3.2346 \(\int \frac{(a+b \sqrt [3]{x})^{15}}{x^2} \, dx\)

Optimal. Leaf size=202 \[ \frac{9009}{2} a^{10} b^5 x^{2/3}+\frac{19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac{5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac{4095}{8} a^4 b^{11} x^{8/3}+\frac{455}{3} a^3 b^{12} x^3+\frac{63}{2} a^2 b^{13} x^{10/3}-\frac{315 a^{13} b^2}{\sqrt [3]{x}}+4095 a^{11} b^4 \sqrt [3]{x}+5005 a^9 b^6 x+455 a^{12} b^3 \log (x)-\frac{45 a^{14} b}{2 x^{2/3}}-\frac{a^{15}}{x}+\frac{45}{11} a b^{14} x^{11/3}+\frac{b^{15} x^4}{4} \]

[Out]

-(a^15/x) - (45*a^14*b)/(2*x^(2/3)) - (315*a^13*b^2)/x^(1/3) + 4095*a^11*b^4*x^(1/3) + (9009*a^10*b^5*x^(2/3))
/2 + 5005*a^9*b^6*x + (19305*a^8*b^7*x^(4/3))/4 + 3861*a^7*b^8*x^(5/3) + (5005*a^6*b^9*x^2)/2 + 1287*a^5*b^10*
x^(7/3) + (4095*a^4*b^11*x^(8/3))/8 + (455*a^3*b^12*x^3)/3 + (63*a^2*b^13*x^(10/3))/2 + (45*a*b^14*x^(11/3))/1
1 + (b^15*x^4)/4 + 455*a^12*b^3*Log[x]

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Rubi [A]  time = 0.118023, antiderivative size = 202, 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} \[ \frac{9009}{2} a^{10} b^5 x^{2/3}+\frac{19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac{5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac{4095}{8} a^4 b^{11} x^{8/3}+\frac{455}{3} a^3 b^{12} x^3+\frac{63}{2} a^2 b^{13} x^{10/3}-\frac{315 a^{13} b^2}{\sqrt [3]{x}}+4095 a^{11} b^4 \sqrt [3]{x}+5005 a^9 b^6 x+455 a^{12} b^3 \log (x)-\frac{45 a^{14} b}{2 x^{2/3}}-\frac{a^{15}}{x}+\frac{45}{11} a b^{14} x^{11/3}+\frac{b^{15} x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^15/x^2,x]

[Out]

-(a^15/x) - (45*a^14*b)/(2*x^(2/3)) - (315*a^13*b^2)/x^(1/3) + 4095*a^11*b^4*x^(1/3) + (9009*a^10*b^5*x^(2/3))
/2 + 5005*a^9*b^6*x + (19305*a^8*b^7*x^(4/3))/4 + 3861*a^7*b^8*x^(5/3) + (5005*a^6*b^9*x^2)/2 + 1287*a^5*b^10*
x^(7/3) + (4095*a^4*b^11*x^(8/3))/8 + (455*a^3*b^12*x^3)/3 + (63*a^2*b^13*x^(10/3))/2 + (45*a*b^14*x^(11/3))/1
1 + (b^15*x^4)/4 + 455*a^12*b^3*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^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (1365 a^{11} b^4+\frac{a^{15}}{x^4}+\frac{15 a^{14} b}{x^3}+\frac{105 a^{13} b^2}{x^2}+\frac{455 a^{12} b^3}{x}+3003 a^{10} b^5 x+5005 a^9 b^6 x^2+6435 a^8 b^7 x^3+6435 a^7 b^8 x^4+5005 a^6 b^9 x^5+3003 a^5 b^{10} x^6+1365 a^4 b^{11} x^7+455 a^3 b^{12} x^8+105 a^2 b^{13} x^9+15 a b^{14} x^{10}+b^{15} x^{11}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{15}}{x}-\frac{45 a^{14} b}{2 x^{2/3}}-\frac{315 a^{13} b^2}{\sqrt [3]{x}}+4095 a^{11} b^4 \sqrt [3]{x}+\frac{9009}{2} a^{10} b^5 x^{2/3}+5005 a^9 b^6 x+\frac{19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac{5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac{4095}{8} a^4 b^{11} x^{8/3}+\frac{455}{3} a^3 b^{12} x^3+\frac{63}{2} a^2 b^{13} x^{10/3}+\frac{45}{11} a b^{14} x^{11/3}+\frac{b^{15} x^4}{4}+455 a^{12} b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0991756, size = 202, normalized size = 1. \[ \frac{9009}{2} a^{10} b^5 x^{2/3}+\frac{19305}{4} a^8 b^7 x^{4/3}+3861 a^7 b^8 x^{5/3}+\frac{5005}{2} a^6 b^9 x^2+1287 a^5 b^{10} x^{7/3}+\frac{4095}{8} a^4 b^{11} x^{8/3}+\frac{455}{3} a^3 b^{12} x^3+\frac{63}{2} a^2 b^{13} x^{10/3}-\frac{315 a^{13} b^2}{\sqrt [3]{x}}+4095 a^{11} b^4 \sqrt [3]{x}+5005 a^9 b^6 x+455 a^{12} b^3 \log (x)-\frac{45 a^{14} b}{2 x^{2/3}}-\frac{a^{15}}{x}+\frac{45}{11} a b^{14} x^{11/3}+\frac{b^{15} x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^15/x^2,x]

