3.803 \(\int \frac{a+b x}{x^4 \left (c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}} \]

[Out]

-a/(8*c^2*x^7*Sqrt[c*x^2]) - b/(7*c^2*x^6*Sqrt[c*x^2])

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Rubi [A]  time = 0.0235344, 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 \[ -\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(x^4*(c*x^2)^(5/2)),x]

[Out]

-a/(8*c^2*x^7*Sqrt[c*x^2]) - b/(7*c^2*x^6*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 9.14298, size = 37, normalized size = 0.9 \[ - \frac{a \sqrt{c x^{2}}}{8 c^{3} x^{9}} - \frac{b \sqrt{c x^{2}}}{7 c^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/x**4/(c*x**2)**(5/2),x)

[Out]

-a*sqrt(c*x**2)/(8*c**3*x**9) - b*sqrt(c*x**2)/(7*c**3*x**8)

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Mathematica [A]  time = 0.012519, size = 24, normalized size = 0.59 \[ \frac{-7 a-8 b x}{56 x^3 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(x^4*(c*x^2)^(5/2)),x]

[Out]

(-7*a - 8*b*x)/(56*x^3*(c*x^2)^(5/2))

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Maple [A]  time = 0.006, size = 21, normalized size = 0.5 \[ -{\frac{8\,bx+7\,a}{56\,{x}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^4/(c*x^2)^(5/2),x)

[Out]

-1/56*(8*b*x+7*a)/x^3/(c*x^2)^(5/2)

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Maxima [A]  time = 1.34083, size = 26, normalized size = 0.63 \[ -\frac{b}{7 \, c^{\frac{5}{2}} x^{7}} - \frac{a}{8 \, c^{\frac{5}{2}} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((c*x^2)^(5/2)*x^4),x, algorithm="maxima")

[Out]

-1/7*b/(c^(5/2)*x^7) - 1/8*a/(c^(5/2)*x^8)

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Fricas [A]  time = 0.206376, size = 31, normalized size = 0.76 \[ -\frac{\sqrt{c x^{2}}{\left (8 \, b x + 7 \, a\right )}}{56 \, c^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/((c*x^2)^(5/2)*x^4),x, algorithm="fricas")

[Out]

-1/56*sqrt(c*x^2)*(8*b*x + 7*a)/(c^3*x^9)

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Sympy [A]  time = 6.92721, size = 37, normalized size = 0.9 \[ - \frac{a}{8 c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b}{7 c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/x**4/(c*x**2)**(5/2),x)

[Out]

-a/(8*c**(5/2)*x**3*(x**2)**(5/2)) - b/(7*c**(5/2)*x**2*(x**2)**(5/2))

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GIAC/XCAS [A]  time = 0.515954, 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^4),x, algorithm="giac")

[Out]

sage0*x