3.1712 \(\int (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

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

[Out]

((A*b - a*B)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + (B*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(3*b^2)

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Rubi [A]  time = 0.021708, 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} \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*b^2) + (B*(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 (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}+\frac{\left (2 A b^2-2 a b B\right ) \int \sqrt{a^2+2 a b x+b^2 x^2} \, dx}{2 b^2}\\ &=\frac{(A b-a B) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac{B \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0155741, size = 45, normalized size = 0.65 \[ \frac{x \sqrt{(a+b x)^2} (3 a (2 A+B x)+b x (3 A+2 B x))}{6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(3*a*(2*A + B*x) + b*x*(3*A + 2*B*x)))/(6*(a + b*x))

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Maple [A]  time = 0.001, size = 42, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,Bb{x}^{2}+3\,Abx+3\,aBx+6\,aA \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((B*x+A)*((b*x+a)^2)^(1/2),x)

[Out]

1/6*x*(2*B*b*x^2+3*A*b*x+3*B*a*x+6*A*a)*((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((B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54198, size = 58, normalized size = 0.84 \begin{align*} \frac{1}{3} \, B b x^{3} + A a x + \frac{1}{2} \,{\left (B a + A b\right )} x^{2} \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*b*x^3 + A*a*x + 1/2*(B*a + A*b)*x^2

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Sympy [A]  time = 0.091475, size = 26, normalized size = 0.38 \begin{align*} A a x + \frac{B b x^{3}}{3} + x^{2} \left (\frac{A b}{2} + \frac{B a}{2}\right ) \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*a*x + B*b*x**3/3 + x**2*(A*b/2 + B*a/2)

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Giac [A]  time = 1.12786, size = 100, normalized size = 1.45 \begin{align*} \frac{1}{3} \, B b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, B a x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A b x^{2} \mathrm{sgn}\left (b x + a\right ) + A a x \mathrm{sgn}\left (b x + a\right ) - \frac{{\left (B a^{3} - 3 \, A a^{2} b\right )} \mathrm{sgn}\left (b x + a\right )}{6 \, b^{2}} \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*b*x^3*sgn(b*x + a) + 1/2*B*a*x^2*sgn(b*x + a) + 1/2*A*b*x^2*sgn(b*x + a) + A*a*x*sgn(b*x + a) - 1/6*(B*a
^3 - 3*A*a^2*b)*sgn(b*x + a)/b^2