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

Optimal. Leaf size=41 \[ -\frac{a}{5 c^2 x^4 \sqrt{c x^2}}-\frac{b}{4 c^2 x^3 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0222753, 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 \[ -\frac{a}{5 c^2 x^4 \sqrt{c x^2}}-\frac{b}{4 c^2 x^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.13288, size = 37, normalized size = 0.9 \[ - \frac{a \sqrt{c x^{2}}}{5 c^{3} x^{6}} - \frac{b \sqrt{c x^{2}}}{4 c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c*x**2)/(5*c**3*x**6) - b*sqrt(c*x**2)/(4*c**3*x**5)

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Mathematica [A]  time = 0.0135106, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c x^2} (4 a+5 b x)}{20 c^3 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(4*a + 5*b*x))/(20*c^3*x^6)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.4 \[ -{\frac{5\,bx+4\,a}{20} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x/(c*x^2)^(5/2),x)

[Out]

-1/20*(5*b*x+4*a)/(c*x^2)^(5/2)

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Maxima [A]  time = 1.33796, size = 26, normalized size = 0.63 \[ -\frac{b}{4 \, c^{\frac{5}{2}} x^{4}} - \frac{a}{5 \, c^{\frac{5}{2}} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*b/(c^(5/2)*x^4) - 1/5*a/(c^(5/2)*x^5)

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Fricas [A]  time = 0.208085, size = 31, normalized size = 0.76 \[ -\frac{\sqrt{c x^{2}}{\left (5 \, b x + 4 \, a\right )}}{20 \, c^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/20*sqrt(c*x^2)*(5*b*x + 4*a)/(c^3*x^6)

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Sympy [A]  time = 3.92021, size = 32, normalized size = 0.78 \[ - \frac{a}{5 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b x}{4 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(5*c**(5/2)*(x**2)**(5/2)) - b*x/(4*c**(5/2)*(x**2)**(5/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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