### 3.1747 $$\int (a^2+2 a b x+b^2 x^2)^p \, dx$$

Optimal. Leaf size=34 $\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)}$

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b*(1 + 2*p))

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Rubi [A]  time = 0.0056333, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {609} $\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^p)/(b*(1 + 2*p))

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

\begin{align*} \int \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (1+2 p)}\\ \end{align*}

Mathematica [A]  time = 0.0111934, size = 23, normalized size = 0.68 $\frac{(a+b x) \left ((a+b x)^2\right )^p}{2 b p+b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((a + b*x)*((a + b*x)^2)^p)/(b + 2*b*p)

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Maple [A]  time = 0.041, size = 35, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{b \left ( 1+2\,p \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^p/b/(1+2*p)

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Maxima [A]  time = 1.11127, size = 34, normalized size = 1. \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p}}{b{\left (2 \, p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*(b*x + a)^(2*p)/(b*(2*p + 1))

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Fricas [A]  time = 1.60567, size = 69, normalized size = 2.03 \begin{align*} \frac{{\left (b x + a\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, b p + b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x + a)*(b^2*x^2 + 2*a*b*x + a^2)^p/(2*b*p + b)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.16335, size = 69, normalized size = 2.03 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b x +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a}{2 \, b p + b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^2*x^2 + 2*a*b*x + a^2)^p*b*x + (b^2*x^2 + 2*a*b*x + a^2)^p*a)/(2*b*p + b)