3.2349 \(\int \frac {(a+b \sqrt [3]{x})^{15}}{x^6} \, dx\)

Optimal. Leaf size=211 \[ -\frac {a^{15}}{5 x^5}-\frac {45 a^{14} b}{14 x^{14/3}}-\frac {315 a^{13} b^2}{13 x^{13/3}}-\frac {455 a^{12} b^3}{4 x^4}-\frac {4095 a^{11} b^4}{11 x^{11/3}}-\frac {9009 a^{10} b^5}{10 x^{10/3}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {19305 a^8 b^7}{8 x^{8/3}}-\frac {19305 a^7 b^8}{7 x^{7/3}}-\frac {5005 a^6 b^9}{2 x^2}-\frac {9009 a^5 b^{10}}{5 x^{5/3}}-\frac {4095 a^4 b^{11}}{4 x^{4/3}}-\frac {455 a^3 b^{12}}{x}-\frac {315 a^2 b^{13}}{2 x^{2/3}}-\frac {45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x) \]

[Out]

-1/5*a^15/x^5-45/14*a^14*b/x^(14/3)-315/13*a^13*b^2/x^(13/3)-455/4*a^12*b^3/x^4-4095/11*a^11*b^4/x^(11/3)-9009
/10*a^10*b^5/x^(10/3)-5005/3*a^9*b^6/x^3-19305/8*a^8*b^7/x^(8/3)-19305/7*a^7*b^8/x^(7/3)-5005/2*a^6*b^9/x^2-90
09/5*a^5*b^10/x^(5/3)-4095/4*a^4*b^11/x^(4/3)-455*a^3*b^12/x-315/2*a^2*b^13/x^(2/3)-45*a*b^14/x^(1/3)+b^15*ln(
x)

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Rubi [A]  time = 0.12, antiderivative size = 211, 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} \[ -\frac {315 a^{13} b^2}{13 x^{13/3}}-\frac {455 a^{12} b^3}{4 x^4}-\frac {4095 a^{11} b^4}{11 x^{11/3}}-\frac {9009 a^{10} b^5}{10 x^{10/3}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {19305 a^8 b^7}{8 x^{8/3}}-\frac {19305 a^7 b^8}{7 x^{7/3}}-\frac {5005 a^6 b^9}{2 x^2}-\frac {9009 a^5 b^{10}}{5 x^{5/3}}-\frac {4095 a^4 b^{11}}{4 x^{4/3}}-\frac {315 a^2 b^{13}}{2 x^{2/3}}-\frac {455 a^3 b^{12}}{x}-\frac {45 a^{14} b}{14 x^{14/3}}-\frac {a^{15}}{5 x^5}-\frac {45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^6,x]

[Out]

