3.1.83 \(\int (a+b x)^5 \, dx\) [83]

Optimal. Leaf size=14 \[ \frac {(a+b x)^6}{6 b} \]

[Out]

1/6*(b*x+a)^6/b

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \begin {gather*} \frac {(a+b x)^6}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5,x]

[Out]

(a + b*x)^6/(6*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int (a+b x)^5 \, dx &=\frac {(a+b x)^6}{6 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {(a+b x)^6}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5,x]

[Out]

(a + b*x)^6/(6*b)

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(14)=28\).
time = 1.89, size = 54, normalized size = 3.86 \begin {gather*} \frac {x \left (6 a^5+15 a^4 b x+20 a^3 b^2 x^2+15 a^2 b^3 x^3+6 a b^4 x^4+b^5 x^5\right )}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^0*(a + b*x)^5,x]')

[Out]

x (6 a ^ 5 + 15 a ^ 4 b x + 20 a ^ 3 b ^ 2 x ^ 2 + 15 a ^ 2 b ^ 3 x ^ 3 + 6 a b ^ 4 x ^ 4 + b ^ 5 x ^ 5) / 6

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 13, normalized size = 0.93

method	result	size
default	\(\frac {\left (b x +a \right )^{6}}{6 b}\)	\(13\)
gosper	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x\)	\(54\)
norman	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x\)	\(54\)
risch	\(\frac {b^{5} x^{6}}{6}+a \,b^{4} x^{5}+\frac {5 a^{2} b^{3} x^{4}}{2}+\frac {10 a^{3} b^{2} x^{3}}{3}+\frac {5 a^{4} b \,x^{2}}{2}+a^{5} x +\frac {a^{6}}{6 b}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

1/6*(b*x+a)^6/b

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).
time = 0.25, size = 53, normalized size = 3.79 \begin {gather*} \frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^5*x^6 + a*b^4*x^5 + 5/2*a^2*b^3*x^4 + 10/3*a^3*b^2*x^3 + 5/2*a^4*b*x^2 + a^5*x

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).
time = 0.30, size = 53, normalized size = 3.79 \begin {gather*} \frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^5*x^6 + a*b^4*x^5 + 5/2*a^2*b^3*x^4 + 10/3*a^3*b^2*x^3 + 5/2*a^4*b*x^2 + a^5*x

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (8) = 16\).
time = 0.03, size = 60, normalized size = 4.29 \begin {gather*} a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5,x)

[Out]

a**5*x + 5*a**4*b*x**2/2 + 10*a**3*b**2*x**3/3 + 5*a**2*b**3*x**4/2 + a*b**4*x**5 + b**5*x**6/6

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {\left (b x+a\right )^{6}}{6 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5,x)

[Out]

1/6*(b*x + a)^6/b

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 53, normalized size = 3.79 \begin {gather*} a^5\,x+\frac {5\,a^4\,b\,x^2}{2}+\frac {10\,a^3\,b^2\,x^3}{3}+\frac {5\,a^2\,b^3\,x^4}{2}+a\,b^4\,x^5+\frac {b^5\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5,x)

[Out]

a^5*x + (b^5*x^6)/6 + (5*a^4*b*x^2)/2 + a*b^4*x^5 + (10*a^3*b^2*x^3)/3 + (5*a^2*b^3*x^4)/2

________________________________________________________________________________________