3.1037 \(\int \frac{a+b x}{(a c-b c x)^6} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a}{5 b c^6 (a-b x)^5}-\frac{1}{4 b c^6 (a-b x)^4} \]

[Out]

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

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Rubi [A]  time = 0.0189225, 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} \[ \frac{2 a}{5 b c^6 (a-b x)^5}-\frac{1}{4 b c^6 (a-b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(a*c - b*c*x)^6,x]

[Out]

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

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)^6} \, dx &=\int \left (\frac{2 a}{c^6 (a-b x)^6}-\frac{1}{c^6 (a-b x)^5}\right ) \, dx\\ &=\frac{2 a}{5 b c^6 (a-b x)^5}-\frac{1}{4 b c^6 (a-b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0121554, size = 27, normalized size = 0.71 \[ -\frac{3 a+5 b x}{20 b c^6 (b x-a)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^6,x]

[Out]

-(3*a + 5*b*x)/(20*b*c^6*(-a + b*x)^5)

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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{6}} \left ( -{\frac{1}{4\,b \left ( bx-a \right ) ^{4}}}-{\frac{2\,a}{5\,b \left ( bx-a \right ) ^{5}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(-b*c*x+a*c)^6,x)

[Out]

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

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Maxima [B]  time = 1.00619, size = 113, normalized size = 2.97 \begin{align*} -\frac{5 \, b x + 3 \, a}{20 \,{\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^6,x, algorithm="maxima")

[Out]

-1/20*(5*b*x + 3*a)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3*b^3*c^6*x^2 + 5*a^4*b^2*c^6*x
- a^5*b*c^6)

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Fricas [B]  time = 1.44014, size = 169, normalized size = 4.45 \begin{align*} -\frac{5 \, b x + 3 \, a}{20 \,{\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^6,x, algorithm="fricas")

[Out]

-1/20*(5*b*x + 3*a)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3*b^3*c^6*x^2 + 5*a^4*b^2*c^6*x
- a^5*b*c^6)

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Sympy [B]  time = 0.617279, size = 88, normalized size = 2.32 \begin{align*} - \frac{3 a + 5 b x}{- 20 a^{5} b c^{6} + 100 a^{4} b^{2} c^{6} x - 200 a^{3} b^{3} c^{6} x^{2} + 200 a^{2} b^{4} c^{6} x^{3} - 100 a b^{5} c^{6} x^{4} + 20 b^{6} c^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)**6,x)

[Out]

-(3*a + 5*b*x)/(-20*a**5*b*c**6 + 100*a**4*b**2*c**6*x - 200*a**3*b**3*c**6*x**2 + 200*a**2*b**4*c**6*x**3 - 1
00*a*b**5*c**6*x**4 + 20*b**6*c**6*x**5)

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Giac [A]  time = 1.05226, size = 34, normalized size = 0.89 \begin{align*} -\frac{5 \, b x + 3 \, a}{20 \,{\left (b x - a\right )}^{5} b c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(-b*c*x+a*c)^6,x, algorithm="giac")

[Out]

-1/20*(5*b*x + 3*a)/((b*x - a)^5*b*c^6)