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

Optimal. Leaf size=114 \[ \frac{x^3 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.155235, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{x^3 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{a A x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b B x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.8452, size = 114, normalized size = 1. \[ \frac{B x^{2} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{8 b} - \frac{a \left (2 a + 2 b x\right ) \left (2 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{8 b^{3}} + \frac{\left (2 A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2*(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(8*b) - a*(2*a + 2*b*x)*(2
*A*b - B*a)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(8*b**3) + (2*A*b - B*a)*(a**2 + 2*
a*b*x + b**2*x**2)**(3/2)/(6*b**3)

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Mathematica [A]  time = 0.0280993, size = 47, normalized size = 0.41 \[ \frac{x^2 \sqrt{(a+b x)^2} (a (6 A+4 B x)+b x (4 A+3 B x))}{12 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[(a + b*x)^2]*(b*x*(4*A + 3*B*x) + a*(6*A + 4*B*x)))/(12*(a + b*x))

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Maple [A]  time = 0.007, size = 44, normalized size = 0.4 \[{\frac{{x}^{2} \left ( 3\,Bb{x}^{2}+4\,Abx+4\,aBx+6\,aA \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267637, size = 36, normalized size = 0.32 \[ \frac{1}{4} \, B b x^{4} + \frac{1}{2} \, A a x^{2} + \frac{1}{3} \,{\left (B a + A b\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.189332, size = 29, normalized size = 0.25 \[ \frac{A a x^{2}}{2} + \frac{B b x^{4}}{4} + x^{3} \left (\frac{A b}{3} + \frac{B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272613, size = 104, normalized size = 0.91 \[ \frac{1}{4} \, B b x^{4}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, B a x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, A b x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, A a x^{2}{\rm sign}\left (b x + a\right ) + \frac{{\left (B a^{4} - 2 \, A a^{3} b\right )}{\rm sign}\left (b x + a\right )}{12 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*b*x^4*sign(b*x + a) + 1/3*B*a*x^3*sign(b*x + a) + 1/3*A*b*x^3*sign(b*x + a
) + 1/2*A*a*x^2*sign(b*x + a) + 1/12*(B*a^4 - 2*A*a^3*b)*sign(b*x + a)/b^3