3.21 \(\int (a+b x)^p \, dx\)

Optimal. Leaf size=18 \[ \frac{(a+b x)^{p+1}}{b (p+1)} \]

[Out]

(a + b*x)^(1 + p)/(b*(1 + p))

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Rubi [A]  time = 0.0040883, antiderivative size = 18, 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} \[ \frac{(a+b x)^{p+1}}{b (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^p,x]

[Out]

(a + b*x)^(1 + p)/(b*(1 + p))

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)^p \, dx &=\frac{(a+b x)^{1+p}}{b (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0097329, size = 17, normalized size = 0.94 \[ \frac{(a+b x)^{p+1}}{b p+b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^p,x]

[Out]

(a + b*x)^(1 + p)/(b + b*p)

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Maple [A]  time = 0.002, size = 19, normalized size = 1.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{1+p}}{b \left ( 1+p \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08196, size = 45, normalized size = 2.5 \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{p}}{b p + b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*(b*x + a)^p/(b*p + b)

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Sympy [A]  time = 0.056502, size = 20, normalized size = 1.11 \begin{align*} \frac{\begin{cases} \frac{\left (a + b x\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + b x \right )} & \text{otherwise} \end{cases}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a + b*x)**(p + 1)/(p + 1), Ne(p, -1)), (log(a + b*x), True))/b

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Giac [A]  time = 1.08902, size = 24, normalized size = 1.33 \begin{align*} \frac{{\left (b x + a\right )}^{p + 1}}{b{\left (p + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)^(p + 1)/(b*(p + 1))