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

Optimal. Leaf size=84 \[ -\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{\sqrt{a+b x} (3 A b-4 a B)}{4 a^2 x}-\frac{A \sqrt{a+b x}}{2 a x^2} \]

[Out]

-(A*Sqrt[a + b*x])/(2*a*x^2) + ((3*A*b - 4*a*B)*Sqrt[a + b*x])/(4*a^2*x) - (b*(3
*A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(5/2))

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Rubi [A]  time = 0.113998, antiderivative size = 84, 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 \[ -\frac{b (3 A b-4 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{\sqrt{a+b x} (3 A b-4 a B)}{4 a^2 x}-\frac{A \sqrt{a+b x}}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(A*Sqrt[a + b*x])/(2*a*x^2) + ((3*A*b - 4*a*B)*Sqrt[a + b*x])/(4*a^2*x) - (b*(3
*A*b - 4*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(5/2))

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Rubi in Sympy [A]  time = 9.27761, size = 75, normalized size = 0.89 \[ - \frac{A \sqrt{a + b x}}{2 a x^{2}} + \frac{\sqrt{a + b x} \left (3 A b - 4 B a\right )}{4 a^{2} x} - \frac{b \left (3 A b - 4 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.128369, size = 70, normalized size = 0.83 \[ \frac{\frac{\sqrt{a} \sqrt{a+b x} (3 A b x-2 a (A+2 B x))}{x^2}+b (4 a B-3 A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*Sqrt[a + b*x]*(3*A*b*x - 2*a*(A + 2*B*x)))/x^2 + b*(-3*A*b + 4*a*B)*Ar
cTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(5/2))

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Maple [A]  time = 0.016, size = 81, normalized size = 1. \[ 2\,b \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( 1/8\,{\frac{ \left ( 3\,Ab-4\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{{a}^{2}}}-1/8\,{\frac{ \left ( 5\,Ab-4\,Ba \right ) \sqrt{bx+a}}{a}} \right ) }-1/8\,{\frac{3\,Ab-4\,Ba}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*((1/8*(3*A*b-4*B*a)/a^2*(b*x+a)^(3/2)-1/8*(5*A*b-4*B*a)/a*(b*x+a)^(1/2))/x^2
/b^2-1/8*(3*A*b-4*B*a)/a^(5/2)*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)/(sqrt(b*x + a)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226759, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (2 \, A a +{\left (4 \, B a - 3 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{a}}{8 \, a^{\frac{5}{2}} x^{2}}, -\frac{{\left (4 \, B a b - 3 \, A b^{2}\right )} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (2 \, A a +{\left (4 \, B a - 3 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{-a}}{4 \, \sqrt{-a} a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*((4*B*a*b - 3*A*b^2)*x^2*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x)
+ 2*(2*A*a + (4*B*a - 3*A*b)*x)*sqrt(b*x + a)*sqrt(a))/(a^(5/2)*x^2), -1/4*((4*B
*a*b - 3*A*b^2)*x^2*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (2*A*a + (4*B*a - 3*A*b
)*x)*sqrt(b*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^2)]

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Sympy [A]  time = 55.0013, size = 156, normalized size = 1.86 \[ - \frac{A}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{A \sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 A b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{3 A b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} - \frac{B \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{a \sqrt{x}} + \frac{B b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(2*sqrt(b)*x**(5/2)*sqrt(a/(b*x) + 1)) + A*sqrt(b)/(4*a*x**(3/2)*sqrt(a/(b*x)
+ 1)) + 3*A*b**(3/2)/(4*a**2*sqrt(x)*sqrt(a/(b*x) + 1)) - 3*A*b**2*asinh(sqrt(a
)/(sqrt(b)*sqrt(x)))/(4*a**(5/2)) - B*sqrt(b)*sqrt(a/(b*x) + 1)/(a*sqrt(x)) + B*
b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(3/2)

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GIAC/XCAS [A]  time = 0.214951, size = 150, normalized size = 1.79 \[ -\frac{\frac{{\left (4 \, B a b^{2} - 3 \, A b^{3}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{4 \,{\left (b x + a\right )}^{\frac{3}{2}} B a b^{2} - 4 \, \sqrt{b x + a} B a^{2} b^{2} - 3 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{3} + 5 \, \sqrt{b x + a} A a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*((4*B*a*b^2 - 3*A*b^3)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (4*(
b*x + a)^(3/2)*B*a*b^2 - 4*sqrt(b*x + a)*B*a^2*b^2 - 3*(b*x + a)^(3/2)*A*b^3 + 5
*sqrt(b*x + a)*A*a*b^3)/(a^2*b^2*x^2))/b