3.1451 \(\int \frac{(a+b x)^5}{(a c+b c x)^{11/2}} \, dx\)

Optimal. Leaf size=20 \[ \frac{2 \sqrt{a c+b c x}}{b c^6} \]

[Out]

(2*Sqrt[a*c + b*c*x])/(b*c^6)

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Rubi [A]  time = 0.0132467, antiderivative size = 20, 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 \[ \frac{2 \sqrt{a c+b c x}}{b c^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a*c + b*c*x)^(11/2),x]

[Out]

(2*Sqrt[a*c + b*c*x])/(b*c^6)

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Rubi in Sympy [A]  time = 4.29449, size = 17, normalized size = 0.85 \[ \frac{2 \sqrt{a c + b c x}}{b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(b*c*x+a*c)**(11/2),x)

[Out]

2*sqrt(a*c + b*c*x)/(b*c**6)

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Mathematica [A]  time = 0.0120304, size = 24, normalized size = 1.2 \[ \frac{2 (a+b x)}{b c^5 \sqrt{c (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a*c + b*c*x)^(11/2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 23, normalized size = 1.2 \[ 2\,{\frac{ \left ( bx+a \right ) ^{6}}{b \left ( bcx+ac \right ) ^{11/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(b*c*x+a*c)^(11/2),x)

[Out]

2*(b*x+a)^6/b/(b*c*x+a*c)^(11/2)

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Maxima [A]  time = 1.33905, size = 24, normalized size = 1.2 \[ \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(11/2),x, algorithm="maxima")

[Out]

2*sqrt(b*c*x + a*c)/(b*c^6)

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Fricas [A]  time = 0.205514, size = 24, normalized size = 1.2 \[ \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(11/2),x, algorithm="fricas")

[Out]

2*sqrt(b*c*x + a*c)/(b*c^6)

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Sympy [A]  time = 22.1551, size = 29, normalized size = 1.45 \[ \begin{cases} \frac{2 \sqrt{a c + b c x}}{b c^{6}} & \text{for}\: b \neq 0 \\\frac{a^{5} x}{\left (a c\right )^{\frac{11}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(b*c*x+a*c)**(11/2),x)

[Out]

Piecewise((2*sqrt(a*c + b*c*x)/(b*c**6), Ne(b, 0)), (a**5*x/(a*c)**(11/2), True)
)

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GIAC/XCAS [A]  time = 0.226985, size = 24, normalized size = 1.2 \[ \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^5/(b*c*x + a*c)^(11/2),x, algorithm="giac")

[Out]

2*sqrt(b*c*x + a*c)/(b*c^6)