3.2313 \(\int (a+b \sqrt [3]{x})^5 x^3 \, dx\)

Optimal. Leaf size=75 \[ \frac {a^5 x^4}{4}+\frac {15}{13} a^4 b x^{13/3}+\frac {15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3} \]

[Out]

1/4*a^5*x^4+15/13*a^4*b*x^(13/3)+15/7*a^3*b^2*x^(14/3)+2*a^2*b^3*x^5+15/16*a*b^4*x^(16/3)+3/17*b^5*x^(17/3)

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Rubi [A]  time = 0.05, antiderivative size = 75, 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} \[ 2 a^2 b^3 x^5+\frac {15}{7} a^3 b^2 x^{14/3}+\frac {15}{13} a^4 b x^{13/3}+\frac {a^5 x^4}{4}+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5*x^3,x]

[Out]

(a^5*x^4)/4 + (15*a^4*b*x^(13/3))/13 + (15*a^3*b^2*x^(14/3))/7 + 2*a^2*b^3*x^5 + (15*a*b^4*x^(16/3))/16 + (3*b
^5*x^(17/3))/17

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 \left (a+b \sqrt [3]{x}\right )^5 x^3 \, dx &=3 \operatorname {Subst}\left (\int x^{11} (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (a^5 x^{11}+5 a^4 b x^{12}+10 a^3 b^2 x^{13}+10 a^2 b^3 x^{14}+5 a b^4 x^{15}+b^5 x^{16}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^5 x^4}{4}+\frac {15}{13} a^4 b x^{13/3}+\frac {15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 75, normalized size = 1.00 \[ \frac {a^5 x^4}{4}+\frac {15}{13} a^4 b x^{13/3}+\frac {15}{7} a^3 b^2 x^{14/3}+2 a^2 b^3 x^5+\frac {15}{16} a b^4 x^{16/3}+\frac {3}{17} b^5 x^{17/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5*x^3,x]

[Out]

(a^5*x^4)/4 + (15*a^4*b*x^(13/3))/13 + (15*a^3*b^2*x^(14/3))/7 + 2*a^2*b^3*x^5 + (15*a*b^4*x^(16/3))/16 + (3*b
^5*x^(17/3))/17

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fricas [A]  time = 0.56, size = 69, normalized size = 0.92 \[ 2 \, a^{2} b^{3} x^{5} + \frac {1}{4} \, a^{5} x^{4} + \frac {3}{119} \, {\left (7 \, b^{5} x^{5} + 85 \, a^{3} b^{2} x^{4}\right )} x^{\frac {2}{3}} + \frac {15}{208} \, {\left (13 \, a b^{4} x^{5} + 16 \, a^{4} b x^{4}\right )} x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^2*b^3*x^5 + 1/4*a^5*x^4 + 3/119*(7*b^5*x^5 + 85*a^3*b^2*x^4)*x^(2/3) + 15/208*(13*a*b^4*x^5 + 16*a^4*b*x^4
)*x^(1/3)

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giac [A]  time = 0.17, size = 57, normalized size = 0.76 \[ \frac {3}{17} \, b^{5} x^{\frac {17}{3}} + \frac {15}{16} \, a b^{4} x^{\frac {16}{3}} + 2 \, a^{2} b^{3} x^{5} + \frac {15}{7} \, a^{3} b^{2} x^{\frac {14}{3}} + \frac {15}{13} \, a^{4} b x^{\frac {13}{3}} + \frac {1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/17*b^5*x^(17/3) + 15/16*a*b^4*x^(16/3) + 2*a^2*b^3*x^5 + 15/7*a^3*b^2*x^(14/3) + 15/13*a^4*b*x^(13/3) + 1/4*
a^5*x^4

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maple [A]  time = 0.00, size = 58, normalized size = 0.77 \[ \frac {3 b^{5} x^{\frac {17}{3}}}{17}+\frac {15 a \,b^{4} x^{\frac {16}{3}}}{16}+2 a^{2} b^{3} x^{5}+\frac {15 a^{3} b^{2} x^{\frac {14}{3}}}{7}+\frac {15 a^{4} b \,x^{\frac {13}{3}}}{13}+\frac {a^{5} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^5*x^4+15/13*a^4*b*x^(13/3)+15/7*a^3*b^2*x^(14/3)+2*a^2*b^3*x^5+15/16*a*b^4*x^(16/3)+3/17*b^5*x^(17/3)

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maxima [B]  time = 0.89, size = 200, normalized size = 2.67 \[ \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17}}{17 \, b^{12}} - \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a}{16 \, b^{12}} + \frac {11 \, {\left (b x^{\frac {1}{3}} + a\right )}^{15} a^{2}}{b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{14} a^{3}}{14 \, b^{12}} + \frac {990 \, {\left (b x^{\frac {1}{3}} + a\right )}^{13} a^{4}}{13 \, b^{12}} - \frac {231 \, {\left (b x^{\frac {1}{3}} + a\right )}^{12} a^{5}}{2 \, b^{12}} + \frac {126 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11} a^{6}}{b^{12}} - \frac {99 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} a^{7}}{b^{12}} + \frac {55 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a^{8}}{b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{9}}{8 \, b^{12}} + \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{10}}{7 \, b^{12}} - \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{11}}{2 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/17*(b*x^(1/3) + a)^17/b^12 - 33/16*(b*x^(1/3) + a)^16*a/b^12 + 11*(b*x^(1/3) + a)^15*a^2/b^12 - 495/14*(b*x^
(1/3) + a)^14*a^3/b^12 + 990/13*(b*x^(1/3) + a)^13*a^4/b^12 - 231/2*(b*x^(1/3) + a)^12*a^5/b^12 + 126*(b*x^(1/
3) + a)^11*a^6/b^12 - 99*(b*x^(1/3) + a)^10*a^7/b^12 + 55*(b*x^(1/3) + a)^9*a^8/b^12 - 165/8*(b*x^(1/3) + a)^8
*a^9/b^12 + 33/7*(b*x^(1/3) + a)^7*a^10/b^12 - 1/2*(b*x^(1/3) + a)^6*a^11/b^12

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mupad [B]  time = 0.03, size = 57, normalized size = 0.76 \[ \frac {a^5\,x^4}{4}+\frac {3\,b^5\,x^{17/3}}{17}+\frac {15\,a^4\,b\,x^{13/3}}{13}+\frac {15\,a\,b^4\,x^{16/3}}{16}+2\,a^2\,b^3\,x^5+\frac {15\,a^3\,b^2\,x^{14/3}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^(1/3))^5,x)

[Out]

(a^5*x^4)/4 + (3*b^5*x^(17/3))/17 + (15*a^4*b*x^(13/3))/13 + (15*a*b^4*x^(16/3))/16 + 2*a^2*b^3*x^5 + (15*a^3*
b^2*x^(14/3))/7

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sympy [A]  time = 2.61, size = 73, normalized size = 0.97 \[ \frac {a^{5} x^{4}}{4} + \frac {15 a^{4} b x^{\frac {13}{3}}}{13} + \frac {15 a^{3} b^{2} x^{\frac {14}{3}}}{7} + 2 a^{2} b^{3} x^{5} + \frac {15 a b^{4} x^{\frac {16}{3}}}{16} + \frac {3 b^{5} x^{\frac {17}{3}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x**4/4 + 15*a**4*b*x**(13/3)/13 + 15*a**3*b**2*x**(14/3)/7 + 2*a**2*b**3*x**5 + 15*a*b**4*x**(16/3)/16 +
3*b**5*x**(17/3)/17

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