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]

(a + b*x)^6/(6*b)

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Rubi [B]  time = 0.014024, 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., 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*}

Mathematica [B]  time = 0.0000577, size = 61, normalized size = 4.36 \[ \frac{5}{2} a^2 b^3 x^4+\frac{10}{3} a^3 b^2 x^3+\frac{5}{2} a^4 b x^2+a^5 x+a b^4 x^5+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]

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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Maple [B]  time = 0.001, size = 54, normalized size = 3.9 \begin{align*}{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{align*}

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 = 1.17789, size = 72, normalized size = 5.14 \begin{align*} \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{align*}

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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Fricas [B]  time = 1.48361, size = 116, normalized size = 8.29 \begin{align*} \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} \end{align*}

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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Sympy [B]  time = 0.070587, size = 60, normalized size = 4.29 \begin{align*} 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{align*}

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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Giac [B]  time = 1.1179, size = 72, normalized size = 5.14 \begin{align*} \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{align*}

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