3.341 \(\int \frac{1}{x^3 \sqrt{a+b x}} \, dx\)

Optimal. Leaf size=68 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{\sqrt{a+b x}}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0554351, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}}+\frac{3 b \sqrt{a+b x}}{4 a^2 x}-\frac{\sqrt{a+b x}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a + b*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.21041, size = 60, normalized size = 0.88 \[ - \frac{\sqrt{a + b x}}{2 a x^{2}} + \frac{3 b \sqrt{a + b x}}{4 a^{2} x} - \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0537338, size = 56, normalized size = 0.82 \[ \frac{\sqrt{a+b x} (3 b x-2 a)}{4 a^2 x^2}-\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[a + b*x]),x]

[Out]

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

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Maple [A]  time = 0.011, size = 66, normalized size = 1. \[ 2\,{b}^{2} \left ( -1/4\,{\frac{\sqrt{bx+a}}{a{b}^{2}{x}^{2}}}-3/4\,{\frac{1}{a} \left ( -1/2\,{\frac{\sqrt{bx+a}}{abx}}+1/2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^(1/2),x)

[Out]

2*b^2*(-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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230155, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{2} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (3 \, b x - 2 \, a\right )} \sqrt{b x + a} \sqrt{a}}{8 \, a^{\frac{5}{2}} x^{2}}, \frac{3 \, b^{2} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (3 \, b x - 2 \, a\right )} \sqrt{b x + a} \sqrt{-a}}{4 \, \sqrt{-a} a^{2} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.6488, size = 102, normalized size = 1.5 \[ - \frac{1}{2 \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{\sqrt{b}}{4 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b^{\frac{3}{2}}}{4 a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.204445, size = 93, normalized size = 1.37 \[ \frac{\frac{3 \, b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3} - 5 \, \sqrt{b x + a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + a)*x^3),x, algorithm="giac")

[Out]

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