3.788 \(\int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{x^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.152469, 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 \[ -\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{\sqrt{x} (a+b x)}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac{2 b B \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.3762, size = 119, normalized size = 1.03 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 a x^{\frac{3}{2}}} - \frac{\left (\frac{4 A b}{3} + 4 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{\sqrt{x} \left (a + b x\right )} + \frac{2 \left (A b + 3 B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0313743, size = 46, normalized size = 0.4 \[ -\frac{2 \sqrt{(a+b x)^2} (a (A+3 B x)+3 b x (A-B x))}{3 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^(5/2),x]

[Out]

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

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Maple [A]  time = 0.007, size = 43, normalized size = 0.4 \[ -{\frac{-6\,Bb{x}^{2}+6\,Abx+6\,aBx+2\,aA}{3\,bx+3\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*((b*x+a)^2)^(1/2)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 0.706863, size = 45, normalized size = 0.39 \[ \frac{2 \,{\left (b x^{2} - a x\right )} B}{x^{\frac{3}{2}}} - \frac{2 \,{\left (3 \, b x^{2} + a x\right )} A}{3 \, x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(5/2),x, algorithm="maxima")

[Out]

2*(b*x^2 - a*x)*B/x^(3/2) - 2/3*(3*b*x^2 + a*x)*A/x^(5/2)

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Fricas [A]  time = 0.3046, size = 36, normalized size = 0.31 \[ \frac{2 \,{\left (3 \, B b x^{2} - A a - 3 \,{\left (B a + A b\right )} x\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.272125, size = 69, normalized size = 0.59 \[ 2 \, B b \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \,{\left (3 \, B a x{\rm sign}\left (b x + a\right ) + 3 \, A b x{\rm sign}\left (b x + a\right ) + A a{\rm sign}\left (b x + a\right )\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(5/2),x, algorithm="giac")

[Out]

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