3.291 \(\int \frac{\sqrt{a+b x}}{x^4} \, dx\)

Optimal. Leaf size=87 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{b^2 \sqrt{a+b x}}{8 a^2 x}-\frac{\sqrt{a+b x}}{3 x^3}-\frac{b \sqrt{a+b x}}{12 a x^2} \]

[Out]

-Sqrt[a + b*x]/(3*x^3) - (b*Sqrt[a + b*x])/(12*a*x^2) + (b^2*Sqrt[a + b*x])/(8*a
^2*x) - (b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(5/2))

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Rubi [A]  time = 0.0787507, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{5/2}}+\frac{b^2 \sqrt{a+b x}}{8 a^2 x}-\frac{\sqrt{a+b x}}{3 x^3}-\frac{b \sqrt{a+b x}}{12 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x^4,x]

[Out]

-Sqrt[a + b*x]/(3*x^3) - (b*Sqrt[a + b*x])/(12*a*x^2) + (b^2*Sqrt[a + b*x])/(8*a
^2*x) - (b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(5/2))

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Rubi in Sympy [A]  time = 10.093, size = 73, normalized size = 0.84 \[ - \frac{\sqrt{a + b x}}{3 x^{3}} - \frac{b \sqrt{a + b x}}{12 a x^{2}} + \frac{b^{2} \sqrt{a + b x}}{8 a^{2} x} - \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**4,x)

[Out]

-sqrt(a + b*x)/(3*x**3) - b*sqrt(a + b*x)/(12*a*x**2) + b**2*sqrt(a + b*x)/(8*a*
*2*x) - b**3*atanh(sqrt(a + b*x)/sqrt(a))/(8*a**(5/2))

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Mathematica [A]  time = 0.0674847, size = 67, normalized size = 0.77 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{5/2}}-\frac{\sqrt{a+b x} \left (8 a^2+2 a b x-3 b^2 x^2\right )}{24 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/x^4,x]

[Out]

-(Sqrt[a + b*x]*(8*a^2 + 2*a*b*x - 3*b^2*x^2))/(24*a^2*x^3) - (b^3*ArcTanh[Sqrt[
a + b*x]/Sqrt[a]])/(8*a^(5/2))

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Maple [A]  time = 0.018, size = 65, normalized size = 0.8 \[ 2\,{b}^{3} \left ({\frac{1}{{b}^{3}{x}^{3}} \left ( 1/16\,{\frac{ \left ( bx+a \right ) ^{5/2}}{{a}^{2}}}-1/6\,{\frac{ \left ( bx+a \right ) ^{3/2}}{a}}-1/16\,\sqrt{bx+a} \right ) }-1/16\,{\frac{1}{{a}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^4,x)

[Out]

2*b^3*((1/16/a^2*(b*x+a)^(5/2)-1/6/a*(b*x+a)^(3/2)-1/16*(b*x+a)^(1/2))/x^3/b^3-1
/16*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(5/2))

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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(sqrt(b*x + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234233, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (3 \, b^{2} x^{2} - 2 \, a b x - 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{a}}{48 \, a^{\frac{5}{2}} x^{3}}, \frac{3 \, b^{3} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (3 \, b^{2} x^{2} - 2 \, a b x - 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/x^4,x, algorithm="fricas")

[Out]

[1/48*(3*b^3*x^3*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + 2*(3*b^2*x^2
- 2*a*b*x - 8*a^2)*sqrt(b*x + a)*sqrt(a))/(a^(5/2)*x^3), 1/24*(3*b^3*x^3*arctan
(a/(sqrt(b*x + a)*sqrt(-a))) + (3*b^2*x^2 - 2*a*b*x - 8*a^2)*sqrt(b*x + a)*sqrt(
-a))/(sqrt(-a)*a^2*x^3)]

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Sympy [A]  time = 18.5323, size = 122, normalized size = 1.4 \[ - \frac{a}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 \sqrt{b}}{12 x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{b^{\frac{3}{2}}}{24 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{b^{\frac{5}{2}}}{8 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/x**4,x)

[Out]

-a/(3*sqrt(b)*x**(7/2)*sqrt(a/(b*x) + 1)) - 5*sqrt(b)/(12*x**(5/2)*sqrt(a/(b*x)
+ 1)) + b**(3/2)/(24*a*x**(3/2)*sqrt(a/(b*x) + 1)) + b**(5/2)/(8*a**2*sqrt(x)*sq
rt(a/(b*x) + 1)) - b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(8*a**(5/2))

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GIAC/XCAS [A]  time = 0.209479, size = 113, normalized size = 1.3 \[ \frac{\frac{3 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} - 8 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} - 3 \, \sqrt{b x + a} a^{2} b^{4}}{a^{2} b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x + a)/x^4,x, algorithm="giac")

[Out]

1/24*(3*b^4*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x + a)^(5/2)*b
^4 - 8*(b*x + a)^(3/2)*a*b^4 - 3*sqrt(b*x + a)*a^2*b^4)/(a^2*b^3*x^3))/b