\(\int (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5) \, dx\) [65]

Optimal result

Integrand size = 49, antiderivative size = 14 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=\frac {(a+b x)^6}{6 b} \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(14)=28\).

Time = 0.01 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.36, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=a^5 x+\frac {5}{2} a^4 b x^2+\frac {10}{3} a^3 b^2 x^3+\frac {5}{2} a^2 b^3 x^4+a b^4 x^5+\frac {b^5 x^6}{6} \]

[In]

Int[a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^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

Rubi steps \begin{align*} \text {integral}& = a^5 x+\frac {5}{2} a^4 b x^2+\frac {10}{3} a^3 b^2 x^3+\frac {5}{2} a^2 b^3 x^4+a b^4 x^5+\frac {b^5 x^6}{6} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(14)=28\).

Time = 0.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 4.36 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=a^5 x+\frac {5}{2} a^4 b x^2+\frac {10}{3} a^3 b^2 x^3+\frac {5}{2} a^2 b^3 x^4+a b^4 x^5+\frac {b^5 x^6}{6} \]

[In]

Integrate[a^5 + 5*a^4*b*x + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a*b^4*x^4 + b^5*x^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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\left (b x +a \right )^{6}}{6 b}\)	\(13\)
norman	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} x^{3} b^{2}+\frac {5}{2} x^{2} a^{4} b +a^{5} x\)	\(54\)
risch	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} x^{3} b^{2}+\frac {5}{2} x^{2} a^{4} b +a^{5} x\)	\(54\)
parallelrisch	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} x^{3} b^{2}+\frac {5}{2} x^{2} a^{4} b +a^{5} x\)	\(54\)
parts	\(\frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} x^{3} b^{2}+\frac {5}{2} x^{2} a^{4} b +a^{5} x\)	\(54\)
gosper	\(\frac {x \left (b^{5} x^{5}+6 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} b^{2} x^{2}+15 x \,a^{4} b +6 a^{5}\right )}{6}\)	\(55\)

[In]

int(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=\frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]

[In]

integrate(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*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] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (8) = 16\).

Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 4.29 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6} \]

[In]

integrate(b**5*x**5+5*a*b**4*x**4+10*a**2*b**3*x**3+10*a**3*b**2*x**2+5*a**4*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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).

Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=\frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]

[In]

integrate(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (12) = 24\).

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=\frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]

[In]

integrate(b^5*x^5+5*a*b^4*x^4+10*a^2*b^3*x^3+10*a^3*b^2*x^2+5*a^4*b*x+a^5,x, algorithm="giac")

[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

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int \left (a^5+5 a^4 b x+10 a^3 b^2 x^2+10 a^2 b^3 x^3+5 a b^4 x^4+b^5 x^5\right ) \, dx=a^5\,x+\frac {5\,a^4\,b\,x^2}{2}+\frac {10\,a^3\,b^2\,x^3}{3}+\frac {5\,a^2\,b^3\,x^4}{2}+a\,b^4\,x^5+\frac {b^5\,x^6}{6} \]

[In]

int(a^5 + b^5*x^5 + 5*a*b^4*x^4 + 10*a^3*b^2*x^2 + 10*a^2*b^3*x^3 + 5*a^4*b*x,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