3.762 \(\int \frac{A+B x}{x^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx\)

Optimal. Leaf size=88 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{3 A b-a B}{a^2 b \sqrt{x}}+\frac{A b-a B}{a b \sqrt{x} (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.107759, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{3 A b-a B}{a^2 b \sqrt{x}}+\frac{A b-a B}{a b \sqrt{x} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.4615, size = 73, normalized size = 0.83 \[ \frac{A b - B a}{a b \sqrt{x} \left (a + b x\right )} - \frac{3 A b - B a}{a^{2} b \sqrt{x}} - \frac{\left (3 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0827098, size = 67, normalized size = 0.76 \[ \frac{(a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}+\frac{-2 a A+a B x-3 A b x}{a^2 \sqrt{x} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*A - 3*A*b*x + a*B*x)/(a^2*Sqrt[x]*(a + b*x)) + ((-3*A*b + a*B)*ArcTan[(Sqr
t[b]*Sqrt[x])/Sqrt[a]])/(a^(5/2)*Sqrt[b])

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Maple [A]  time = 0.025, size = 87, normalized size = 1. \[ -2\,{\frac{A}{{a}^{2}\sqrt{x}}}-{\frac{Ab}{{a}^{2} \left ( bx+a \right ) }\sqrt{x}}+{\frac{B}{a \left ( bx+a \right ) }\sqrt{x}}-3\,{\frac{Ab}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }+{\frac{B}{a}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317087, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B a^{2} - 3 \, A a b +{\left (B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, a b \sqrt{x} - \sqrt{-a b}{\left (b x - a\right )}}{b x + a}\right ) + 2 \,{\left (2 \, A a -{\left (B a - 3 \, A b\right )} x\right )} \sqrt{-a b}}{2 \,{\left (a^{2} b x + a^{3}\right )} \sqrt{-a b} \sqrt{x}}, -\frac{{\left (B a^{2} - 3 \, A a b +{\left (B a b - 3 \, A b^{2}\right )} x\right )} \sqrt{x} \arctan \left (\frac{a}{\sqrt{a b} \sqrt{x}}\right ) +{\left (2 \, A a -{\left (B a - 3 \, A b\right )} x\right )} \sqrt{a b}}{{\left (a^{2} b x + a^{3}\right )} \sqrt{a b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((B*a^2 - 3*A*a*b + (B*a*b - 3*A*b^2)*x)*sqrt(x)*log(-(2*a*b*sqrt(x) - sqr
t(-a*b)*(b*x - a))/(b*x + a)) + 2*(2*A*a - (B*a - 3*A*b)*x)*sqrt(-a*b))/((a^2*b*
x + a^3)*sqrt(-a*b)*sqrt(x)), -((B*a^2 - 3*A*a*b + (B*a*b - 3*A*b^2)*x)*sqrt(x)*
arctan(a/(sqrt(a*b)*sqrt(x))) + (2*A*a - (B*a - 3*A*b)*x)*sqrt(a*b))/((a^2*b*x +
a^3)*sqrt(a*b)*sqrt(x))]

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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)/x**(3/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285337, size = 81, normalized size = 0.92 \[ \frac{{\left (B a - 3 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{B a x - 3 \, A b x - 2 \, A a}{{\left (b x^{\frac{3}{2}} + a \sqrt{x}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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