3.787 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=116 \[ \frac{2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b B x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.152162, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b B x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.4261, size = 117, normalized size = 1.01 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a \sqrt{x}} + \frac{\sqrt{x} \left (4 A b + \frac{4 B a}{3}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a + b x} + \frac{2 \sqrt{x} \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0350177, size = 47, normalized size = 0.41 \[ \frac{2 \sqrt{(a+b x)^2} (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt{x} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 0.4 \[ -{\frac{-2\,Bb{x}^{2}-6\,Abx-6\,aBx+6\,aA}{3\,bx+3\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.70617, size = 45, normalized size = 0.39 \[ \frac{2 \,{\left (b x^{2} + 3 \, a x\right )} B}{3 \, \sqrt{x}} + \frac{2 \,{\left (b x^{2} - a x\right )} A}{x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.302811, size = 35, normalized size = 0.3 \[ \frac{2 \,{\left (B b x^{2} - 3 \, A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt((a + b*x)**2)/x**(3/2), x)

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GIAC/XCAS [A]  time = 0.270054, size = 72, normalized size = 0.62 \[ \frac{2}{3} \, B b x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, B a \sqrt{x}{\rm sign}\left (b x + a\right ) + 2 \, A b \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \, A a{\rm sign}\left (b x + a\right )}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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