3.1546 \(\int (d+e x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=69 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]

[Out]

((b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + (e*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(3*b^2)

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Rubi [A]  time = 0.0208345, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {640, 609} \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + (e*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(3*b^2)

Rule 640

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

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

Mathematica [A]  time = 0.0172388, size = 45, normalized size = 0.65 \[ \frac{x \sqrt{(a+b x)^2} (3 a (2 d+e x)+b x (3 d+2 e x))}{6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(3*a*(2*d + e*x) + b*x*(3*d + 2*e*x)))/(6*(a + b*x))

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Maple [A]  time = 0.041, size = 42, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,be{x}^{2}+3\,aex+3\,bdx+6\,ad \right ) }{6\,bx+6\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x*(2*b*e*x^2+3*a*e*x+3*b*d*x+6*a*d)*((b*x+a)^2)^(1/2)/(b*x+a)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51045, size = 58, normalized size = 0.84 \begin{align*} \frac{1}{3} \, b e x^{3} + a d x + \frac{1}{2} \,{\left (b d + a e\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.115544, size = 26, normalized size = 0.38 \begin{align*} a d x + \frac{b e x^{3}}{3} + x^{2} \left (\frac{a e}{2} + \frac{b d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*d*x + b*e*x**3/3 + x**2*(a*e/2 + b*d/2)

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Giac [A]  time = 1.12668, size = 70, normalized size = 1.01 \begin{align*} \frac{1}{3} \, b x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, b d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a x^{2} e \mathrm{sgn}\left (b x + a\right ) + a d x \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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