-a^15/(5*x^5) - (45*a^14*b)/(14*x^(14/3)) - (315*a^13*b^2)/(13*x^(13/3)) - (455*a^12*b^3)/(4*x^4) - (4095*a^11
*b^4)/(11*x^(11/3)) - (9009*a^10*b^5)/(10*x^(10/3)) - (5005*a^9*b^6)/(3*x^3) - (19305*a^8*b^7)/(8*x^(8/3)) - (
19305*a^7*b^8)/(7*x^(7/3)) - (5005*a^6*b^9)/(2*x^2) - (9009*a^5*b^10)/(5*x^(5/3)) - (4095*a^4*b^11)/(4*x^(4/3)
) - (455*a^3*b^12)/x - (315*a^2*b^13)/(2*x^(2/3)) - (45*a*b^14)/x^(1/3) + b^15*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^6} \, dx &=3 \operatorname {Subst}\left (\int \frac {(a+b x)^{15}}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {a^{15}}{x^{16}}+\frac {15 a^{14} b}{x^{15}}+\frac {105 a^{13} b^2}{x^{14}}+\frac {455 a^{12} b^3}{x^{13}}+\frac {1365 a^{11} b^4}{x^{12}}+\frac {3003 a^{10} b^5}{x^{11}}+\frac {5005 a^9 b^6}{x^{10}}+\frac {6435 a^8 b^7}{x^9}+\frac {6435 a^7 b^8}{x^8}+\frac {5005 a^6 b^9}{x^7}+\frac {3003 a^5 b^{10}}{x^6}+\frac {1365 a^4 b^{11}}{x^5}+\frac {455 a^3 b^{12}}{x^4}+\frac {105 a^2 b^{13}}{x^3}+\frac {15 a b^{14}}{x^2}+\frac {b^{15}}{x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^{15}}{5 x^5}-\frac {45 a^{14} b}{14 x^{14/3}}-\frac {315 a^{13} b^2}{13 x^{13/3}}-\frac {455 a^{12} b^3}{4 x^4}-\frac {4095 a^{11} b^4}{11 x^{11/3}}-\frac {9009 a^{10} b^5}{10 x^{10/3}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {19305 a^8 b^7}{8 x^{8/3}}-\frac {19305 a^7 b^8}{7 x^{7/3}}-\frac {5005 a^6 b^9}{2 x^2}-\frac {9009 a^5 b^{10}}{5 x^{5/3}}-\frac {4095 a^4 b^{11}}{4 x^{4/3}}-\frac {455 a^3 b^{12}}{x}-\frac {315 a^2 b^{13}}{2 x^{2/3}}-\frac {45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 211, normalized size = 1.00 \[ -\frac {a^{15}}{5 x^5}-\frac {45 a^{14} b}{14 x^{14/3}}-\frac {315 a^{13} b^2}{13 x^{13/3}}-\frac {455 a^{12} b^3}{4 x^4}-\frac {4095 a^{11} b^4}{11 x^{11/3}}-\frac {9009 a^{10} b^5}{10 x^{10/3}}-\frac {5005 a^9 b^6}{3 x^3}-\frac {19305 a^8 b^7}{8 x^{8/3}}-\frac {19305 a^7 b^8}{7 x^{7/3}}-\frac {5005 a^6 b^9}{2 x^2}-\frac {9009 a^5 b^{10}}{5 x^{5/3}}-\frac {4095 a^4 b^{11}}{4 x^{4/3}}-\frac {455 a^3 b^{12}}{x}-\frac {315 a^2 b^{13}}{2 x^{2/3}}-\frac {45 a b^{14}}{\sqrt [3]{x}}+b^{15} \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^6,x]

[Out]

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

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fricas [A]  time = 0.75, size = 173, normalized size = 0.82 \[ \frac {360360 \, b^{15} x^{5} \log \left (x^{\frac {1}{3}}\right ) - 54654600 \, a^{3} b^{12} x^{4} - 300600300 \, a^{6} b^{9} x^{3} - 200400200 \, a^{9} b^{6} x^{2} - 13663650 \, a^{12} b^{3} x - 24024 \, a^{15} - 594 \, {\left (9100 \, a b^{14} x^{4} + 207025 \, a^{4} b^{11} x^{3} + 557700 \, a^{7} b^{8} x^{2} + 182182 \, a^{10} b^{5} x + 4900 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} - 351 \, {\left (53900 \, a^{2} b^{13} x^{4} + 616616 \, a^{5} b^{10} x^{3} + 825825 \, a^{8} b^{7} x^{2} + 127400 \, a^{11} b^{4} x + 1100 \, a^{14} b\right )} x^{\frac {1}{3}}}{120120 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^6,x, algorithm="fricas")

[Out]

1/120120*(360360*b^15*x^5*log(x^(1/3)) - 54654600*a^3*b^12*x^4 - 300600300*a^6*b^9*x^3 - 200400200*a^9*b^6*x^2
- 13663650*a^12*b^3*x - 24024*a^15 - 594*(9100*a*b^14*x^4 + 207025*a^4*b^11*x^3 + 557700*a^7*b^8*x^2 + 182182
*a^10*b^5*x + 4900*a^13*b^2)*x^(2/3) - 351*(53900*a^2*b^13*x^4 + 616616*a^5*b^10*x^3 + 825825*a^8*b^7*x^2 + 12
7400*a^11*b^4*x + 1100*a^14*b)*x^(1/3))/x^5

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giac [A]  time = 0.18, size = 167, normalized size = 0.79 \[ b^{15} \log \left ({\left | x \right |}\right ) - \frac {5405400 \, a b^{14} x^{\frac {14}{3}} + 18918900 \, a^{2} b^{13} x^{\frac {13}{3}} + 54654600 \, a^{3} b^{12} x^{4} + 122972850 \, a^{4} b^{11} x^{\frac {11}{3}} + 216432216 \, a^{5} b^{10} x^{\frac {10}{3}} + 300600300 \, a^{6} b^{9} x^{3} + 331273800 \, a^{7} b^{8} x^{\frac {8}{3}} + 289864575 \, a^{8} b^{7} x^{\frac {7}{3}} + 200400200 \, a^{9} b^{6} x^{2} + 108216108 \, a^{10} b^{5} x^{\frac {5}{3}} + 44717400 \, a^{11} b^{4} x^{\frac {4}{3}} + 13663650 \, a^{12} b^{3} x + 2910600 \, a^{13} b^{2} x^{\frac {2}{3}} + 386100 \, a^{14} b x^{\frac {1}{3}} + 24024 \, a^{15}}{120120 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^6,x, algorithm="giac")

