3.2812 \(\int \left (\frac{c}{a+b x}\right )^{5/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0251999, 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 \[ -\frac{2 c^2 \sqrt{\frac{c}{a+b x}}}{3 b (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(c/(a + b*x))^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.59199, size = 24, normalized size = 0.8 \[ - \frac{2 c^{2} \sqrt{\frac{c}{a + b x}}}{3 b \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0194639, size = 21, normalized size = 0.7 \[ -\frac{2 c \left (\frac{c}{a+b x}\right )^{3/2}}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(c/(a + b*x))^(3/2))/(3*b)

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Maple [A]  time = 0.002, size = 22, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{3\,b} \left ({\frac{c}{bx+a}} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a))^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.37615, size = 23, normalized size = 0.77 \[ -\frac{2 \, c \left (\frac{c}{b x + a}\right )^{\frac{3}{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*c*(c/(b*x + a))^(3/2)/b

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Fricas [A]  time = 0.216045, size = 36, normalized size = 1.2 \[ -\frac{2 \, c^{2} \sqrt{\frac{c}{b x + a}}}{3 \,{\left (b^{2} x + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.79229, size = 51, normalized size = 1.7 \[ \begin{cases} - \frac{2 a c^{\frac{5}{2}} \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}}{3 b} - \frac{2 c^{\frac{5}{2}} x \left (\frac{1}{a + b x}\right )^{\frac{5}{2}}}{3} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a}\right )^{\frac{5}{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a))**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217741, size = 32, normalized size = 1.07 \[ -\frac{2 \, c^{4}{\rm sign}\left (b x + a\right )}{3 \,{\left (b c x + a c\right )}^{\frac{3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*c^4*sign(b*x + a)/((b*c*x + a*c)^(3/2)*b)