3.83 \(\int (a+b x)^5 \, dx\)

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

[Out]

(a + b*x)^6/(6*b)

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Rubi [A]  time = 0.0017074, antiderivative size = 14, normalized size of antiderivative = 1., 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} \[ \frac{(a+b x)^6}{6 b} \]

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 = 0.0058494, size = 14, normalized size = 1. \[ \frac{(a+b x)^6}{6 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5,x]

[Out]

(a + b*x)^6/(6*b)

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Maple [A]  time = 0., size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) ^{6}}{6\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.05434, 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*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 = 1.35106, 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*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 = 0.076919, 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*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 [A]  time = 1.19522, size = 16, normalized size = 1.14 \begin{align*} \frac{{\left (b x + a\right )}^{6}}{6 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b*x + a)^6/b