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

Optimal. Leaf size=23 \[ -\frac{2}{3 b d (d (a+b x)+c)^{3/2}} \]

[Out]

-2/(3*b*d*(c + d*(a + b*x))^(3/2))

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Rubi [A]  time = 0.0241968, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2}{3 b d (d (a+b x)+c)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*(a + b*x))^(-5/2),x]

[Out]

-2/(3*b*d*(c + d*(a + b*x))^(3/2))

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Rubi in Sympy [A]  time = 2.13926, size = 19, normalized size = 0.83 \[ - \frac{2}{3 b d \left (c + d \left (a + b x\right )\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(3*b*d*(c + d*(a + b*x))**(3/2))

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Mathematica [A]  time = 0.0172023, size = 23, normalized size = 1. \[ -\frac{2}{3 b d (d (a+b x)+c)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(3*b*d*(c + d*(a + b*x))^(3/2))

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Maple [A]  time = 0.004, size = 20, normalized size = 0.9 \[ -{\frac{2}{3\,db} \left ( bdx+ad+c \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+d*(b*x+a))^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.32207, size = 26, normalized size = 1.13 \[ -\frac{2}{3 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{3}{2}} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(((b*x + a)*d + c)^(3/2)*b*d)

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Fricas [A]  time = 0.200409, size = 46, normalized size = 2. \[ -\frac{2}{3 \,{\left (b^{2} d^{2} x + a b d^{2} + b c d\right )} \sqrt{b d x + a d + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 19.7246, size = 102, normalized size = 4.43 \[ \begin{cases} \frac{x}{c^{\frac{5}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x}{\left (a d + c\right )^{\frac{5}{2}}} & \text{for}\: b = 0 \\\frac{x}{c^{\frac{5}{2}}} & \text{for}\: d = 0 \\- \frac{2 \sqrt{a d + b d x + c}}{3 a^{2} b d^{3} + 6 a b^{2} d^{3} x + 6 a b c d^{2} + 3 b^{3} d^{3} x^{2} + 6 b^{2} c d^{2} x + 3 b c^{2} d} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((x/c**(5/2), Eq(b, 0) & Eq(d, 0)), (x/(a*d + c)**(5/2), Eq(b, 0)), (x/
c**(5/2), Eq(d, 0)), (-2*sqrt(a*d + b*d*x + c)/(3*a**2*b*d**3 + 6*a*b**2*d**3*x
+ 6*a*b*c*d**2 + 3*b**3*d**3*x**2 + 6*b**2*c*d**2*x + 3*b*c**2*d), True))

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GIAC/XCAS [A]  time = 0.209909, size = 26, normalized size = 1.13 \[ -\frac{2}{3 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{3}{2}} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3/(((b*x + a)*d + c)^(3/2)*b*d)