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

Optimal. Leaf size=49 \[ -\frac{2 \sqrt{x} (2 A b-a B)}{a^2 \sqrt{a+b x}}-\frac{2 A}{a \sqrt{x} \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0654186, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 \sqrt{x} (2 A b-a B)}{a^2 \sqrt{a+b x}}-\frac{2 A}{a \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.24888, size = 46, normalized size = 0.94 \[ - \frac{2 A}{a \sqrt{x} \sqrt{a + b x}} - \frac{4 \sqrt{x} \left (A b - \frac{B a}{2}\right )}{a^{2} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0448325, size = 33, normalized size = 0.67 \[ \frac{2 (-a A+a B x-2 A b x)}{a^2 \sqrt{x} \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 30, normalized size = 0.6 \[ -2\,{\frac{2\,Abx-Bax+Aa}{\sqrt{x}\sqrt{bx+a}{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34386, size = 74, normalized size = 1.51 \[ \frac{2 \, B x}{\sqrt{b x^{2} + a x} a} - \frac{4 \, A b x}{\sqrt{b x^{2} + a x} a^{2}} - \frac{2 \, A}{\sqrt{b x^{2} + a x} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.23391, size = 41, normalized size = 0.84 \[ -\frac{2 \,{\left (A a -{\left (B a - 2 \, A b\right )} x\right )}}{\sqrt{b x + a} a^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.234274, size = 126, normalized size = 2.57 \[ -\frac{2 \, \sqrt{b x + a} A b^{2}}{\sqrt{{\left (b x + a\right )} b - a b} a^{2}{\left | b \right |}} + \frac{4 \,{\left (B a b^{\frac{3}{2}} - A b^{\frac{5}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b*x + a)*A*b^2/(sqrt((b*x + a)*b - a*b)*a^2*abs(b)) + 4*(B*a*b^(3/2) - A
*b^(5/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*a*abs(b))