3.1534 \(\int (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.144299, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}{3 e^2 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.9171, size = 73, normalized size = 0.79 \[ \frac{\left (d + e x\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 e} + \frac{\left (d + e x\right )^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 e^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.045707, size = 67, normalized size = 0.73 \[ \frac{x \sqrt{(a+b x)^2} \left (4 a \left (3 d^2+3 d e x+e^2 x^2\right )+b x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )}{12 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 66, normalized size = 0.7 \[{\frac{x \left ( 3\,b{e}^{2}{x}^{3}+4\,{x}^{2}a{e}^{2}+8\,{x}^{2}bde+12\,adex+6\,xb{d}^{2}+12\,a{d}^{2} \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x*(3*b*e^2*x^3+4*a*e^2*x^2+8*b*d*e*x^2+12*a*d*e*x+6*b*d^2*x+12*a*d^2)*((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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.202831, size = 65, normalized size = 0.71 \[ \frac{1}{4} \, b e^{2} x^{4} + a d^{2} x + \frac{1}{3} \,{\left (2 \, b d e + a e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.213802, size = 49, normalized size = 0.53 \[ a d^{2} x + \frac{b e^{2} x^{4}}{4} + x^{3} \left (\frac{a e^{2}}{3} + \frac{2 b d e}{3}\right ) + x^{2} \left (a d e + \frac{b d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209629, size = 115, normalized size = 1.25 \[ \frac{1}{4} \, b x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, b d x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a x^{3} e^{2}{\rm sign}\left (b x + a\right ) + a d x^{2} e{\rm sign}\left (b x + a\right ) + a d^{2} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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