\(\int (a+b x)^m (a+b (2+m) x) \, dx\) [1844]

Optimal result

Integrand size = 16, antiderivative size = 11 \[ \int (a+b x)^m (a+b (2+m) x) \, dx=x (a+b x)^{1+m} \]

[Out]

x*(b*x+a)^(1+m)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {34} \[ \int (a+b x)^m (a+b (2+m) x) \, dx=x (a+b x)^{m+1} \]

[In]

Int[(a + b*x)^m*(a + b*(2 + m)*x),x]

[Out]

x*(a + b*x)^(1 + m)

Rule 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[d*x*((a + b*x)^(m + 1)/(b*(m + 2))), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

Rubi steps \begin{align*} \text {integral}& = x (a+b x)^{1+m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int (a+b x)^m (a+b (2+m) x) \, dx=x (a+b x)^{1+m} \]

[In]

Integrate[(a + b*x)^m*(a + b*(2 + m)*x),x]

[Out]

x*(a + b*x)^(1 + m)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
gosper	\(x \left (b x +a \right )^{1+m}\)	\(12\)
risch	\(\left (b x +a \right )^{m} x \left (b x +a \right )\)	\(15\)
norman	\(a x \,{\mathrm e}^{m \ln \left (b x +a \right )}+b \,x^{2} {\mathrm e}^{m \ln \left (b x +a \right )}\)	\(28\)
parallelrisch	\(\frac {x^{2} \left (b x +a \right )^{m} b^{2}+x \left (b x +a \right )^{m} a b}{b}\)	\(31\)

[In]

int((b*x+a)^m*(a+b*(2+m)*x),x,method=_RETURNVERBOSE)

[Out]

x*(b*x+a)^(1+m)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int (a+b x)^m (a+b (2+m) x) \, dx={\left (b x^{2} + a x\right )} {\left (b x + a\right )}^{m} \]

[In]

integrate((b*x+a)^m*(a+b*(2+m)*x),x, algorithm="fricas")

[Out]

(b*x^2 + a*x)*(b*x + a)^m

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82 \[ \int (a+b x)^m (a+b (2+m) x) \, dx=a x \left (a + b x\right )^{m} + b x^{2} \left (a + b x\right )^{m} \]

[In]

integrate((b*x+a)**m*(a+b*(2+m)*x),x)

[Out]

a*x*(a + b*x)**m + b*x**2*(a + b*x)**m

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (11) = 22\).

Time = 0.21 (sec) , antiderivative size = 106, normalized size of antiderivative = 9.64 \[ \int (a+b x)^m (a+b (2+m) x) \, dx=\frac {{\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} m}{{\left (m^{2} + 3 \, m + 2\right )} b} + \frac {2 \, {\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m}}{{\left (m^{2} + 3 \, m + 2\right )} b} + \frac {{\left (b x + a\right )}^{m + 1} a}{b {\left (m + 1\right )}} \]

[In]

integrate((b*x+a)^m*(a+b*(2+m)*x),x, algorithm="maxima")

[Out]

(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*m/((m^2 + 3*m + 2)*b) + 2*(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x
+ a)^m/((m^2 + 3*m + 2)*b) + (b*x + a)^(m + 1)*a/(b*(m + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (11) = 22\).

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int (a+b x)^m (a+b (2+m) x) \, dx={\left (b x + a\right )}^{m} b x^{2} + {\left (b x + a\right )}^{m} a x \]

[In]

integrate((b*x+a)^m*(a+b*(2+m)*x),x, algorithm="giac")

[Out]

(b*x + a)^m*b*x^2 + (b*x + a)^m*a*x

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int (a+b x)^m (a+b (2+m) x) \, dx=x\,{\left (a+b\,x\right )}^{m+1} \]

[In]

int((a + b*x*(m + 2))*(a + b*x)^m,x)

[Out]

x*(a + b*x)^(m + 1)