3.2312 \(\int (a+b \sqrt [3]{x})^5 x^4 \, dx\)

Optimal. Leaf size=77 \[ \frac{5}{3} a^2 b^3 x^6+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{15}{16} a^4 b x^{16/3}+\frac{a^5 x^5}{5}+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3} \]

[Out]

(a^5*x^5)/5 + (15*a^4*b*x^(16/3))/16 + (30*a^3*b^2*x^(17/3))/17 + (5*a^2*b^3*x^6)/3 + (15*a*b^4*x^(19/3))/19 +
(3*b^5*x^(20/3))/20

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Rubi [A]  time = 0.0506198, antiderivative size = 77, 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} \[ \frac{5}{3} a^2 b^3 x^6+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{15}{16} a^4 b x^{16/3}+\frac{a^5 x^5}{5}+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5*x^4,x]

[Out]

(a^5*x^5)/5 + (15*a^4*b*x^(16/3))/16 + (30*a^3*b^2*x^(17/3))/17 + (5*a^2*b^3*x^6)/3 + (15*a*b^4*x^(19/3))/19 +
(3*b^5*x^(20/3))/20

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 \left (a+b \sqrt [3]{x}\right )^5 x^4 \, dx &=3 \operatorname{Subst}\left (\int x^{14} (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^5 x^{14}+5 a^4 b x^{15}+10 a^3 b^2 x^{16}+10 a^2 b^3 x^{17}+5 a b^4 x^{18}+b^5 x^{19}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^5 x^5}{5}+\frac{15}{16} a^4 b x^{16/3}+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{5}{3} a^2 b^3 x^6+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3}\\ \end{align*}

Mathematica [A]  time = 0.0377595, size = 77, normalized size = 1. \[ \frac{5}{3} a^2 b^3 x^6+\frac{30}{17} a^3 b^2 x^{17/3}+\frac{15}{16} a^4 b x^{16/3}+\frac{a^5 x^5}{5}+\frac{15}{19} a b^4 x^{19/3}+\frac{3}{20} b^5 x^{20/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5*x^4,x]

[Out]

(a^5*x^5)/5 + (15*a^4*b*x^(16/3))/16 + (30*a^3*b^2*x^(17/3))/17 + (5*a^2*b^3*x^6)/3 + (15*a*b^4*x^(19/3))/19 +
(3*b^5*x^(20/3))/20

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Maple [A]  time = 0.001, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{15\,{a}^{4}b}{16}{x}^{{\frac{16}{3}}}}+{\frac{30\,{a}^{3}{b}^{2}}{17}{x}^{{\frac{17}{3}}}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{6}}{3}}+{\frac{15\,a{b}^{4}}{19}{x}^{{\frac{19}{3}}}}+{\frac{3\,{b}^{5}}{20}{x}^{{\frac{20}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5*x^4,x)

[Out]

1/5*a^5*x^5+15/16*a^4*b*x^(16/3)+30/17*a^3*b^2*x^(17/3)+5/3*a^2*b^3*x^6+15/19*a*b^4*x^(19/3)+3/20*b^5*x^(20/3)

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Maxima [B]  time = 0.963746, size = 339, normalized size = 4.4 \begin{align*} \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20}}{20 \, b^{15}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a}{19 \, b^{15}} + \frac{91 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{2}}{6 \, b^{15}} - \frac{1092 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{3}}{17 \, b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{4}}{16 \, b^{15}} - \frac{2002 \,{\left (b x^{\frac{1}{3}} + a\right )}^{15} a^{5}}{5 \, b^{15}} + \frac{1287 \,{\left (b x^{\frac{1}{3}} + a\right )}^{14} a^{6}}{2 \, b^{15}} - \frac{792 \,{\left (b x^{\frac{1}{3}} + a\right )}^{13} a^{7}}{b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{12} a^{8}}{4 \, b^{15}} - \frac{546 \,{\left (b x^{\frac{1}{3}} + a\right )}^{11} a^{9}}{b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{10} a^{10}}{10 \, b^{15}} - \frac{364 \,{\left (b x^{\frac{1}{3}} + a\right )}^{9} a^{11}}{3 \, b^{15}} + \frac{273 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a^{12}}{8 \, b^{15}} - \frac{6 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{13}}{b^{15}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{14}}{2 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5*x^4,x, algorithm="maxima")