[Out]

-(a^15/x) - (45*a^14*b)/(2*x^(2/3)) - (315*a^13*b^2)/x^(1/3) + 4095*a^11*b^4*x^(1/3) + (9009*a^10*b^5*x^(2/3))
/2 + 5005*a^9*b^6*x + (19305*a^8*b^7*x^(4/3))/4 + 3861*a^7*b^8*x^(5/3) + (5005*a^6*b^9*x^2)/2 + 1287*a^5*b^10*
x^(7/3) + (4095*a^4*b^11*x^(8/3))/8 + (455*a^3*b^12*x^3)/3 + (63*a^2*b^13*x^(10/3))/2 + (45*a*b^14*x^(11/3))/1
1 + (b^15*x^4)/4 + 455*a^12*b^3*Log[x]

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Maple [A]  time = 0.006, size = 165, normalized size = 0.8 \begin{align*} -{\frac{{a}^{15}}{x}}-{\frac{45\,{a}^{14}b}{2}{x}^{-{\frac{2}{3}}}}-315\,{\frac{{a}^{13}{b}^{2}}{\sqrt [3]{x}}}+4095\,{a}^{11}{b}^{4}\sqrt [3]{x}+{\frac{9009\,{a}^{10}{b}^{5}}{2}{x}^{{\frac{2}{3}}}}+5005\,{a}^{9}{b}^{6}x+{\frac{19305\,{a}^{8}{b}^{7}}{4}{x}^{{\frac{4}{3}}}}+3861\,{a}^{7}{b}^{8}{x}^{5/3}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{2}}{2}}+1287\,{a}^{5}{b}^{10}{x}^{7/3}+{\frac{4095\,{a}^{4}{b}^{11}}{8}{x}^{{\frac{8}{3}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{3}}{3}}+{\frac{63\,{a}^{2}{b}^{13}}{2}{x}^{{\frac{10}{3}}}}+{\frac{45\,a{b}^{14}}{11}{x}^{{\frac{11}{3}}}}+{\frac{{b}^{15}{x}^{4}}{4}}+455\,{a}^{12}{b}^{3}\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^2,x)

[Out]

-a^15/x-45/2*a^14*b/x^(2/3)-315*a^13*b^2/x^(1/3)+4095*a^11*b^4*x^(1/3)+9009/2*a^10*b^5*x^(2/3)+5005*a^9*b^6*x+
19305/4*a^8*b^7*x^(4/3)+3861*a^7*b^8*x^(5/3)+5005/2*a^6*b^9*x^2+1287*a^5*b^10*x^(7/3)+4095/8*a^4*b^11*x^(8/3)+
455/3*a^3*b^12*x^3+63/2*a^2*b^13*x^(10/3)+45/11*a*b^14*x^(11/3)+1/4*b^15*x^4+455*a^12*b^3*ln(x)

