3.1452 \(\int \frac{(a+b x)^5}{(a c+b c x)^{13/2}} \, dx\)

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

[Out]

-2/(b*c^6*Sqrt[a*c + b*c*x])

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Rubi [A]  time = 0.0045802, 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} \[ -\frac{2}{b c^6 \sqrt{a c+b c x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^(13/2),x]

[Out]

-2/(b*c^6*Sqrt[a*c + b*c*x])

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)^{13/2}} \, dx &=\frac{\int \frac{1}{(a c+b c x)^{3/2}} \, dx}{c^5}\\ &=-\frac{2}{b c^6 \sqrt{a c+b c x}}\\ \end{align*}

Mathematica [A]  time = 0.0091496, size = 24, normalized size = 1.2 \[ -\frac{2 (a+b x)}{b c^5 (c (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^(13/2),x]

[Out]

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

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Maple [A]  time = 0.001, size = 23, normalized size = 1.2 \begin{align*} -2\,{\frac{ \left ( bx+a \right ) ^{6}}{b \left ( bcx+ac \right ) ^{13/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^(13/2),x)

[Out]

-2*(b*x+a)^6/b/(b*c*x+a*c)^(13/2)

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Maxima [A]  time = 0.955574, 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)^(13/2),x, algorithm="maxima")

[Out]

-2/(sqrt(b*c*x + a*c)*b*c^6)

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Fricas [A]  time = 1.97097, size = 59, normalized size = 2.95 \begin{align*} -\frac{2 \, \sqrt{b c x + a c}}{b^{2} c^{7} x + a b c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^(13/2),x, algorithm="fricas")

[Out]

-2*sqrt(b*c*x + a*c)/(b^2*c^7*x + a*b*c^7)

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Sympy [A]  time = 39.472, size = 48, normalized size = 2.4 \begin{align*} \begin{cases} - \frac{2 \sqrt{a c + b c x}}{a b c^{7} + b^{2} c^{7} x} & \text{for}\: a \neq 0 \\- \frac{2}{b^{\frac{3}{2}} c^{\frac{13}{2}} \sqrt{x}} & \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)**(13/2),x)

[Out]

Piecewise((-2*sqrt(a*c + b*c*x)/(a*b*c**7 + b**2*c**7*x), Ne(a, 0)), (-2/(b**(3/2)*c**(13/2)*sqrt(x)), True))

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Giac [A]  time = 1.05479, 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)^(13/2),x, algorithm="giac")

[Out]

-2/(sqrt(b*c*x + a*c)*b*c^6)