[Out]

b^15*log(abs(x)) - 1/120120*(5405400*a*b^14*x^(14/3) + 18918900*a^2*b^13*x^(13/3) + 54654600*a^3*b^12*x^4 + 12
2972850*a^4*b^11*x^(11/3) + 216432216*a^5*b^10*x^(10/3) + 300600300*a^6*b^9*x^3 + 331273800*a^7*b^8*x^(8/3) +
289864575*a^8*b^7*x^(7/3) + 200400200*a^9*b^6*x^2 + 108216108*a^10*b^5*x^(5/3) + 44717400*a^11*b^4*x^(4/3) + 1
3663650*a^12*b^3*x + 2910600*a^13*b^2*x^(2/3) + 386100*a^14*b*x^(1/3) + 24024*a^15)/x^5

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maple [A]  time = 0.01, size = 166, normalized size = 0.79 \[ b^{15} \ln \relax (x )-\frac {45 a \,b^{14}}{x^{\frac {1}{3}}}-\frac {315 a^{2} b^{13}}{2 x^{\frac {2}{3}}}-\frac {455 a^{3} b^{12}}{x}-\frac {4095 a^{4} b^{11}}{4 x^{\frac {4}{3}}}-\frac {9009 a^{5} b^{10}}{5 x^{\frac {5}{3}}}-\frac {5005 a^{6} b^{9}}{2 x^{2}}-\frac {19305 a^{7} b^{8}}{7 x^{\frac {7}{3}}}-\frac {19305 a^{8} b^{7}}{8 x^{\frac {8}{3}}}-\frac {5005 a^{9} b^{6}}{3 x^{3}}-\frac {9009 a^{10} b^{5}}{10 x^{\frac {10}{3}}}-\frac {4095 a^{11} b^{4}}{11 x^{\frac {11}{3}}}-\frac {455 a^{12} b^{3}}{4 x^{4}}-\frac {315 a^{13} b^{2}}{13 x^{\frac {13}{3}}}-\frac {45 a^{14} b}{14 x^{\frac {14}{3}}}-\frac {a^{15}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^15/x^6,x)

[Out]

-1/5*a^15/x^5-45/14*a^14*b/x^(14/3)-315/13*a^13*b^2/x^(13/3)-455/4*a^12*b^3/x^4-4095/11*a^11*b^4/x^(11/3)-9009
/10*a^10*b^5/x^(10/3)-5005/3*a^9*b^6/x^3-19305/8*a^8*b^7/x^(8/3)-19305/7*a^7*b^8/x^(7/3)-5005/2*a^6*b^9/x^2-90
09/5*a^5*b^10/x^(5/3)-4095/4*a^4*b^11/x^(4/3)-455*a^3*b^12/x-315/2*a^2*b^13/x^(2/3)-45*a*b^14/x^(1/3)+b^15*ln(
x)

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maxima [A]  time = 0.90, size = 166, normalized size = 0.79 \[ b^{15} \log \relax (x) - \frac {5405400 \, a b^{14} x^{\frac {14}{3}} + 18918900 \, a^{2} b^{13} x^{\frac {13}{3}} + 54654600 \, a^{3} b^{12} x^{4} + 122972850 \, a^{4} b^{11} x^{\frac {11}{3}} + 216432216 \, a^{5} b^{10} x^{\frac {10}{3}} + 300600300 \, a^{6} b^{9} x^{3} + 331273800 \, a^{7} b^{8} x^{\frac {8}{3}} + 289864575 \, a^{8} b^{7} x^{\frac {7}{3}} + 200400200 \, a^{9} b^{6} x^{2} + 108216108 \, a^{10} b^{5} x^{\frac {5}{3}} + 44717400 \, a^{11} b^{4} x^{\frac {4}{3}} + 13663650 \, a^{12} b^{3} x + 2910600 \, a^{13} b^{2} x^{\frac {2}{3}} + 386100 \, a^{14} b x^{\frac {1}{3}} + 24024 \, a^{15}}{120120 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^6,x, algorithm="maxima")

