3.67 \(\int \frac{1}{x \left (a x+b x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{128 b^2 (a+2 b x)}{15 a^5 \sqrt{a x+b x^2}}+\frac{16 b (a+2 b x)}{15 a^3 \left (a x+b x^2\right )^{3/2}}-\frac{2}{5 a x \left (a x+b x^2\right )^{3/2}} \]

[Out]

-2/(5*a*x*(a*x + b*x^2)^(3/2)) + (16*b*(a + 2*b*x))/(15*a^3*(a*x + b*x^2)^(3/2))
- (128*b^2*(a + 2*b*x))/(15*a^5*Sqrt[a*x + b*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0731734, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{128 b^2 (a+2 b x)}{15 a^5 \sqrt{a x+b x^2}}+\frac{16 b (a+2 b x)}{15 a^3 \left (a x+b x^2\right )^{3/2}}-\frac{2}{5 a x \left (a x+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a*x + b*x^2)^(5/2)),x]

[Out]

-2/(5*a*x*(a*x + b*x^2)^(3/2)) + (16*b*(a + 2*b*x))/(15*a^3*(a*x + b*x^2)^(3/2))
- (128*b^2*(a + 2*b*x))/(15*a^5*Sqrt[a*x + b*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.6077, size = 75, normalized size = 0.94 \[ - \frac{2}{5 a x \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{16 b \left (a + 2 b x\right )}{15 a^{3} \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{64 b^{2} \left (2 a + 4 b x\right )}{15 a^{5} \sqrt{a x + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x**2+a*x)**(5/2),x)

[Out]

-2/(5*a*x*(a*x + b*x**2)**(3/2)) + 16*b*(a + 2*b*x)/(15*a**3*(a*x + b*x**2)**(3/
2)) - 64*b**2*(2*a + 4*b*x)/(15*a**5*sqrt(a*x + b*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0510494, size = 62, normalized size = 0.78 \[ -\frac{2 \left (3 a^4-8 a^3 b x+48 a^2 b^2 x^2+192 a b^3 x^3+128 b^4 x^4\right )}{15 a^5 x (x (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a*x + b*x^2)^(5/2)),x]

[Out]

(-2*(3*a^4 - 8*a^3*b*x + 48*a^2*b^2*x^2 + 192*a*b^3*x^3 + 128*b^4*x^4))/(15*a^5*
x*(x*(a + b*x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 63, normalized size = 0.8 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 128\,{b}^{4}{x}^{4}+192\,a{b}^{3}{x}^{3}+48\,{b}^{2}{x}^{2}{a}^{2}-8\,bx{a}^{3}+3\,{a}^{4} \right ) }{15\,{a}^{5}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^2+a*x)^(5/2),x)

[Out]

-2/15*(b*x+a)*(128*b^4*x^4+192*a*b^3*x^3+48*a^2*b^2*x^2-8*a^3*b*x+3*a^4)/a^5/(b*
x^2+a*x)^(5/2)

_______________________________________________________________________________________

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^2 + a*x)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.222511, size = 97, normalized size = 1.21 \[ -\frac{2 \,{\left (128 \, b^{4} x^{4} + 192 \, a b^{3} x^{3} + 48 \, a^{2} b^{2} x^{2} - 8 \, a^{3} b x + 3 \, a^{4}\right )}}{15 \,{\left (a^{5} b x^{3} + a^{6} x^{2}\right )} \sqrt{b x^{2} + a x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a*x)^(5/2)*x),x, algorithm="fricas")

[Out]

-2/15*(128*b^4*x^4 + 192*a*b^3*x^3 + 48*a^2*b^2*x^2 - 8*a^3*b*x + 3*a^4)/((a^5*b
*x^3 + a^6*x^2)*sqrt(b*x^2 + a*x))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**2+a*x)**(5/2),x)

[Out]

Integral(1/(x*(x*(a + b*x))**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a x\right )}^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a*x)^(5/2)*x),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a*x)^(5/2)*x), x)