3.1959 \(\int (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(3*b)

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Rubi [A]  time = 0.0077252, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {629} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/(3*b)

Rule 629

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

Rubi steps

\begin{align*} \int (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0089149, size = 18, normalized size = 0.67 \[ \frac{\left ((a+b x)^2\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^2)^(3/2)/(3*b)

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Maple [A]  time = 0.001, size = 38, normalized size = 1.4 \begin{align*}{\frac{x \left ({b}^{2}{x}^{2}+3\,abx+3\,{a}^{2} \right ) }{3\,bx+3\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*((b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.984666, size = 19, normalized size = 0.7 \begin{align*} \frac{{\left ({\left (b x + a\right )}^{2}\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*((b*x + a)^2)^(3/2)/b

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Fricas [A]  time = 1.77892, size = 42, normalized size = 1.56 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

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Sympy [A]  time = 0.081853, size = 19, normalized size = 0.7 \begin{align*} a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)**2)**(1/2),x)

[Out]

a**2*x + a*b*x**2 + b**2*x**3/3

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Giac [A]  time = 1.12292, size = 24, normalized size = 0.89 \begin{align*} \frac{{\left (b x + a\right )}^{3} \mathrm{sgn}\left (b x + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/3*(b*x + a)^3*sgn(b*x + a)/b