3.1958 \(\int (a+b x) (d+e x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)}{3 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^2} \]

[Out]

((b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^2) + (e*(a + b*x)^3
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^2)

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Rubi [A]  time = 0.170191, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)}{3 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^2) + (e*(a + b*x)^3
*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*b^2)

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Rubi in Sympy [A]  time = 18.0937, size = 60, normalized size = 0.77 \[ \frac{\left (d + e x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 b} - \frac{\left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)*((b*x+a)**2)**(1/2),x)

[Out]

(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(4*b) - (a*e - b*d)*(a**2 + 2*a*b*
x + b**2*x**2)**(3/2)/(12*b**2)

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Mathematica [A]  time = 0.0389653, size = 64, normalized size = 0.82 \[ \frac{x \sqrt{(a+b x)^2} \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right )}{12 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 66, normalized size = 0.9 \[{\frac{x \left ( 3\,{b}^{2}e{x}^{3}+8\,{x}^{2}bea+4\,{x}^{2}{b}^{2}d+6\,xe{a}^{2}+12\,abdx+12\,{a}^{2}d \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x*(3*b^2*e*x^3+8*a*b*e*x^2+4*b^2*d*x^2+6*a^2*e*x+12*a*b*d*x+12*a^2*d)*((b*x
+a)^2)^(1/2)/(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282393, size = 65, normalized size = 0.83 \[ \frac{1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac{1}{3} \,{\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d),x, algorithm="fricas")

[Out]

1/4*b^2*e*x^4 + a^2*d*x + 1/3*(b^2*d + 2*a*b*e)*x^3 + 1/2*(2*a*b*d + a^2*e)*x^2

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Sympy [A]  time = 0.209033, size = 49, normalized size = 0.63 \[ a^{2} d x + \frac{b^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a b e}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*d*x + b**2*e*x**4/4 + x**3*(2*a*b*e/3 + b**2*d/3) + x**2*(a**2*e/2 + a*b*d)

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GIAC/XCAS [A]  time = 0.280409, size = 119, normalized size = 1.53 \[ \frac{1}{4} \, b^{2} x^{4} e{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d x^{3}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, a b x^{3} e{\rm sign}\left (b x + a\right ) + a b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a^{2} d x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d),x, algorithm="giac")

[Out]

1/4*b^2*x^4*e*sign(b*x + a) + 1/3*b^2*d*x^3*sign(b*x + a) + 2/3*a*b*x^3*e*sign(b
*x + a) + a*b*d*x^2*sign(b*x + a) + 1/2*a^2*x^2*e*sign(b*x + a) + a^2*d*x*sign(b
*x + a)