3.2813 \(\int \left (\frac{c}{a+b x}\right )^{3/2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

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Rubi [A]  time = 0.0190732, antiderivative size = 19, 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 \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x))^(3/2),x]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

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Rubi in Sympy [A]  time = 1.98011, size = 15, normalized size = 0.79 \[ - \frac{2 c \sqrt{\frac{c}{a + b x}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*sqrt(c/(a + b*x))/b

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Mathematica [A]  time = 0.00808533, size = 19, normalized size = 1. \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

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Maple [A]  time = 0.005, size = 22, normalized size = 1.2 \[ -2\,{\frac{bx+a}{b} \left ({\frac{c}{bx+a}} \right ) ^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a))^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.34978, size = 23, normalized size = 1.21 \[ -\frac{2 \, c \sqrt{\frac{c}{b x + a}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*sqrt(c/(b*x + a))/b

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Fricas [A]  time = 0.215668, size = 23, normalized size = 1.21 \[ -\frac{2 \, c \sqrt{\frac{c}{b x + a}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c*sqrt(c/(b*x + a))/b

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Sympy [A]  time = 2.61381, size = 48, normalized size = 2.53 \[ \begin{cases} - \frac{2 a c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}}{b} - 2 c^{\frac{3}{2}} x \left (\frac{1}{a + b x}\right )^{\frac{3}{2}} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a))**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220024, size = 32, normalized size = 1.68 \[ -\frac{2 \, c^{2}{\rm sign}\left (b x + a\right )}{\sqrt{b c x + a c} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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