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

Optimal result

Integrand size = 7, antiderivative size = 18 \[ \int (a+b x)^n \, dx=\frac {(a+b x)^{1+n}}{b (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int (a+b x)^n \, dx=\frac {(a+b x)^{n+1}}{b (n+1)} \]

[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*} \text {integral}& = \frac {(a+b x)^{1+n}}{b (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (a+b x)^n \, dx=\frac {(a+b x)^{1+n}}{b (1+n)} \]

[In]

Integrate[(a + b*x)^n,x]

[Out]

(a + b*x)^(1 + n)/(b*(1 + n))

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 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	\(\frac {\left (b x +a \right ) \left (b x +a \right )^{n}}{b \left (1+n \right )}\)	\(22\)
parallelrisch	\(\frac {x \left (b x +a \right )^{n} a b +\left (b x +a \right )^{n} a^{2}}{\left (1+n \right ) a b}\)	\(36\)
norman	\(\frac {x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{1+n}+\frac {a \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (1+n \right )}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (a+b x)^n \, dx=\frac {{\left (b x + a\right )} {\left (b x + a\right )}^{n}}{b n + b} \]

[In]

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

[Out]

(b*x + a)*(b*x + a)^n/(b*n + b)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (a+b x)^n \, dx=\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (a+b x)^n \, dx=\frac {{\left (b x + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

[In]

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

[Out]

(b*x + a)^(n + 1)/(b*(n + 1))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (a+b x)^n \, dx=\frac {{\left (b x + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]

[In]

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

[Out]

(b*x + a)^(n + 1)/(b*(n + 1))

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int (a+b x)^n \, dx=\frac {{\left (a+b\,x\right )}^{n+1}}{b\,\left (n+1\right )} \]

[In]

int((a + b*x)^n,x)

[Out]

(a + b*x)^(n + 1)/(b*(n + 1))