3.1439 \(\int \frac{1}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{2}{3 d (c+d x)^{3/2}} \]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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Rubi [A]  time = 0.0072233, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{2}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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Rubi in Sympy [A]  time = 1.29436, size = 14, normalized size = 0.88 \[ - \frac{2}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*d*(c + d*x)**(3/2))

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Mathematica [A]  time = 0.00546563, size = 16, normalized size = 1. \[ -\frac{2}{3 d (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(-5/2),x]

[Out]

-2/(3*d*(c + d*x)^(3/2))

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[ -{\frac{2}{3\,d} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x+c)^(5/2),x)

[Out]

-2/3/d/(d*x+c)^(3/2)

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Maxima [A]  time = 1.34382, size = 16, normalized size = 1. \[ -\frac{2}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/((d*x + c)^(3/2)*d)

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Fricas [A]  time = 0.20619, size = 27, normalized size = 1.69 \[ -\frac{2}{3 \,{\left (d^{2} x + c d\right )} \sqrt{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/((d^2*x + c*d)*sqrt(d*x + c))

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Sympy [A]  time = 0.046771, size = 14, normalized size = 0.88 \[ - \frac{2}{3 d \left (c + d x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x+c)**(5/2),x)

[Out]

-2/(3*d*(c + d*x)**(3/2))

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GIAC/XCAS [A]  time = 0.220043, size = 16, normalized size = 1. \[ -\frac{2}{3 \,{\left (d x + c\right )}^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/((d*x + c)^(3/2)*d)