3.1047 \(\int \frac{(a+b x)^2}{(a c-b c x)^6} \, dx\)

Optimal. Leaf size=57 \[ \frac{4 a^2}{5 b c^6 (a-b x)^5}-\frac{a}{b c^6 (a-b x)^4}+\frac{1}{3 b c^6 (a-b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0248059, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{4 a^2}{5 b c^6 (a-b x)^5}-\frac{a}{b c^6 (a-b x)^4}+\frac{1}{3 b c^6 (a-b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^6,x]

[Out]

(4*a^2)/(5*b*c^6*(a - b*x)^5) - a/(b*c^6*(a - b*x)^4) + 1/(3*b*c^6*(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)^2}{(a c-b c x)^6} \, dx &=\int \left (\frac{4 a^2}{c^6 (a-b x)^6}-\frac{4 a}{c^6 (a-b x)^5}+\frac{1}{c^6 (a-b x)^4}\right ) \, dx\\ &=\frac{4 a^2}{5 b c^6 (a-b x)^5}-\frac{a}{b c^6 (a-b x)^4}+\frac{1}{3 b c^6 (a-b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0182162, size = 38, normalized size = 0.67 \[ -\frac{2 a^2+5 a b x+5 b^2 x^2}{15 b c^6 (b x-a)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^6,x]

[Out]

-(2*a^2 + 5*a*b*x + 5*b^2*x^2)/(15*b*c^6*(-a + b*x)^5)

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Maple [A]  time = 0.005, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{6}} \left ( -{\frac{a}{b \left ( bx-a \right ) ^{4}}}-{\frac{1}{3\,b \left ( bx-a \right ) ^{3}}}-{\frac{4\,{a}^{2}}{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)^2/(-b*c*x+a*c)^6,x)

[Out]

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

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Maxima [A]  time = 1.0339, size = 128, normalized size = 2.25 \begin{align*} -\frac{5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \,{\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)^2/(-b*c*x+a*c)^6,x, algorithm="maxima")

[Out]

-1/15*(5*b^2*x^2 + 5*a*b*x + 2*a^2)/(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 [A]  time = 1.50276, size = 190, normalized size = 3.33 \begin{align*} -\frac{5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \,{\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)^2/(-b*c*x+a*c)^6,x, algorithm="fricas")

[Out]

-1/15*(5*b^2*x^2 + 5*a*b*x + 2*a^2)/(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 = 0.651973, size = 100, normalized size = 1.75 \begin{align*} - \frac{2 a^{2} + 5 a b x + 5 b^{2} x^{2}}{- 15 a^{5} b c^{6} + 75 a^{4} b^{2} c^{6} x - 150 a^{3} b^{3} c^{6} x^{2} + 150 a^{2} b^{4} c^{6} x^{3} - 75 a b^{5} c^{6} x^{4} + 15 b^{6} c^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/(-b*c*x+a*c)**6,x)

[Out]

-(2*a**2 + 5*a*b*x + 5*b**2*x**2)/(-15*a**5*b*c**6 + 75*a**4*b**2*c**6*x - 150*a**3*b**3*c**6*x**2 + 150*a**2*
b**4*c**6*x**3 - 75*a*b**5*c**6*x**4 + 15*b**6*c**6*x**5)

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Giac [A]  time = 1.06136, size = 49, normalized size = 0.86 \begin{align*} -\frac{5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \,{\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)^2/(-b*c*x+a*c)^6,x, algorithm="giac")

[Out]

-1/15*(5*b^2*x^2 + 5*a*b*x + 2*a^2)/((b*x - a)^5*b*c^6)