3.661 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^4} \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.150538, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{2 a^2 x^2}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 a^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.6283, size = 73, normalized size = 0.97 \[ - \frac{A \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3 a^{2} x^{3}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0291988, size = 46, normalized size = 0.61 \[ -\frac{\sqrt{(a+b x)^2} (a (2 A+3 B x)+3 b x (A+2 B x))}{6 x^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[ -{\frac{6\,Bb{x}^{2}+3\,Abx+3\,aBx+2\,aA}{6\,{x}^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283428, size = 36, normalized size = 0.48 \[ -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

-1/6*(6*B*b*x^2 + 2*A*a + 3*(B*a + A*b)*x)/x^3

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Sympy [A]  time = 1.81732, size = 31, normalized size = 0.41 \[ - \frac{2 A a + 6 B b x^{2} + x \left (3 A b + 3 B a\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.268335, size = 104, normalized size = 1.39 \[ -\frac{{\left (3 \, B a b^{2} - A b^{3}\right )}{\rm sign}\left (b x + a\right )}{6 \, a^{2}} - \frac{6 \, B b x^{2}{\rm sign}\left (b x + a\right ) + 3 \, B a x{\rm sign}\left (b x + a\right ) + 3 \, A b x{\rm sign}\left (b x + a\right ) + 2 \, A a{\rm sign}\left (b x + a\right )}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

-1/6*(3*B*a*b^2 - A*b^3)*sign(b*x + a)/a^2 - 1/6*(6*B*b*x^2*sign(b*x + a) + 3*B*
a*x*sign(b*x + a) + 3*A*b*x*sign(b*x + a) + 2*A*a*sign(b*x + a))/x^3