3.1035 \(\int \frac{a+b x}{(a c-b c x)^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0187421, 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} \[ \frac{2 a}{3 b c^4 (a-b x)^3}-\frac{1}{2 b c^4 (a-b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0113416, size = 25, normalized size = 0.66 \[ -\frac{a+3 b x}{6 b c^4 (b x-a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^4,x]

[Out]

-(a + 3*b*x)/(6*b*c^4*(-a + b*x)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-b*c*x+a*c)^4,x)

[Out]

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

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Maxima [A]  time = 1.00579, size = 73, normalized size = 1.92 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48177, size = 108, normalized size = 2.84 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.486193, size = 56, normalized size = 1.47 \begin{align*} - \frac{a + 3 b x}{- 6 a^{3} b c^{4} + 18 a^{2} b^{2} c^{4} x - 18 a b^{3} c^{4} x^{2} + 6 b^{4} c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)**4,x)

[Out]

-(a + 3*b*x)/(-6*a**3*b*c**4 + 18*a**2*b**2*c**4*x - 18*a*b**3*c**4*x**2 + 6*b**4*c**4*x**3)

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Giac [A]  time = 1.05926, size = 31, normalized size = 0.82 \begin{align*} -\frac{3 \, b x + a}{6 \,{\left (b x - a\right )}^{3} b c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

-1/6*(3*b*x + a)/((b*x - a)^3*b*c^4)