3.1026 \(\int \frac{(a+b x)^5}{(a c+b c x)^8} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{2 b c^8 (a+b x)^2} \]

[Out]

-1/(2*b*c^8*(a + b*x)^2)

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Rubi [A]  time = 0.0038258, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {21, 32} \[ -\frac{1}{2 b c^8 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^8,x]

[Out]

-1/(2*b*c^8*(a + b*x)^2)

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)^8} \, dx &=\frac{\int \frac{1}{(a+b x)^3} \, dx}{c^8}\\ &=-\frac{1}{2 b c^8 (a+b x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0041017, size = 17, normalized size = 1. \[ -\frac{1}{2 b c^8 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^8,x]

[Out]

-1/(2*b*c^8*(a + b*x)^2)

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,b{c}^{8} \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^8,x)

[Out]

-1/2/b/c^8/(b*x+a)^2

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Maxima [B]  time = 1.0553, size = 45, normalized size = 2.65 \begin{align*} -\frac{1}{2 \,{\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^8,x, algorithm="maxima")

[Out]

-1/2/(b^3*c^8*x^2 + 2*a*b^2*c^8*x + a^2*b*c^8)

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Fricas [B]  time = 1.45614, size = 65, normalized size = 3.82 \begin{align*} -\frac{1}{2 \,{\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^8,x, algorithm="fricas")

[Out]

-1/2/(b^3*c^8*x^2 + 2*a*b^2*c^8*x + a^2*b*c^8)

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Sympy [B]  time = 0.385585, size = 36, normalized size = 2.12 \begin{align*} - \frac{1}{2 a^{2} b c^{8} + 4 a b^{2} c^{8} x + 2 b^{3} c^{8} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**8,x)

[Out]

-1/(2*a**2*b*c**8 + 4*a*b**2*c**8*x + 2*b**3*c**8*x**2)

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Giac [A]  time = 1.04775, size = 20, normalized size = 1.18 \begin{align*} -\frac{1}{2 \,{\left (b x + a\right )}^{2} b c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^8,x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b*c^8)