frac {455 a^{3} b^{12} x^{2}}{2} + 45 a^{2} b^{13} x^{\frac {7}{3}} + \frac {45 a b^{14} x^{\frac {8}{3}}}{8} + \frac {b^{15} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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