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

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

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Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2}}{3 b} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 18, normalized size = 0.67 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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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fricas [A]  time = 0.42, size = 20, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end {gather*}

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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giac [A]  time = 0.15, size = 18, normalized size = 0.67 \begin {gather*} \frac {{\left (b x + a\right )}^{3} \mathrm {sgn}\left (b x + a\right )}{3 \, b} \end {gather*}

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

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maple [A]  time = 0.05, size = 38, normalized size = 1.41 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{3 b x +3 a} \end {gather*}

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 = 0.70, size = 14, normalized size = 0.52 \begin {gather*} \frac {{\left ({\left (b x + a\right )}^{2}\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}

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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mupad [B]  time = 2.17, size = 76, normalized size = 2.81 \begin {gather*} \frac {\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {a\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)^2)^(1/2)*(a + b*x),x)

[Out]

((8*b^2*(a^2 + b^2*x^2) - 12*a^2*b^2 + 4*a*b^3*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(24*b^3) + (a*((a + b*x)^2)
^(1/2)*(a + b*x))/(2*b)

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sympy [A]  time = 0.09, size = 19, normalized size = 0.70 \begin {gather*} a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3} \end {gather*}

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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