3.1450 \(\int \frac{(a+b x)^5}{(a c+b c x)^{9/2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{2 (a c+b c x)^{3/2}}{3 b c^6} \]

[Out]

(2*(a*c + b*c*x)^(3/2))/(3*b*c^6)

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Rubi [A]  time = 0.0133833, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (a c+b c x)^{3/2}}{3 b c^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + b*c*x)^(9/2),x]

[Out]

(2*(a*c + b*c*x)^(3/2))/(3*b*c^6)

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Rubi in Sympy [A]  time = 4.36494, size = 19, normalized size = 0.86 \[ \frac{2 \left (a c + b c x\right )^{\frac{3}{2}}}{3 b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(b*c*x+a*c)**(9/2),x)

[Out]

2*(a*c + b*c*x)**(3/2)/(3*b*c**6)

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Mathematica [A]  time = 0.0205586, size = 26, normalized size = 1.18 \[ \frac{2 (a+b x) \sqrt{c (a+b x)}}{3 b c^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + b*c*x)^(9/2),x]

[Out]

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

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Maple [A]  time = 0.004, size = 23, normalized size = 1.1 \[{\frac{2\, \left ( bx+a \right ) ^{6}}{3\,b} \left ( bcx+ac \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^(9/2),x)

[Out]

2/3*(b*x+a)^6/b/(b*c*x+a*c)^(9/2)

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Maxima [A]  time = 1.32261, size = 24, normalized size = 1.09 \[ \frac{2 \,{\left (b c x + a c\right )}^{\frac{3}{2}}}{3 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(9/2),x, algorithm="maxima")

[Out]

2/3*(b*c*x + a*c)^(3/2)/(b*c^6)

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Fricas [A]  time = 0.202029, size = 31, normalized size = 1.41 \[ \frac{2 \, \sqrt{b c x + a c}{\left (b x + a\right )}}{3 \, b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(9/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 11.6554, size = 53, normalized size = 2.41 \[ \begin{cases} \frac{2 a \sqrt{a c + b c x}}{3 b c^{5}} + \frac{2 x \sqrt{a c + b c x}}{3 c^{5}} & \text{for}\: b \neq 0 \\\frac{a^{5} x}{\left (a c\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(b*c*x+a*c)**(9/2),x)

[Out]

Piecewise((2*a*sqrt(a*c + b*c*x)/(3*b*c**5) + 2*x*sqrt(a*c + b*c*x)/(3*c**5), Ne
(b, 0)), (a**5*x/(a*c)**(9/2), True))

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GIAC/XCAS [A]  time = 0.22483, size = 73, normalized size = 3.32 \[ \frac{2 \,{\left (3 \, \sqrt{b c x + a c} a - \frac{3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}}{c}\right )}}{3 \, b c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(9/2),x, algorithm="giac")

[Out]

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