3.1025 \(\int \frac{(a+b x)^5}{(a c+b c x)^7} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{b c^7 (a+b x)} \]

[Out]

-(1/(b*c^7*(a + b*x)))

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Rubi [A]  time = 0.0041574, antiderivative size = 15, 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} \[ -\frac{1}{b c^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^7,x]

[Out]

-(1/(b*c^7*(a + b*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)^7} \, dx &=\frac{\int \frac{1}{(a+b x)^2} \, dx}{c^7}\\ &=-\frac{1}{b c^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0021952, size = 15, normalized size = 1. \[ -\frac{1}{b c^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^7,x]

[Out]

-(1/(b*c^7*(a + b*x)))

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Maple [A]  time = 0.002, size = 16, normalized size = 1.1 \begin{align*} -{\frac{1}{b{c}^{7} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/(b*c*x+a*c)^7,x)

[Out]

-1/b/c^7/(b*x+a)

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Maxima [A]  time = 1.03331, size = 26, normalized size = 1.73 \begin{align*} -\frac{1}{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)^7,x, algorithm="maxima")

[Out]

-1/(b^2*c^7*x + a*b*c^7)

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Fricas [A]  time = 1.50465, size = 35, normalized size = 2.33 \begin{align*} -\frac{1}{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)^7,x, algorithm="fricas")

[Out]

-1/(b^2*c^7*x + a*b*c^7)

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Sympy [A]  time = 0.42156, size = 17, normalized size = 1.13 \begin{align*} - \frac{1}{a b c^{7} + b^{2} c^{7} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**7,x)

[Out]

-1/(a*b*c**7 + b**2*c**7*x)

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Giac [A]  time = 1.05078, size = 20, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b x + a\right )} 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)^7,x, algorithm="giac")

[Out]

-1/((b*x + a)*b*c^7)