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

Optimal. Leaf size=73 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3 A b-2 a B}{a^2 \sqrt{a+b x}}-\frac{A}{a x \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.110473, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3 A b-2 a B}{a^2 \sqrt{a+b x}}-\frac{A}{a x \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.81852, size = 66, normalized size = 0.9 \[ - \frac{A}{a x \sqrt{a + b x}} - \frac{2 \left (\frac{3 A b}{2} - B a\right )}{a^{2} \sqrt{a + b x}} + \frac{2 \left (\frac{3 A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.118068, size = 63, normalized size = 0.86 \[ \frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{-a A+2 a B x-3 A b x}{a^2 x \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 67, normalized size = 0.9 \[ -2\,{\frac{Ab-Ba}{{a}^{2}\sqrt{bx+a}}}-2\,{\frac{1}{{a}^{2}} \left ( 1/2\,{\frac{A\sqrt{bx+a}}{x}}-1/2\,{\frac{3\,Ab-2\,Ba}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234111, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, B a - 3 \, A b\right )} \sqrt{b x + a} x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (A a -{\left (2 \, B a - 3 \, A b\right )} x\right )} \sqrt{a}}{2 \, \sqrt{b x + a} a^{\frac{5}{2}} x}, \frac{{\left (2 \, B a - 3 \, A b\right )} \sqrt{b x + a} x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) -{\left (A a -{\left (2 \, B a - 3 \, A b\right )} x\right )} \sqrt{-a}}{\sqrt{b x + a} \sqrt{-a} a^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*((2*B*a - 3*A*b)*sqrt(b*x + a)*x*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a
)*a)/x) + 2*(A*a - (2*B*a - 3*A*b)*x)*sqrt(a))/(sqrt(b*x + a)*a^(5/2)*x), ((2*B*
a - 3*A*b)*sqrt(b*x + a)*x*arctan(a/(sqrt(b*x + a)*sqrt(-a))) - (A*a - (2*B*a -
3*A*b)*x)*sqrt(-a))/(sqrt(b*x + a)*sqrt(-a)*a^2*x)]

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Sympy [A]  time = 23.839, size = 224, normalized size = 3.07 \[ A \left (- \frac{1}{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{5}{2}}}\right ) + B \left (\frac{2 a^{3} \sqrt{1 + \frac{b x}{a}}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{3} \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{2} b x \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{2} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*(-1/(a*sqrt(b)*x**(3/2)*sqrt(a/(b*x) + 1)) - 3*sqrt(b)/(a**2*sqrt(x)*sqrt(a/(b
*x) + 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(5/2)) + B*(2*a**3*sqrt(1 +
b*x/a)/(a**(9/2) + a**(7/2)*b*x) + a**3*log(b*x/a)/(a**(9/2) + a**(7/2)*b*x) - 2
*a**3*log(sqrt(1 + b*x/a) + 1)/(a**(9/2) + a**(7/2)*b*x) + a**2*b*x*log(b*x/a)/(
a**(9/2) + a**(7/2)*b*x) - 2*a**2*b*x*log(sqrt(1 + b*x/a) + 1)/(a**(9/2) + a**(7
/2)*b*x))

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GIAC/XCAS [A]  time = 0.216325, size = 117, normalized size = 1.6 \[ \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{2 \,{\left (b x + a\right )} B a - 2 \, B a^{2} - 3 \,{\left (b x + a\right )} A b + 2 \, A a b}{{\left ({\left (b x + a\right )}^{\frac{3}{2}} - \sqrt{b x + a} a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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