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 = 0.0041583, 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, Rules used = {21, 32} \[ \frac{2 \sqrt{a c+b c x}}{b c^6} \]

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)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{(a c+b c x)^{11/2}} \, dx &=\frac{\int \frac{1}{\sqrt{a c+b c x}} \, dx}{c^5}\\ &=\frac{2 \sqrt{a c+b c x}}{b c^6}\\ \end{align*}

Mathematica [A]  time = 0.0071777, 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 = 0.002, size = 23, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( bx+a \right ) ^{6}}{b \left ( bcx+ac \right ) ^{11/2}}} \end{align*}

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 = 0.963576, size = 24, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \end{align*}

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 = 2.02837, size = 39, normalized size = 1.95 \begin{align*} \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \end{align*}

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 = 14.4151, size = 29, normalized size = 1.45 \begin{align*} \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} \end{align*}

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 [A]  time = 1.05408, size = 24, normalized size = 1.2 \begin{align*} \frac{2 \, \sqrt{b c x + a c}}{b c^{6}} \end{align*}

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)