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Maxima [A]  time = 1.06476, size = 225, normalized size = 1.11 \begin{align*} \frac{1}{4} \, b^{15} x^{4} + \frac{45}{11} \, a b^{14} x^{\frac{11}{3}} + \frac{63}{2} \, a^{2} b^{13} x^{\frac{10}{3}} + \frac{455}{3} \, a^{3} b^{12} x^{3} + \frac{4095}{8} \, a^{4} b^{11} x^{\frac{8}{3}} + 1287 \, a^{5} b^{10} x^{\frac{7}{3}} + \frac{5005}{2} \, a^{6} b^{9} x^{2} + 3861 \, a^{7} b^{8} x^{\frac{5}{3}} + \frac{19305}{4} \, a^{8} b^{7} x^{\frac{4}{3}} + 5005 \, a^{9} b^{6} x + 455 \, a^{12} b^{3} \log \left (x\right ) + \frac{9009}{2} \, a^{10} b^{5} x^{\frac{2}{3}} + 4095 \, a^{11} b^{4} x^{\frac{1}{3}} - \frac{630 \, a^{13} b^{2} x^{\frac{2}{3}} + 45 \, a^{14} b x^{\frac{1}{3}} + 2 \, a^{15}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^2,x, algorithm="maxima")

[Out]

1/4*b^15*x^4 + 45/11*a*b^14*x^(11/3) + 63/2*a^2*b^13*x^(10/3) + 455/3*a^3*b^12*x^3 + 4095/8*a^4*b^11*x^(8/3) +
1287*a^5*b^10*x^(7/3) + 5005/2*a^6*b^9*x^2 + 3861*a^7*b^8*x^(5/3) + 19305/4*a^8*b^7*x^(4/3) + 5005*a^9*b^6*x
+ 455*a^12*b^3*log(x) + 9009/2*a^10*b^5*x^(2/3) + 4095*a^11*b^4*x^(1/3) - 1/2*(630*a^13*b^2*x^(2/3) + 45*a^14*
b*x^(1/3) + 2*a^15)/x

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Fricas [A]  time = 1.49757, size = 452, normalized size = 2.24 \begin{align*} \frac{66 \, b^{15} x^{5} + 40040 \, a^{3} b^{12} x^{4} + 660660 \, a^{6} b^{9} x^{3} + 1321320 \, a^{9} b^{6} x^{2} + 360360 \, a^{12} b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 264 \, a^{15} + 27 \,{\left (40 \, a b^{14} x^{4} + 5005 \, a^{4} b^{11} x^{3} + 37752 \, a^{7} b^{8} x^{2} + 44044 \, a^{10} b^{5} x - 3080 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 594 \,{\left (14 \, a^{2} b^{13} x^{4} + 572 \, a^{5} b^{10} x^{3} + 2145 \, a^{8} b^{7} x^{2} + 1820 \, a^{11} b^{4} x - 10 \, a^{14} b\right )} x^{\frac{1}{3}}}{264 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^2,x, algorithm="fricas")

[Out]

1/264*(66*b^15*x^5 + 40040*a^3*b^12*x^4 + 660660*a^6*b^9*x^3 + 1321320*a^9*b^6*x^2 + 360360*a^12*b^3*x*log(x^(
1/3)) - 264*a^15 + 27*(40*a*b^14*x^4 + 5005*a^4*b^11*x^3 + 37752*a^7*b^8*x^2 + 44044*a^10*b^5*x - 3080*a^13*b^
2)*x^(2/3) + 594*(14*a^2*b^13*x^4 + 572*a^5*b^10*x^3 + 2145*a^8*b^7*x^2 + 1820*a^11*b^4*x - 10*a^14*b)*x^(1/3)
)/x

