3.1710 \(\int (A+B x) (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=158 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (B d-A e)}{3 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^3 (a+b x)} \]

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a +
b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4
*e^3*(a + b*x)) + (b*B*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*
x))

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Rubi [A]  time = 0.332536, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3 (a+b x)}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e) (B d-A e)}{3 e^3 (a+b x)}+\frac{b B \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)*(B*d - A*e)*(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^3*(a +
b*x)) - ((2*b*B*d - A*b*e - a*B*e)*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4
*e^3*(a + b*x)) + (b*B*(d + e*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^3*(a + b*
x))

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Rubi in Sympy [A]  time = 58.6752, size = 162, normalized size = 1.03 \[ \frac{B \left (2 a + 2 b x\right ) \left (d + e x\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{10 b e} + \frac{\left (d + e x\right )^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 3 B a e - 2 B b d\right )}{20 b e^{2}} + \frac{\left (d + e x\right )^{3} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \left (5 A b e - 3 B a e - 2 B b d\right )}{60 b e^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(2*a + 2*b*x)*(d + e*x)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(10*b*e) + (d + e*
x)**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e - 3*B*a*e - 2*B*b*d)/(20*b*e**2)
+ (d + e*x)**3*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)*(5*A*b*e - 3*B*a*e
- 2*B*b*d)/(60*b*e**3*(a + b*x))

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Mathematica [A]  time = 0.0871051, size = 120, normalized size = 0.76 \[ \frac{x \sqrt{(a+b x)^2} \left (5 a \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )\right )}{60 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[(a + b*x)^2]*(5*a*(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)) + b*x*(5*A*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 2*B*x*(10*d^2 + 15*d*
e*x + 6*e^2*x^2))))/(60*(a + b*x))

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Maple [A]  time = 0.007, size = 128, normalized size = 0.8 \[{\frac{x \left ( 12\,B{e}^{2}b{x}^{4}+15\,{x}^{3}Ab{e}^{2}+15\,{x}^{3}aB{e}^{2}+30\,{x}^{3}Bbde+20\,{x}^{2}A{e}^{2}a+40\,{x}^{2}Abde+40\,{x}^{2}aBde+20\,{x}^{2}Bb{d}^{2}+60\,xaAde+30\,xAb{d}^{2}+30\,xBa{d}^{2}+60\,aA{d}^{2} \right ) }{60\,bx+60\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*x*(12*B*b*e^2*x^4+15*A*b*e^2*x^3+15*B*a*e^2*x^3+30*B*b*d*e*x^3+20*A*a*e^2*x
^2+40*A*b*d*e*x^2+40*B*a*d*e*x^2+20*B*b*d^2*x^2+60*A*a*d*e*x+30*A*b*d^2*x+30*B*a
*d^2*x+60*A*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)*(B*x + A)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274247, size = 126, normalized size = 0.8 \[ \frac{1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{2} + A a e^{2} + 2 \,{\left (B a + A b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a d e +{\left (B a + A b\right )} d^{2}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.274925, size = 116, normalized size = 0.73 \[ A a d^{2} x + \frac{B b e^{2} x^{5}}{5} + x^{4} \left (\frac{A b e^{2}}{4} + \frac{B a e^{2}}{4} + \frac{B b d e}{2}\right ) + x^{3} \left (\frac{A a e^{2}}{3} + \frac{2 A b d e}{3} + \frac{2 B a d e}{3} + \frac{B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac{A b d^{2}}{2} + \frac{B a d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285846, size = 250, normalized size = 1.58 \[ \frac{1}{5} \, B b x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B b d x^{4} e{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, B b d^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, B a x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, A b x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a d x^{3} e{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A b d x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, B a d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A b d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, A a x^{3} e^{2}{\rm sign}\left (b x + a\right ) + A a d x^{2} e{\rm sign}\left (b x + a\right ) + A a d^{2} x{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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