[Out]

b^15*log(x) - 1/120120*(5405400*a*b^14*x^(14/3) + 18918900*a^2*b^13*x^(13/3) + 54654600*a^3*b^12*x^4 + 1229728
50*a^4*b^11*x^(11/3) + 216432216*a^5*b^10*x^(10/3) + 300600300*a^6*b^9*x^3 + 331273800*a^7*b^8*x^(8/3) + 28986
4575*a^8*b^7*x^(7/3) + 200400200*a^9*b^6*x^2 + 108216108*a^10*b^5*x^(5/3) + 44717400*a^11*b^4*x^(4/3) + 136636
50*a^12*b^3*x + 2910600*a^13*b^2*x^(2/3) + 386100*a^14*b*x^(1/3) + 24024*a^15)/x^5

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mupad [B]  time = 1.28, size = 168, normalized size = 0.80 \[ 3\,b^{15}\,\ln \left (x^{1/3}\right )-\frac {a^{15}}{5\,x^5}-\frac {45\,a\,b^{14}}{x^{1/3}}-\frac {45\,a^{14}\,b}{14\,x^{14/3}}-\frac {455\,a^3\,b^{12}}{x}-\frac {5005\,a^6\,b^9}{2\,x^2}-\frac {5005\,a^9\,b^6}{3\,x^3}-\frac {455\,a^{12}\,b^3}{4\,x^4}-\frac {315\,a^2\,b^{13}}{2\,x^{2/3}}-\frac {4095\,a^4\,b^{11}}{4\,x^{4/3}}-\frac {9009\,a^5\,b^{10}}{5\,x^{5/3}}-\frac {19305\,a^7\,b^8}{7\,x^{7/3}}-\frac {19305\,a^8\,b^7}{8\,x^{8/3}}-\frac {9009\,a^{10}\,b^5}{10\,x^{10/3}}-\frac {4095\,a^{11}\,b^4}{11\,x^{11/3}}-\frac {315\,a^{13}\,b^2}{13\,x^{13/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^15/x^6,x)

[Out]

3*b^15*log(x^(1/3)) - a^15/(5*x^5) - (45*a*b^14)/x^(1/3) - (45*a^14*b)/(14*x^(14/3)) - (455*a^3*b^12)/x - (500
5*a^6*b^9)/(2*x^2) - (5005*a^9*b^6)/(3*x^3) - (455*a^12*b^3)/(4*x^4) - (315*a^2*b^13)/(2*x^(2/3)) - (4095*a^4*
b^11)/(4*x^(4/3)) - (9009*a^5*b^10)/(5*x^(5/3)) - (19305*a^7*b^8)/(7*x^(7/3)) - (19305*a^8*b^7)/(8*x^(8/3)) -
(9009*a^10*b^5)/(10*x^(10/3)) - (4095*a^11*b^4)/(11*x^(11/3)) - (315*a^13*b^2)/(13*x^(13/3))

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sympy [A]  time = 7.90, size = 212, normalized size = 1.00 \[ - \frac {a^{15}}{5 x^{5}} - \frac {45 a^{14} b}{14 x^{\frac {14}{3}}} - \frac {315 a^{13} b^{2}}{13 x^{\frac {13}{3}}} - \frac {455 a^{12} b^{3}}{4 x^{4}} - \frac {4095 a^{11} b^{4}}{11 x^{\frac {11}{3}}} - \frac {9009 a^{10} b^{5}}{10 x^{\frac {10}{3}}} - \frac {5005 a^{9} b^{6}}{3 x^{3}} - \frac {19305 a^{8} b^{7}}{8 x^{\frac {8}{3}}} - \frac {19305 a^{7} b^{8}}{7 x^{\frac {7}{3}}} - \frac {5005 a^{6} b^{9}}{2 x^{2}} - \frac {9009 a^{5} b^{10}}{5 x^{\frac {5}{3}}} - \frac {4095 a^{4} b^{11}}{4 x^{\frac {4}{3}}} - \frac {455 a^{3} b^{12}}{x} - \frac {315 a^{2} b^{13}}{2 x^{\frac {2}{3}}} - \frac {45 a b^{14}}{\sqrt [3]{x}} + b^{15} \log {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15/x**6,x)

[Out]

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

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