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

Optimal. Leaf size=114 \[ \frac{x^3 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

[Out]

(a*A*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + ((A*b + a*B)*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a
+ b*x)) + (b*B*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x))

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Rubi [A]  time = 0.0456135, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {770, 76} \[ \frac{x^3 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*(a + b*x)) + ((A*b + a*B)*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a
+ b*x)) + (b*B*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x (A+B x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int x \left (a b+b^2 x\right ) (A+B x) \, dx}{a b+b^2 x}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a A b x+b (A b+a B) x^2+b^2 B x^3\right ) \, dx}{a b+b^2 x}\\ &=\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{(A b+a B) x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0167732, size = 47, normalized size = 0.41 \[ \frac{x^2 \sqrt{(a+b x)^2} (a (6 A+4 B x)+b x (4 A+3 B x))}{12 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*Sqrt[(a + b*x)^2]*(b*x*(4*A + 3*B*x) + a*(6*A + 4*B*x)))/(12*(a + b*x))

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Maple [A]  time = 0.009, size = 44, normalized size = 0.4 \begin{align*}{\frac{{x}^{2} \left ( 3\,Bb{x}^{2}+4\,Abx+4\,aBx+6\,aA \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^2*(3*B*b*x^2+4*A*b*x+4*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(x*(B*x+A)*((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2566, size = 66, normalized size = 0.58 \begin{align*} \frac{1}{4} \, B b x^{4} + \frac{1}{2} \, A a x^{2} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*b*x^4 + 1/2*A*a*x^2 + 1/3*(B*a + A*b)*x^3

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Sympy [A]  time = 0.122032, size = 29, normalized size = 0.25 \begin{align*} \frac{A a x^{2}}{2} + \frac{B b x^{4}}{4} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a*x**2/2 + B*b*x**4/4 + x**3*(A*b/3 + B*a/3)

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Giac [A]  time = 1.15756, size = 104, normalized size = 0.91 \begin{align*} \frac{1}{4} \, B b x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, B a x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, A b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, A a x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{{\left (B a^{4} - 2 \, A a^{3} b\right )} \mathrm{sgn}\left (b x + a\right )}{12 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*b*x^4*sgn(b*x + a) + 1/3*B*a*x^3*sgn(b*x + a) + 1/3*A*b*x^3*sgn(b*x + a) + 1/2*A*a*x^2*sgn(b*x + a) + 1/
12*(B*a^4 - 2*A*a^3*b)*sgn(b*x + a)/b^3