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

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

[Out]

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

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Rubi [A]  time = 0.0225028, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a}{4 c^2 x^3 \sqrt{c x^2}}-\frac{b}{3 c^2 x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.21177, size = 37, normalized size = 0.9 \[ - \frac{a \sqrt{c x^{2}}}{4 c^{3} x^{5}} - \frac{b \sqrt{c x^{2}}}{3 c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0082818, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c x^2} (3 a+4 b x)}{12 c^3 x^5} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(3*a + 4*b*x))/(12*c^3*x^5)

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Maple [A]  time = 0.006, size = 19, normalized size = 0.5 \[ -{\frac{x \left ( 4\,bx+3\,a \right ) }{12} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(c*x^2)^(5/2),x)

[Out]

-1/12*x*(4*b*x+3*a)/(c*x^2)^(5/2)

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Maxima [A]  time = 1.32344, size = 31, normalized size = 0.76 \[ -\frac{b}{3 \, \left (c x^{2}\right )^{\frac{3}{2}} c} - \frac{a}{4 \, c^{\frac{5}{2}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b/((c*x^2)^(3/2)*c) - 1/4*a/(c^(5/2)*x^4)

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Fricas [A]  time = 0.208374, size = 31, normalized size = 0.76 \[ -\frac{\sqrt{c x^{2}}{\left (4 \, b x + 3 \, a\right )}}{12 \, c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(c*x^2)*(4*b*x + 3*a)/(c^3*x^5)

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Sympy [A]  time = 3.21976, size = 36, normalized size = 0.88 \[ - \frac{a x}{4 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b x^{2}}{3 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.519605, 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, algorithm="giac")

[Out]

sage0*x