3.8.35 \(\int (a+b x)^n \, dx\) [735]

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

[Out]

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

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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)^{n+1}}{b (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n,x]

[Out]

(a + b*x)^(1 + n)/(b*(1 + n))

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

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n,x]

[Out]

(a + b*x)^(1 + n)/(b + b*n)

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: } \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^0*(a + b*x)^n,x]')

[Out]

cought exception:

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Maple [A]
time = 0.11, size = 19, normalized size = 1.06

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (b x + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)^(n + 1)/(b*(n + 1))

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Fricas [A]
time = 0.31, size = 20, normalized size = 1.11 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{n}}{b n + b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*(b*x + a)^n/(b*n + b)

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Sympy [A]
time = 0.03, size = 20, normalized size = 1.11 \begin {gather*} \frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \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)**n,x)

[Out]

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

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Giac [A]
time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} \frac {\left (b x+a\right )^{n+1}}{b \left (n+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n,x)

[Out]

(b*x + a)^(n + 1)/(b*(n + 1))

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Mupad [B]
time = 0.20, size = 18, normalized size = 1.00 \begin {gather*} \frac {{\left (a+b\,x\right )}^{n+1}}{b\,\left (n+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n,x)

[Out]

(a + b*x)^(n + 1)/(b*(n + 1))

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