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

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

[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]

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Rubi [A]  time = 0.115621, 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} \[ -\frac{315 a^{13} b^2}{4 x^{4/3}}-\frac{4095 a^{11} b^4}{2 x^{2/3}}+\frac{19305}{2} a^7 b^8 x^{2/3}+\frac{9009}{4} a^5 b^{10} x^{4/3}+819 a^4 b^{11} x^{5/3}+\frac{455}{2} a^3 b^{12} x^2+45 a^2 b^{13} x^{7/3}-\frac{455 a^{12} b^3}{x}-\frac{9009 a^{10} b^5}{\sqrt [3]{x}}+19305 a^8 b^7 \sqrt [3]{x}+5005 a^6 b^9 x+5005 a^9 b^6 \log (x)-\frac{9 a^{14} b}{x^{5/3}}-\frac{a^{15}}{2 x^2}+\frac{45}{8} a b^{14} x^{8/3}+\frac{b^{15} x^3}{3} \]

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 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^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*}

Mathematica [A]  time = 0.102595, size = 200, normalized size = 1. \[ -\frac{315 a^{13} b^2}{4 x^{4/3}}-\frac{4095 a^{11} b^4}{2 x^{2/3}}+\frac{19305}{2} a^7 b^8 x^{2/3}+\frac{9009}{4} a^5 b^{10} x^{4/3}+819 a^4 b^{11} x^{5/3}+\frac{455}{2} a^3 b^{12} x^2+45 a^2 b^{13} x^{7/3}-\frac{455 a^{12} b^3}{x}-\frac{9009 a^{10} b^5}{\sqrt [3]{x}}+19305 a^8 b^7 \sqrt [3]{x}+5005 a^6 b^9 x+5005 a^9 b^6 \log (x)-\frac{9 a^{14} b}{x^{5/3}}-\frac{a^{15}}{2 x^2}+\frac{45}{8} a b^{14} x^{8/3}+\frac{b^{15} x^3}{3} \]

Antiderivative was successfully verified.

[In]

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) + 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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Maple [A]  time = 0.008, size = 165, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{2\,{x}^{2}}}-9\,{\frac{{a}^{14}b}{{x}^{5/3}}}-{\frac{315\,{a}^{13}{b}^{2}}{4}{x}^{-{\frac{4}{3}}}}-455\,{\frac{{a}^{12}{b}^{3}}{x}}-{\frac{4095\,{a}^{11}{b}^{4}}{2}{x}^{-{\frac{2}{3}}}}-9009\,{\frac{{a}^{10}{b}^{5}}{\sqrt [3]{x}}}+19305\,{a}^{8}{b}^{7}\sqrt [3]{x}+{\frac{19305\,{a}^{7}{b}^{8}}{2}{x}^{{\frac{2}{3}}}}+5005\,{a}^{6}{b}^{9}x+{\frac{9009\,{a}^{5}{b}^{10}}{4}{x}^{{\frac{4}{3}}}}+819\,{a}^{4}{b}^{11}{x}^{5/3}+{\frac{455\,{a}^{3}{b}^{12}{x}^{2}}{2}}+45\,{a}^{2}{b}^{13}{x}^{7/3}+{\frac{45\,a{b}^{14}}{8}{x}^{{\frac{8}{3}}}}+{\frac{{b}^{15}{x}^{3}}{3}}+5005\,{a}^{9}{b}^{6}\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^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 = 0.955111, size = 223, normalized size = 1.12 \begin{align*} \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 (x\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}} \end{align*}

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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Fricas [A]  time = 1.48218, size = 436, normalized size = 2.18 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 7.32309, size = 202, normalized size = 1.01 \begin{align*} - \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{\left (x \right )} + 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} \end{align*}

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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Giac [A]  time = 1.18066, size = 224, normalized size = 1.12 \begin{align*} \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}} \end{align*}

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