3.2348 \(\int \frac{(a+b \sqrt [3]{x})^{15}}{x^4} \, dx\)

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

[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) +
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]

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Rubi [A]  time = 0.121691, antiderivative size = 200, normalized size of antiderivative = 1., 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} \[ -\frac{45 a^{13} b^2}{x^{7/3}}-\frac{455 a^{12} b^3}{2 x^2}-\frac{819 a^{11} b^4}{x^{5/3}}-\frac{9009 a^{10} b^5}{4 x^{4/3}}-\frac{19305 a^8 b^7}{2 x^{2/3}}+\frac{4095}{2} a^4 b^{11} x^{2/3}+\frac{315}{4} a^2 b^{13} x^{4/3}-\frac{5005 a^9 b^6}{x}-\frac{19305 a^7 b^8}{\sqrt [3]{x}}+9009 a^5 b^{10} \sqrt [3]{x}+455 a^3 b^{12} x+5005 a^6 b^9 \log (x)-\frac{45 a^{14} b}{8 x^{8/3}}-\frac{a^{15}}{3 x^3}+9 a b^{14} x^{5/3}+\frac{b^{15} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(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) +
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 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]]

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])

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x^4} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{10}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{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]  time = 0.108953, size = 200, normalized size = 1. \[ -\frac{45 a^{13} b^2}{x^{7/3}}-\frac{455 a^{12} b^3}{2 x^2}-\frac{819 a^{11} b^4}{x^{5/3}}-\frac{9009 a^{10} b^5}{4 x^{4/3}}-\frac{19305 a^8 b^7}{2 x^{2/3}}+\frac{4095}{2} a^4 b^{11} x^{2/3}+\frac{315}{4} a^2 b^{13} x^{4/3}-\frac{5005 a^9 b^6}{x}-\frac{19305 a^7 b^8}{\sqrt [3]{x}}+9009 a^5 b^{10} \sqrt [3]{x}+455 a^3 b^{12} x+5005 a^6 b^9 \log (x)-\frac{45 a^{14} b}{8 x^{8/3}}-\frac{a^{15}}{3 x^3}+9 a b^{14} x^{5/3}+\frac{b^{15} x^2}{2} \]

Antiderivative was successfully verified.

[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) +
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]

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Maple [A]  time = 0.011, size = 165, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{3\,{x}^{3}}}-{\frac{45\,{a}^{14}b}{8}{x}^{-{\frac{8}{3}}}}-45\,{\frac{{a}^{13}{b}^{2}}{{x}^{7/3}}}-{\frac{455\,{a}^{12}{b}^{3}}{2\,{x}^{2}}}-819\,{\frac{{a}^{11}{b}^{4}}{{x}^{5/3}}}-{\frac{9009\,{a}^{10}{b}^{5}}{4}{x}^{-{\frac{4}{3}}}}-5005\,{\frac{{a}^{9}{b}^{6}}{x}}-{\frac{19305\,{a}^{8}{b}^{7}}{2}{x}^{-{\frac{2}{3}}}}-19305\,{\frac{{a}^{7}{b}^{8}}{\sqrt [3]{x}}}+9009\,{a}^{5}{b}^{10}\sqrt [3]{x}+{\frac{4095\,{a}^{4}{b}^{11}}{2}{x}^{{\frac{2}{3}}}}+455\,{a}^{3}{b}^{12}x+{\frac{315\,{a}^{2}{b}^{13}}{4}{x}^{{\frac{4}{3}}}}+9\,a{b}^{14}{x}^{5/3}+{\frac{{b}^{15}{x}^{2}}{2}}+5005\,{a}^{6}{b}^{9}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 0.966674, size = 223, normalized size = 1.12 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.51091, size = 436, normalized size = 2.18 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 6.58083, size = 202, normalized size = 1.01 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.58748, size = 224, normalized size = 1.12 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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