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

Optimal. Leaf size=41 \[ -\frac{a}{6 c^2 x^5 \sqrt{c x^2}}-\frac{b}{5 c^2 x^4 \sqrt{c x^2}} \]

[Out]

-a/(6*c^2*x^5*Sqrt[c*x^2]) - b/(5*c^2*x^4*Sqrt[c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0232365, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a}{6 c^2 x^5 \sqrt{c x^2}}-\frac{b}{5 c^2 x^4 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-a/(6*c^2*x^5*Sqrt[c*x^2]) - b/(5*c^2*x^4*Sqrt[c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.21953, size = 37, normalized size = 0.9 \[ - \frac{a \sqrt{c x^{2}}}{6 c^{3} x^{7}} - \frac{b \sqrt{c x^{2}}}{5 c^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c*x**2)/(6*c**3*x**7) - b*sqrt(c*x**2)/(5*c**3*x**6)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0124851, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c x^2} (5 a+6 b x)}{30 c^3 x^7} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(5*a + 6*b*x))/(30*c^3*x^7)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 21, normalized size = 0.5 \[ -{\frac{6\,bx+5\,a}{30\,x} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(6*b*x+5*a)/x/(c*x^2)^(5/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.31975, size = 26, normalized size = 0.63 \[ -\frac{b}{5 \, c^{\frac{5}{2}} x^{5}} - \frac{a}{6 \, c^{\frac{5}{2}} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*b/(c^(5/2)*x^5) - 1/6*a/(c^(5/2)*x^6)

_______________________________________________________________________________________

Fricas [A]  time = 0.209571, size = 31, normalized size = 0.76 \[ -\frac{\sqrt{c x^{2}}{\left (6 \, b x + 5 \, a\right )}}{30 \, c^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*sqrt(c*x^2)*(6*b*x + 5*a)/(c^3*x^7)

_______________________________________________________________________________________

Sympy [A]  time = 5.10562, size = 32, normalized size = 0.78 \[ - \frac{a}{6 c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b}{5 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(6*c**(5/2)*x*(x**2)**(5/2)) - b/(5*c**(5/2)*(x**2)**(5/2))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.535343, size = 4, normalized size = 0.1 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x