3.342 \(\int \frac{1}{x^4 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=90 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{\sqrt{a+b x}}{3 a x^3} \]

[Out]

-Sqrt[a + b*x]/(3*a*x^3) + (5*b*Sqrt[a + b*x])/(12*a^2*x^2) - (5*b^2*Sqrt[a + b*
x])/(8*a^3*x) + (5*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(7/2))

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Rubi [A]  time = 0.0794137, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{5 b^2 \sqrt{a+b x}}{8 a^3 x}+\frac{5 b \sqrt{a+b x}}{12 a^2 x^2}-\frac{\sqrt{a+b x}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[a + b*x]),x]

[Out]

-Sqrt[a + b*x]/(3*a*x^3) + (5*b*Sqrt[a + b*x])/(12*a^2*x^2) - (5*b^2*Sqrt[a + b*
x])/(8*a^3*x) + (5*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 10.2058, size = 82, normalized size = 0.91 \[ - \frac{\sqrt{a + b x}}{3 a x^{3}} + \frac{5 b \sqrt{a + b x}}{12 a^{2} x^{2}} - \frac{5 b^{2} \sqrt{a + b x}}{8 a^{3} x} + \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(3*a*x**3) + 5*b*sqrt(a + b*x)/(12*a**2*x**2) - 5*b**2*sqrt(a + b
*x)/(8*a**3*x) + 5*b**3*atanh(sqrt(a + b*x)/sqrt(a))/(8*a**(7/2))

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Mathematica [A]  time = 0.0782442, size = 67, normalized size = 0.74 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{\sqrt{a+b x} \left (8 a^2-10 a b x+15 b^2 x^2\right )}{24 a^3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[a + b*x]),x]

[Out]

-(Sqrt[a + b*x]*(8*a^2 - 10*a*b*x + 15*b^2*x^2))/(24*a^3*x^3) + (5*b^3*ArcTanh[S
qrt[a + b*x]/Sqrt[a]])/(8*a^(7/2))

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Maple [A]  time = 0.01, size = 90, normalized size = 1. \[ 2\,{b}^{3} \left ( -1/6\,{\frac{\sqrt{bx+a}}{a{x}^{3}{b}^{3}}}-5/6\,{\frac{1}{a} \left ( -1/4\,{\frac{\sqrt{bx+a}}{a{b}^{2}{x}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x+a)^(1/2),x)

[Out]

2*b^3*(-1/6*(b*x+a)^(1/2)/a/x^3/b^3-5/6/a*(-1/4*(b*x+a)^(1/2)/a/x^2/b^2-3/4/a*(-
1/2*(b*x+a)^(1/2)/a/x/b+1/2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(3/2))))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240065, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} x^{3} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (15 \, b^{2} x^{2} - 10 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{a}}{48 \, a^{\frac{7}{2}} x^{3}}, -\frac{15 \, b^{3} x^{3} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (15 \, b^{2} x^{2} - 10 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-a}}{24 \, \sqrt{-a} a^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^4),x, algorithm="fricas")

[Out]

[1/48*(15*b^3*x^3*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) - 2*(15*b^2*x
^2 - 10*a*b*x + 8*a^2)*sqrt(b*x + a)*sqrt(a))/(a^(7/2)*x^3), -1/24*(15*b^3*x^3*a
rctan(a/(sqrt(b*x + a)*sqrt(-a))) + (15*b^2*x^2 - 10*a*b*x + 8*a^2)*sqrt(b*x + a
)*sqrt(-a))/(sqrt(-a)*a^3*x^3)]

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Sympy [A]  time = 10.3305, size = 129, normalized size = 1.43 \[ - \frac{1}{3 \sqrt{b} x^{\frac{7}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{\sqrt{b}}{12 a x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 b^{\frac{3}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{5 b^{\frac{5}{2}}}{8 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{5 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205175, size = 113, normalized size = 1.26 \[ -\frac{\frac{15 \, b^{4} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{15 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{4} - 40 \,{\left (b x + a\right )}^{\frac{3}{2}} a b^{4} + 33 \, \sqrt{b x + a} a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^4),x, algorithm="giac")

[Out]

-1/24*(15*b^4*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (15*(b*x + a)^(5/2
)*b^4 - 40*(b*x + a)^(3/2)*a*b^4 + 33*sqrt(b*x + a)*a^2*b^4)/(a^3*b^3*x^3))/b