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Sympy [A]  time = 6.95619, size = 204, normalized size = 1.01 \begin{align*} - \frac{a^{15}}{x} - \frac{45 a^{14} b}{2 x^{\frac{2}{3}}} - \frac{315 a^{13} b^{2}}{\sqrt [3]{x}} + 455 a^{12} b^{3} \log{\left (x \right )} + 4095 a^{11} b^{4} \sqrt [3]{x} + \frac{9009 a^{10} b^{5} x^{\frac{2}{3}}}{2} + 5005 a^{9} b^{6} x + \frac{19305 a^{8} b^{7} x^{\frac{4}{3}}}{4} + 3861 a^{7} b^{8} x^{\frac{5}{3}} + \frac{5005 a^{6} b^{9} x^{2}}{2} + 1287 a^{5} b^{10} x^{\frac{7}{3}} + \frac{4095 a^{4} b^{11} x^{\frac{8}{3}}}{8} + \frac{455 a^{3} b^{12} x^{3}}{3} + \frac{63 a^{2} b^{13} x^{\frac{10}{3}}}{2} + \frac{45 a b^{14} x^{\frac{11}{3}}}{11} + \frac{b^{15} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**15/x**2,x)

[Out]

-a**15/x - 45*a**14*b/(2*x**(2/3)) - 315*a**13*b**2/x**(1/3) + 455*a**12*b**3*log(x) + 4095*a**11*b**4*x**(1/3
) + 9009*a**10*b**5*x**(2/3)/2 + 5005*a**9*b**6*x + 19305*a**8*b**7*x**(4/3)/4 + 3861*a**7*b**8*x**(5/3) + 500
5*a**6*b**9*x**2/2 + 1287*a**5*b**10*x**(7/3) + 4095*a**4*b**11*x**(8/3)/8 + 455*a**3*b**12*x**3/3 + 63*a**2*b
**13*x**(10/3)/2 + 45*a*b**14*x**(11/3)/11 + b**15*x**4/4

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Giac [A]  time = 1.16957, size = 227, normalized size = 1.12 \begin{align*} \frac{1}{4} \, b^{15} x^{4} + \frac{45}{11} \, a b^{14} x^{\frac{11}{3}} + \frac{63}{2} \, a^{2} b^{13} x^{\frac{10}{3}} + \frac{455}{3} \, a^{3} b^{12} x^{3} + \frac{4095}{8} \, a^{4} b^{11} x^{\frac{8}{3}} + 1287 \, a^{5} b^{10} x^{\frac{7}{3}} + \frac{5005}{2} \, a^{6} b^{9} x^{2} + 3861 \, a^{7} b^{8} x^{\frac{5}{3}} + \frac{19305}{4} \, a^{8} b^{7} x^{\frac{4}{3}} + 5005 \, a^{9} b^{6} x + 455 \, a^{12} b^{3} \log \left ({\left | x \right |}\right ) + \frac{9009}{2} \, a^{10} b^{5} x^{\frac{2}{3}} + 4095 \, a^{11} b^{4} x^{\frac{1}{3}} - \frac{630 \, a^{13} b^{2} x^{\frac{2}{3}} + 45 \, a^{14} b x^{\frac{1}{3}} + 2 \, a^{15}}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^15/x^2,x, algorithm="giac")

[Out]

1/4*b^15*x^4 + 45/11*a*b^14*x^(11/3) + 63/2*a^2*b^13*x^(10/3) + 455/3*a^3*b^12*x^3 + 4095/8*a^4*b^11*x^(8/3) +
1287*a^5*b^10*x^(7/3) + 5005/2*a^6*b^9*x^2 + 3861*a^7*b^8*x^(5/3) + 19305/4*a^8*b^7*x^(4/3) + 5005*a^9*b^6*x
+ 455*a^12*b^3*log(abs(x)) + 9009/2*a^10*b^5*x^(2/3) + 4095*a^11*b^4*x^(1/3) - 1/2*(630*a^13*b^2*x^(2/3) + 45*
a^14*b*x^(1/3) + 2*a^15)/x