3.1045 \(\int \frac{(a+b x)^2}{(a c-b c x)^4} \, dx\)

Optimal. Leaf size=28 \[ \frac{(a+b x)^3}{6 a b c^4 (a-b x)^3} \]

[Out]

(a + b*x)^3/(6*a*b*c^4*(a - b*x)^3)

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Rubi [A]  time = 0.0051261, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ \frac{(a+b x)^3}{6 a b c^4 (a-b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^4,x]

[Out]

(a + b*x)^3/(6*a*b*c^4*(a - b*x)^3)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0182353, size = 31, normalized size = 1.11 \[ -\frac{a^2+3 b^2 x^2}{3 b c^4 (b x-a)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^4,x]

[Out]

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

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Maple [A]  time = 0.004, size = 52, normalized size = 1.9 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{4\,{a}^{2}}{3\,b \left ( bx-a \right ) ^{3}}}-{\frac{1}{b \left ( bx-a \right ) }}-2\,{\frac{a}{b \left ( bx-a \right ) ^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/(-b*c*x+a*c)^4,x)

[Out]

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

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Maxima [B]  time = 1.0375, size = 81, normalized size = 2.89 \begin{align*} -\frac{3 \, b^{2} x^{2} + a^{2}}{3 \,{\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)^2/(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

-1/3*(3*b^2*x^2 + a^2)/(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 [B]  time = 1.55696, size = 116, normalized size = 4.14 \begin{align*} -\frac{3 \, b^{2} x^{2} + a^{2}}{3 \,{\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)^2/(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

-1/3*(3*b^2*x^2 + a^2)/(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 [B]  time = 0.507735, size = 61, normalized size = 2.18 \begin{align*} - \frac{a^{2} + 3 b^{2} x^{2}}{- 3 a^{3} b c^{4} + 9 a^{2} b^{2} c^{4} x - 9 a b^{3} c^{4} x^{2} + 3 b^{4} c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/(-b*c*x+a*c)**4,x)

[Out]

-(a**2 + 3*b**2*x**2)/(-3*a**3*b*c**4 + 9*a**2*b**2*c**4*x - 9*a*b**3*c**4*x**2 + 3*b**4*c**4*x**3)

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Giac [A]  time = 1.06528, size = 39, normalized size = 1.39 \begin{align*} -\frac{3 \, b^{2} x^{2} + a^{2}}{3 \,{\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)^2/(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

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