3.2816 \(\int \frac{1}{\left (\frac{c}{a+b x}\right )^{3/2}} \, dx\)

Optimal. Leaf size=30 \[ \frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}} \]

[Out]

(2*(a + b*x)^2)/(5*b*c*Sqrt[c/(a + b*x)])

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Rubi [A]  time = 0.02392, 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 \[ \frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x))^(-3/2),x]

[Out]

(2*(a + b*x)^2)/(5*b*c*Sqrt[c/(a + b*x)])

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Rubi in Sympy [A]  time = 2.56321, size = 24, normalized size = 0.8 \[ \frac{2 \sqrt{\frac{c}{a + b x}} \left (a + b x\right )^{3}}{5 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c/(b*x+a))**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0221646, size = 21, normalized size = 0.7 \[ \frac{2 c}{5 b \left (\frac{c}{a+b x}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c/(a + b*x))^(-3/2),x]

[Out]

(2*c)/(5*b*(c/(a + b*x))^(5/2))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{5\,b} \left ({\frac{c}{bx+a}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a))^(3/2),x)

[Out]

2/5*(b*x+a)/b/(c/(b*x+a))^(3/2)

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Maxima [A]  time = 1.34105, size = 23, normalized size = 0.77 \[ \frac{2 \, c}{5 \, b \left (\frac{c}{b x + a}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a))^(-3/2),x, algorithm="maxima")

[Out]

2/5*c/(b*(c/(b*x + a))^(5/2))

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Fricas [A]  time = 0.215612, size = 47, normalized size = 1.57 \[ \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}}{5 \, b c \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a))^(-3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.11551, size = 49, normalized size = 1.63 \[ \begin{cases} \frac{2 a}{5 b c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}} + \frac{2 x}{5 c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\frac{c}{a}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c/(b*x+a))**(3/2),x)

[Out]

Piecewise((2*a/(5*b*c**(3/2)*(1/(a + b*x))**(3/2)) + 2*x/(5*c**(3/2)*(1/(a + b*x
))**(3/2)), Ne(b, 0)), (x/(c/a)**(3/2), True))

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GIAC/XCAS [A]  time = 0.217871, size = 35, normalized size = 1.17 \[ \frac{2 \,{\left (b x + a\right )}^{2}}{5 \, b c \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x + a))^(-3/2),x, algorithm="giac")

[Out]

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