[Out]

3/20*(b*x^(1/3) + a)^20/b^15 - 42/19*(b*x^(1/3) + a)^19*a/b^15 + 91/6*(b*x^(1/3) + a)^18*a^2/b^15 - 1092/17*(b
*x^(1/3) + a)^17*a^3/b^15 + 3003/16*(b*x^(1/3) + a)^16*a^4/b^15 - 2002/5*(b*x^(1/3) + a)^15*a^5/b^15 + 1287/2*
(b*x^(1/3) + a)^14*a^6/b^15 - 792*(b*x^(1/3) + a)^13*a^7/b^15 + 3003/4*(b*x^(1/3) + a)^12*a^8/b^15 - 546*(b*x^
(1/3) + a)^11*a^9/b^15 + 3003/10*(b*x^(1/3) + a)^10*a^10/b^15 - 364/3*(b*x^(1/3) + a)^9*a^11/b^15 + 273/8*(b*x
^(1/3) + a)^8*a^12/b^15 - 6*(b*x^(1/3) + a)^7*a^13/b^15 + 1/2*(b*x^(1/3) + a)^6*a^14/b^15

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Fricas [A]  time = 1.49557, size = 169, normalized size = 2.19 \begin{align*} \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{3}{340} \,{\left (17 \, b^{5} x^{6} + 200 \, a^{3} b^{2} x^{5}\right )} x^{\frac{2}{3}} + \frac{15}{304} \,{\left (16 \, a b^{4} x^{6} + 19 \, a^{4} b x^{5}\right )} x^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5*x^4,x, algorithm="fricas")

[Out]

5/3*a^2*b^3*x^6 + 1/5*a^5*x^5 + 3/340*(17*b^5*x^6 + 200*a^3*b^2*x^5)*x^(2/3) + 15/304*(16*a*b^4*x^6 + 19*a^4*b
*x^5)*x^(1/3)

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Sympy [A]  time = 3.3816, size = 75, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{5}}{5} + \frac{15 a^{4} b x^{\frac{16}{3}}}{16} + \frac{30 a^{3} b^{2} x^{\frac{17}{3}}}{17} + \frac{5 a^{2} b^{3} x^{6}}{3} + \frac{15 a b^{4} x^{\frac{19}{3}}}{19} + \frac{3 b^{5} x^{\frac{20}{3}}}{20} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5*x**4,x)

[Out]

a**5*x**5/5 + 15*a**4*b*x**(16/3)/16 + 30*a**3*b**2*x**(17/3)/17 + 5*a**2*b**3*x**6/3 + 15*a*b**4*x**(19/3)/19
+ 3*b**5*x**(20/3)/20

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Giac [A]  time = 1.16931, size = 77, normalized size = 1. \begin{align*} \frac{3}{20} \, b^{5} x^{\frac{20}{3}} + \frac{15}{19} \, a b^{4} x^{\frac{19}{3}} + \frac{5}{3} \, a^{2} b^{3} x^{6} + \frac{30}{17} \, a^{3} b^{2} x^{\frac{17}{3}} + \frac{15}{16} \, a^{4} b x^{\frac{16}{3}} + \frac{1}{5} \, a^{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5*x^4,x, algorithm="giac")

[Out]

3/20*b^5*x^(20/3) + 15/19*a*b^4*x^(19/3) + 5/3*a^2*b^3*x^6 + 30/17*a^3*b^2*x^(17/3) + 15/16*a^4*b*x^(16/3) + 1
/5*a^5*x^5