3.1844 \(\int (a+b x)^m (a+b (2+m) x) \, dx\)

Optimal. Leaf size=11 \[ x (a+b x)^{m+1} \]

[Out]

x*(a + b*x)^(1 + m)

________________________________________________________________________________________

Rubi [A]  time = 0.0022903, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {34} \[ x (a+b x)^{m+1} \]

Antiderivative was successfully verified.

[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*} \int (a+b x)^m (a+b (2+m) x) \, dx &=x (a+b x)^{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0108006, size = 11, normalized size = 1. \[ x (a+b x)^{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(a + b*x)^(1 + m)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 12, normalized size = 1.1 \begin{align*} x \left ( bx+a \right ) ^{1+m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.71111, size = 36, normalized size = 3.27 \begin{align*}{\left (b x^{2} + a x\right )}{\left (b x + a\right )}^{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]  time = 0.251909, size = 20, normalized size = 1.82 \begin{align*} a x \left (a + b x\right )^{m} + b x^{2} \left (a + b x\right )^{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [B]  time = 1.06955, size = 31, normalized size = 2.82 \begin{align*}{\left (b x + a\right )}^{m} b x^{2} +{\left (b x + a\right )}^{m} a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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