3.350 \(\int \frac{1}{x^3 (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{2}{a x^2 \sqrt{a+b x}} \]

[Out]

2/(a*x^2*Sqrt[a + b*x]) - (5*Sqrt[a + b*x])/(2*a^2*x^2) + (15*b*Sqrt[a + b*x])/(
4*a^3*x) - (15*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.0754344, antiderivative size = 85, 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 \[ -\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{15 b \sqrt{a+b x}}{4 a^3 x}-\frac{5 \sqrt{a+b x}}{2 a^2 x^2}+\frac{2}{a x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x)^(3/2)),x]

[Out]

2/(a*x^2*Sqrt[a + b*x]) - (5*Sqrt[a + b*x])/(2*a^2*x^2) + (15*b*Sqrt[a + b*x])/(
4*a^3*x) - (15*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.4421, size = 78, normalized size = 0.92 \[ \frac{2}{a x^{2} \sqrt{a + b x}} - \frac{5 \sqrt{a + b x}}{2 a^{2} x^{2}} + \frac{15 b \sqrt{a + b x}}{4 a^{3} x} - \frac{15 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/(a*x**2*sqrt(a + b*x)) - 5*sqrt(a + b*x)/(2*a**2*x**2) + 15*b*sqrt(a + b*x)/(4
*a**3*x) - 15*b**2*atanh(sqrt(a + b*x)/sqrt(a))/(4*a**(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0956186, size = 67, normalized size = 0.79 \[ \frac{-2 a^2+5 a b x+15 b^2 x^2}{4 a^3 x^2 \sqrt{a+b x}}-\frac{15 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{4 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2 + 5*a*b*x + 15*b^2*x^2)/(4*a^3*x^2*Sqrt[a + b*x]) - (15*b^2*ArcTanh[Sqrt
[a + b*x]/Sqrt[a]])/(4*a^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 67, normalized size = 0.8 \[ 2\,{b}^{2} \left ({\frac{1}{\sqrt{bx+a}{a}^{3}}}+{\frac{1}{{a}^{3}} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ({\frac{7\, \left ( bx+a \right ) ^{3/2}}{8}}-{\frac{9\,a\sqrt{bx+a}}{8}} \right ) }-{\frac{15}{8\,\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x+a)^(3/2),x)

[Out]

2*b^2*(1/a^3/(b*x+a)^(1/2)+1/a^3*((7/8*(b*x+a)^(3/2)-9/8*a*(b*x+a)^(1/2))/x^2/b^
2-15/8*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)))

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.281535, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b x + a} b^{2} x^{2} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \,{\left (15 \, b^{2} x^{2} + 5 \, a b x - 2 \, a^{2}\right )} \sqrt{a}}{8 \, \sqrt{b x + a} a^{\frac{7}{2}} x^{2}}, \frac{15 \, \sqrt{b x + a} b^{2} x^{2} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (15 \, b^{2} x^{2} + 5 \, a b x - 2 \, a^{2}\right )} \sqrt{-a}}{4 \, \sqrt{b x + a} \sqrt{-a} a^{3} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^3),x, algorithm="fricas")

[Out]

[1/8*(15*sqrt(b*x + a)*b^2*x^2*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x)
+ 2*(15*b^2*x^2 + 5*a*b*x - 2*a^2)*sqrt(a))/(sqrt(b*x + a)*a^(7/2)*x^2), 1/4*(15
*sqrt(b*x + a)*b^2*x^2*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + (15*b^2*x^2 + 5*a*b*
x - 2*a^2)*sqrt(-a))/(sqrt(b*x + a)*sqrt(-a)*a^3*x^2)]

_______________________________________________________________________________________

Sympy [A]  time = 19.274, size = 107, normalized size = 1.26 \[ - \frac{1}{2 a \sqrt{b} x^{\frac{5}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{5 \sqrt{b}}{4 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{15 b^{\frac{3}{2}}}{4 a^{3} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} - \frac{15 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{4 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(2*a*sqrt(b)*x**(5/2)*sqrt(a/(b*x) + 1)) + 5*sqrt(b)/(4*a**2*x**(3/2)*sqrt(a/
(b*x) + 1)) + 15*b**(3/2)/(4*a**3*sqrt(x)*sqrt(a/(b*x) + 1)) - 15*b**2*asinh(sqr
t(a)/(sqrt(b)*sqrt(x)))/(4*a**(7/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205279, size = 108, normalized size = 1.27 \[ \frac{15 \, b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a} a^{3}} + \frac{2 \, b^{2}}{\sqrt{b x + a} a^{3}} + \frac{7 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2} - 9 \, \sqrt{b x + a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x^3),x, algorithm="giac")

[Out]

15/4*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + 2*b^2/(sqrt(b*x + a)*a^
3) + 1/4*(7*(b*x + a)^(3/2)*b^2 - 9*sqrt(b*x + a)*a*b^2)/(a^3*b^2*x^2)