3.2817 \(\int \frac{1}{\left (\frac{c}{a+b x}\right )^{5/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 (a+b x)^3}{7 b c^2 \sqrt{\frac{c}{a+b x}}} \]

[Out]

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

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Rubi [A]  time = 0.0243568, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 (a+b x)^3}{7 b c^2 \sqrt{\frac{c}{a+b x}}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x))^(-5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.5368, size = 24, normalized size = 0.8 \[ \frac{2 \sqrt{\frac{c}{a + b x}} \left (a + b x\right )^{4}}{7 b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c/(a + b*x))*(a + b*x)**4/(7*b*c**3)

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Mathematica [A]  time = 0.0271589, size = 21, normalized size = 0.7 \[ \frac{2 c}{7 b \left (\frac{c}{a+b x}\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c)/(7*b*(c/(a + b*x))^(7/2))

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Maple [A]  time = 0.003, size = 22, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{7\,b} \left ({\frac{c}{bx+a}} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a))^(5/2),x)

[Out]

2/7*(b*x+a)/b/(c/(b*x+a))^(5/2)

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Maxima [A]  time = 1.37362, size = 23, normalized size = 0.77 \[ \frac{2 \, c}{7 \, b \left (\frac{c}{b x + a}\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*c/(b*(c/(b*x + a))^(7/2))

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Fricas [A]  time = 0.214024, size = 62, normalized size = 2.07 \[ \frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )}}{7 \, b c^{2} \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)/(b*c^2*sqrt(c/(b*x + a)))

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Sympy [A]  time = 6.42452, size = 49, normalized size = 1.63 \[ \begin{cases} \frac{2 a}{7 b c^{\frac{5}{2}} \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}} + \frac{2 x}{7 c^{\frac{5}{2}} \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\frac{c}{a}\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a/(7*b*c**(5/2)*(1/(a + b*x))**(5/2)) + 2*x/(7*c**(5/2)*(1/(a + b*x
))**(5/2)), Ne(b, 0)), (x/(c/a)**(5/2), True))

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GIAC/XCAS [A]  time = 0.217307, size = 35, normalized size = 1.17 \[ \frac{2 \,{\left (b x + a\right )}^{3}}{7 \, b c^{2} \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(b*x + a)^3/(b*c^2*sqrt(c/(b*x + a)))