3.1.21 \(\int (a+b x)^p \, dx\) [21]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(a+b x)^{p+1}}{b (p+1)} \end {gather*}

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*}

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Mathematica [A]
time = 0.02, size = 17, normalized size = 0.94 \begin {gather*} \frac {(a+b x)^{1+p}}{b+b p} \end {gather*}

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 = 0.02, size = 19, normalized size = 1.06

method	result	size
gosper	\(\frac {\left (b x +a \right )^{1+p}}{b \left (1+p \right )}\)	\(19\)
default	\(\frac {\left (b x +a \right )^{1+p}}{b \left (1+p \right )}\)	\(19\)
risch	\(\frac {\left (b x +a \right ) \left (b x +a \right )^{p}}{b \left (1+p \right )}\)	\(22\)
norman	\(\frac {x \,{\mathrm e}^{p \ln \left (b x +a \right )}}{1+p}+\frac {a \,{\mathrm e}^{p \ln \left (b x +a \right )}}{b \left (1+p \right )}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^p,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.50, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (b x + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)^(p + 1)/(b*(p + 1))

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Fricas [A]
time = 0.51, size = 20, normalized size = 1.11 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{p}}{b p + b} \end {gather*}

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 = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \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 {gather*}

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 = 0.45, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (b x + a\right )}^{p + 1}}{b {\left (p + 1\right )}} \end {gather*}

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))

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Mupad [B]
time = 0.22, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,x\right )}^{p+1}}{b\,\left (p+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^p,x)

[Out]

(a + b*x)^(p + 1)/(b*(p + 1))

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