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

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

[Out]

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

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Rubi [B]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 4.36, number of steps used = 1, number of rules used = 0, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \frac {5}{2} a^2 b^3 x^4+\frac {10}{3} a^3 b^2 x^3+\frac {5}{2} a^4 b x^2+a^5 x+a b^4 x^5+\frac {b^5 x^6}{6} \]

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 61, normalized size = 4.36 \[ a^5 x+\frac {5}{2} a^4 b x^2+\frac {10}{3} a^3 b^2 x^3+\frac {5}{2} a^2 b^3 x^4+a b^4 x^5+\frac {b^5 x^6}{6} \]

Antiderivative was successfully verified.

[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

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fricas [B]  time = 0.56, size = 53, normalized size = 3.79 \[ \frac {1}{6} x^{6} b^{5} + x^{5} b^{4} a + \frac {5}{2} x^{4} b^{3} a^{2} + \frac {10}{3} x^{3} b^{2} a^{3} + \frac {5}{2} x^{2} b a^{4} + x a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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giac [B]  time = 0.29, size = 53, normalized size = 3.79 \[ \frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.00, size = 54, normalized size = 3.86 \[ \frac {1}{6} b^{5} x^{6}+a \,b^{4} x^{5}+\frac {5}{2} a^{2} b^{3} x^{4}+\frac {10}{3} a^{3} b^{2} x^{3}+\frac {5}{2} a^{4} b \,x^{2}+a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

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

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maxima [B]  time = 0.46, size = 53, normalized size = 3.79 \[ \frac {1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac {5}{2} \, a^{2} b^{3} x^{4} + \frac {10}{3} \, a^{3} b^{2} x^{3} + \frac {5}{2} \, a^{4} b x^{2} + a^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.07, size = 60, normalized size = 4.29 \[ a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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