3.2316 \(\int (a+b \sqrt [3]{x})^5 \, dx\)

Optimal. Leaf size=59 \[ \frac {a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]

[Out]

1/2*a^2*(a+b*x^(1/3))^6/b^3-6/7*a*(a+b*x^(1/3))^7/b^3+3/8*(a+b*x^(1/3))^8/b^3

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5,x]

[Out]

(a^2*(a + b*x^(1/3))^6)/(2*b^3) - (6*a*(a + b*x^(1/3))^7)/(7*b^3) + (3*(a + b*x^(1/3))^8)/(8*b^3)

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 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps

\begin {align*} \int \left (a+b \sqrt [3]{x}\right )^5 \, dx &=3 \operatorname {Subst}\left (\int x^2 (a+b x)^5 \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^5}{b^2}-\frac {2 a (a+b x)^6}{b^2}+\frac {(a+b x)^7}{b^2}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {a^2 \left (a+b \sqrt [3]{x}\right )^6}{2 b^3}-\frac {6 a \left (a+b \sqrt [3]{x}\right )^7}{7 b^3}+\frac {3 \left (a+b \sqrt [3]{x}\right )^8}{8 b^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 41, normalized size = 0.69 \[ \frac {\left (a+b \sqrt [3]{x}\right )^6 \left (a^2-6 a b \sqrt [3]{x}+21 b^2 x^{2/3}\right )}{56 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5,x]

[Out]

((a + b*x^(1/3))^6*(a^2 - 6*a*b*x^(1/3) + 21*b^2*x^(2/3)))/(56*b^3)

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 61, normalized size = 1.03 \[ 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac {3}{8} \, {\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac {2}{3}} + \frac {15}{28} \, {\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a^2*b^3*x^2 + a^5*x + 3/8*(b^5*x^2 + 16*a^3*b^2*x)*x^(2/3) + 15/28*(4*a*b^4*x^2 + 7*a^4*b*x)*x^(1/3)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 54, normalized size = 0.92 \[ \frac {3}{8} \, b^{5} x^{\frac {8}{3}} + \frac {15}{7} \, a b^{4} x^{\frac {7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac {5}{3}} + \frac {15}{4} \, a^{4} b x^{\frac {4}{3}} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*b^5*x^(8/3) + 15/7*a*b^4*x^(7/3) + 5*a^2*b^3*x^2 + 6*a^3*b^2*x^(5/3) + 15/4*a^4*b*x^(4/3) + a^5*x

________________________________________________________________________________________

maple [A]  time = 0.00, size = 55, normalized size = 0.93 \[ \frac {3 b^{5} x^{\frac {8}{3}}}{8}+\frac {15 a \,b^{4} x^{\frac {7}{3}}}{7}+5 a^{2} b^{3} x^{2}+6 a^{3} b^{2} x^{\frac {5}{3}}+\frac {15 a^{4} b \,x^{\frac {4}{3}}}{4}+a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^5*x+3/8*b^5*x^(8/3)+15/7*a*b^4*x^(7/3)+5*a^2*b^3*x^2+6*a^3*b^2*x^(5/3)+15/4*a^4*b*x^(4/3)

________________________________________________________________________________________

maxima [A]  time = 0.88, size = 54, normalized size = 0.92 \[ \frac {3}{8} \, b^{5} x^{\frac {8}{3}} + \frac {15}{7} \, a b^{4} x^{\frac {7}{3}} + 5 \, a^{2} b^{3} x^{2} + 6 \, a^{3} b^{2} x^{\frac {5}{3}} + \frac {15}{4} \, a^{4} b x^{\frac {4}{3}} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*b^5*x^(8/3) + 15/7*a*b^4*x^(7/3) + 5*a^2*b^3*x^2 + 6*a^3*b^2*x^(5/3) + 15/4*a^4*b*x^(4/3) + a^5*x

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 54, normalized size = 0.92 \[ a^5\,x+\frac {3\,b^5\,x^{8/3}}{8}+\frac {15\,a^4\,b\,x^{4/3}}{4}+\frac {15\,a\,b^4\,x^{7/3}}{7}+5\,a^2\,b^3\,x^2+6\,a^3\,b^2\,x^{5/3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^5,x)

[Out]

a^5*x + (3*b^5*x^(8/3))/8 + (15*a^4*b*x^(4/3))/4 + (15*a*b^4*x^(7/3))/7 + 5*a^2*b^3*x^2 + 6*a^3*b^2*x^(5/3)

________________________________________________________________________________________

sympy [A]  time = 2.08, size = 68, normalized size = 1.15 \[ a^{5} x + \frac {15 a^{4} b x^{\frac {4}{3}}}{4} + 6 a^{3} b^{2} x^{\frac {5}{3}} + 5 a^{2} b^{3} x^{2} + \frac {15 a b^{4} x^{\frac {7}{3}}}{7} + \frac {3 b^{5} x^{\frac {8}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**5*x + 15*a**4*b*x**(4/3)/4 + 6*a**3*b**2*x**(5/3) + 5*a**2*b**3*x**2 + 15*a*b**4*x**(7/3)/7 + 3*b**5*x**(8/
3)/8

________________________________________________________________________________________