3.275 \(\int \frac{1}{x^{5/2} \sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{16 b^2 \sqrt{a x^2+b x^3}}{15 a^3 x^{3/2}}+\frac{8 b \sqrt{a x^2+b x^3}}{15 a^2 x^{5/2}}-\frac{2 \sqrt{a x^2+b x^3}}{5 a x^{7/2}} \]

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(5*a*x^(7/2)) + (8*b*Sqrt[a*x^2 + b*x^3])/(15*a^2*x^(5/
2)) - (16*b^2*Sqrt[a*x^2 + b*x^3])/(15*a^3*x^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.200527, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{16 b^2 \sqrt{a x^2+b x^3}}{15 a^3 x^{3/2}}+\frac{8 b \sqrt{a x^2+b x^3}}{15 a^2 x^{5/2}}-\frac{2 \sqrt{a x^2+b x^3}}{5 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*Sqrt[a*x^2 + b*x^3]),x]

[Out]

(-2*Sqrt[a*x^2 + b*x^3])/(5*a*x^(7/2)) + (8*b*Sqrt[a*x^2 + b*x^3])/(15*a^2*x^(5/
2)) - (16*b^2*Sqrt[a*x^2 + b*x^3])/(15*a^3*x^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.1327, size = 78, normalized size = 0.91 \[ - \frac{2 \sqrt{a x^{2} + b x^{3}}}{5 a x^{\frac{7}{2}}} + \frac{8 b \sqrt{a x^{2} + b x^{3}}}{15 a^{2} x^{\frac{5}{2}}} - \frac{16 b^{2} \sqrt{a x^{2} + b x^{3}}}{15 a^{3} x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(b*x**3+a*x**2)**(1/2),x)

[Out]

-2*sqrt(a*x**2 + b*x**3)/(5*a*x**(7/2)) + 8*b*sqrt(a*x**2 + b*x**3)/(15*a**2*x**
(5/2)) - 16*b**2*sqrt(a*x**2 + b*x**3)/(15*a**3*x**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0345092, size = 44, normalized size = 0.51 \[ -\frac{2 \sqrt{x^2 (a+b x)} \left (3 a^2-4 a b x+8 b^2 x^2\right )}{15 a^3 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*Sqrt[a*x^2 + b*x^3]),x]

[Out]

(-2*Sqrt[x^2*(a + b*x)]*(3*a^2 - 4*a*b*x + 8*b^2*x^2))/(15*a^3*x^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 46, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 8\,{b}^{2}{x}^{2}-4\,abx+3\,{a}^{2} \right ) }{15\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(b*x^3+a*x^2)^(1/2),x)

[Out]

-2/15*(b*x+a)*(8*b^2*x^2-4*a*b*x+3*a^2)/x^(3/2)/a^3/(b*x^3+a*x^2)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.41875, size = 62, normalized size = 0.72 \[ -\frac{2 \,{\left (\frac{15 \, \sqrt{b x + a} b^{2}}{\sqrt{x}} - \frac{10 \,{\left (b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} + \frac{3 \,{\left (b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{15 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a*x^2)*x^(5/2)),x, algorithm="maxima")

[Out]

-2/15*(15*sqrt(b*x + a)*b^2/sqrt(x) - 10*(b*x + a)^(3/2)*b/x^(3/2) + 3*(b*x + a)
^(5/2)/x^(5/2))/a^3

_______________________________________________________________________________________

Fricas [A]  time = 0.215156, size = 54, normalized size = 0.63 \[ -\frac{2 \,{\left (8 \, b^{2} x^{2} - 4 \, a b x + 3 \, a^{2}\right )} \sqrt{b x^{3} + a x^{2}}}{15 \, a^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a*x^2)*x^(5/2)),x, algorithm="fricas")

[Out]

-2/15*(8*b^2*x^2 - 4*a*b*x + 3*a^2)*sqrt(b*x^3 + a*x^2)/(a^3*x^(7/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**(5/2)/(b*x**3+a*x**2)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224477, size = 104, normalized size = 1.21 \[ \frac{32 \,{\left (10 \,{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{4} - 5 \, a{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} + a^{2}\right )} b^{\frac{5}{2}}}{15 \,{\left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a*x^2)*x^(5/2)),x, algorithm="giac")

[Out]

32/15*(10*(sqrt(b)*sqrt(x) - sqrt(b*x + a))^4 - 5*a*(sqrt(b)*sqrt(x) - sqrt(b*x
+ a))^2 + a^2)*b^(5/2)/((sqrt(b)*sqrt(x) - sqrt(b*x + a))^2 - a)^5