3.1535 \(\int (d+e x) \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

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 = 0.0756078, 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 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{2 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]

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)

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Rubi in Sympy [A]  time = 10.4858, size = 66, normalized size = 0.96 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.030735, 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 = 0.003, size = 42, normalized size = 0.6 \[{\frac{x \left ( 2\,be{x}^{2}+3\,aex+3\,xbd+6\,ad \right ) }{6\,bx+6\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

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]  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)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.202046, size = 32, normalized size = 0.46 \[ \frac{1}{3} \, b e x^{3} + a d x + \frac{1}{2} \,{\left (b d + a e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(e*x + d),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 = 0.167453, size = 26, normalized size = 0.38 \[ a d x + \frac{b e x^{3}}{3} + x^{2} \left (\frac{a e}{2} + \frac{b d}{2}\right ) \]

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/XCAS [A]  time = 0.210176, size = 70, normalized size = 1.01 \[ \frac{1}{3} \, b x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a x^{2} e{\rm sign}\left (b x + a\right ) + a d x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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