3.1533 \(\int (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=92 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)} \]

[Out]

-((b*d - a*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^2*(a + b*x)) + (b*
(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^2*(a + b*x))

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Rubi [A]  time = 0.116782, 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 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

-((b*d - a*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^2*(a + b*x)) + (b*
(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^2*(a + b*x))

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Rubi in Sympy [A]  time = 13.0178, size = 73, normalized size = 0.79 \[ \frac{\left (d + e x\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{20 e^{2} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*((b*x+a)**2)**(1/2),x)

[Out]

(d + e*x)**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(5*e) + (d + e*x)**4*(a*e - b*d)*s
qrt(a**2 + 2*a*b*x + b**2*x**2)/(20*e**2*(a + b*x))

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Mathematica [A]  time = 0.0546361, size = 89, normalized size = 0.97 \[ \frac{x \sqrt{(a+b x)^2} \left (5 a \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )}{20 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x*Sqrt[(a + b*x)^2]*(5*a*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*x*(10*
d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3)))/(20*(a + b*x))

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Maple [A]  time = 0.005, size = 90, normalized size = 1. \[{\frac{x \left ( 4\,b{e}^{3}{x}^{4}+5\,{x}^{3}a{e}^{3}+15\,{x}^{3}bd{e}^{2}+20\,ad{e}^{2}{x}^{2}+20\,b{d}^{2}e{x}^{2}+30\,xa{d}^{2}e+10\,xb{d}^{3}+20\,a{d}^{3} \right ) }{20\,bx+20\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*((b*x+a)^2)^(1/2),x)

[Out]

1/20*x*(4*b*e^3*x^4+5*a*e^3*x^3+15*b*d*e^2*x^3+20*a*d*e^2*x^2+20*b*d^2*e*x^2+30*
a*d^2*e*x+10*b*d^3*x+20*a*d^3)*((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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.201542, size = 93, normalized size = 1.01 \[ \frac{1}{5} \, b e^{3} x^{5} + a d^{3} x + \frac{1}{4} \,{\left (3 \, b d e^{2} + a e^{3}\right )} x^{4} +{\left (b d^{2} e + a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

1/5*b*e^3*x^5 + a*d^3*x + 1/4*(3*b*d*e^2 + a*e^3)*x^4 + (b*d^2*e + a*d*e^2)*x^3
+ 1/2*(b*d^3 + 3*a*d^2*e)*x^2

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Sympy [A]  time = 0.251069, size = 73, normalized size = 0.79 \[ a d^{3} x + \frac{b e^{3} x^{5}}{5} + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 b d e^{2}}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e\right ) + x^{2} \left (\frac{3 a d^{2} e}{2} + \frac{b d^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*((b*x+a)**2)**(1/2),x)

[Out]

a*d**3*x + b*e**3*x**5/5 + x**4*(a*e**3/4 + 3*b*d*e**2/4) + x**3*(a*d*e**2 + b*d
**2*e) + x**2*(3*a*d**2*e/2 + b*d**3/2)

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GIAC/XCAS [A]  time = 0.211994, size = 159, normalized size = 1.73 \[ \frac{1}{5} \, b x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, b d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + b d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a x^{4} e^{3}{\rm sign}\left (b x + a\right ) + a d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a d^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a d^{3} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/5*b*x^5*e^3*sign(b*x + a) + 3/4*b*d*x^4*e^2*sign(b*x + a) + b*d^2*x^3*e*sign(b
*x + a) + 1/2*b*d^3*x^2*sign(b*x + a) + 1/4*a*x^4*e^3*sign(b*x + a) + a*d*x^3*e^
2*sign(b*x + a) + 3/2*a*d^2*x^2*e*sign(b*x + a) + a*d^3*x*sign(b*x + a)