3.1547 \(\int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.074788, 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 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.25341, size = 66, normalized size = 0.96 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(5*b**2) - (2*a + 2*b*x)*(a*e - b*d)*(a**2
+ 2*a*b*x + b**2*x**2)**(3/2)/(8*b**2)

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Mathematica [A]  time = 0.0606221, size = 83, normalized size = 1.2 \[ \frac{x \sqrt{(a+b x)^2} \left (10 a^3 (2 d+e x)+10 a^2 b x (3 d+2 e x)+5 a b^2 x^2 (4 d+3 e x)+b^3 x^3 (5 d+4 e x)\right )}{20 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 90, normalized size = 1.3 \[{\frac{x \left ( 4\,e{b}^{3}{x}^{4}+15\,{x}^{3}ae{b}^{2}+5\,{x}^{3}d{b}^{3}+20\,{a}^{2}be{x}^{2}+20\,a{b}^{2}d{x}^{2}+10\,x{a}^{3}e+30\,x{a}^{2}bd+20\,{a}^{3}d \right ) }{20\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204713, size = 93, normalized size = 1.35 \[ \frac{1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac{1}{4} \,{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} +{\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212023, size = 167, normalized size = 2.42 \[ \frac{1}{5} \, b^{3} x^{5} e{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d x^{4}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a b^{2} x^{4} e{\rm sign}\left (b x + a\right ) + a b^{2} d x^{3}{\rm sign}\left (b x + a\right ) + a^{2} b x^{3} e{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{3} x^{2} e{\rm sign}\left (b x + a\right ) + a^{3} d x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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