3.1036 \(\int \frac{a+b x}{(a c-b c x)^5} \, dx\)

Optimal. Leaf size=38 \[ \frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3} \]

[Out]

a/(2*b*c^5*(a - b*x)^4) - 1/(3*b*c^5*(a - b*x)^3)

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Rubi [A]  time = 0.018712, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ \frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^5,x]

[Out]

a/(2*b*c^5*(a - b*x)^4) - 1/(3*b*c^5*(a - b*x)^3)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b x}{(a c-b c x)^5} \, dx &=\int \left (\frac{2 a}{c^5 (a-b x)^5}-\frac{1}{c^5 (a-b x)^4}\right ) \, dx\\ &=\frac{a}{2 b c^5 (a-b x)^4}-\frac{1}{3 b c^5 (a-b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0105647, size = 24, normalized size = 0.63 \[ \frac{a+2 b x}{6 b c^5 (a-b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + 2*b*x)/(6*b*c^5*(a - b*x)^4)

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Maple [A]  time = 0.005, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{2\,b \left ( bx-a \right ) ^{4}}}+{\frac{1}{3\,b \left ( bx-a \right ) ^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(1/2/b*a/(b*x-a)^4+1/3/b/(b*x-a)^3)

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Maxima [A]  time = 1.01628, size = 90, normalized size = 2.37 \begin{align*} \frac{2 \, b x + a}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*x + a)/(b^5*c^5*x^4 - 4*a*b^4*c^5*x^3 + 6*a^2*b^3*c^5*x^2 - 4*a^3*b^2*c^5*x + a^4*b*c^5)

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Fricas [A]  time = 1.45573, size = 134, normalized size = 3.53 \begin{align*} \frac{2 \, b x + a}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b*x + a)/(b^5*c^5*x^4 - 4*a*b^4*c^5*x^3 + 6*a^2*b^3*c^5*x^2 - 4*a^3*b^2*c^5*x + a^4*b*c^5)

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Sympy [B]  time = 0.52499, size = 70, normalized size = 1.84 \begin{align*} \frac{a + 2 b x}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a + 2*b*x)/(6*a**4*b*c**5 - 24*a**3*b**2*c**5*x + 36*a**2*b**3*c**5*x**2 - 24*a*b**4*c**5*x**3 + 6*b**5*c**5*
x**4)

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Giac [A]  time = 1.09206, size = 54, normalized size = 1.42 \begin{align*} \frac{a}{2 \,{\left (b c x - a c\right )}^{4} b c} + \frac{1}{3 \,{\left (b c x - a c\right )}^{3} b c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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