3.1046 \(\int \frac{(a+b x)^2}{(a c-b c x)^5} \, dx\)

Optimal. Leaf size=56 \[ \frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0237935, antiderivative size = 56, 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} \[ \frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.0119657, size = 35, normalized size = 0.62 \[ \frac{a^2+2 a b x+3 b^2 x^2}{6 b c^5 (a-b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/(a*c - b*c*x)^5,x]

[Out]

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

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Maple [A]  time = 0.004, size = 51, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{{a}^{2}}{b \left ( bx-a \right ) ^{4}}}+{\frac{4\,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)^2/(-b*c*x+a*c)^5,x)

[Out]

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

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Maxima [A]  time = 1.06078, size = 105, normalized size = 1.88 \begin{align*} \frac{3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{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)^2/(-b*c*x+a*c)^5,x, algorithm="maxima")

[Out]

1/6*(3*b^2*x^2 + 2*a*b*x + a^2)/(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 = 1.48695, size = 155, normalized size = 2.77 \begin{align*} \frac{3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{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)^2/(-b*c*x+a*c)^5,x, algorithm="fricas")

[Out]

1/6*(3*b^2*x^2 + 2*a*b*x + a^2)/(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 [A]  time = 0.584813, size = 82, normalized size = 1.46 \begin{align*} \frac{a^{2} + 2 a b x + 3 b^{2} x^{2}}{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)**2/(-b*c*x+a*c)**5,x)

[Out]

(a**2 + 2*a*b*x + 3*b**2*x**2)/(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 = 1.05837, size = 86, normalized size = 1.54 \begin{align*} \frac{\frac{6 \, a^{2}}{{\left (b c x - a c\right )}^{4} b} + \frac{8 \, a}{{\left (b c x - a c\right )}^{3} b c} + \frac{3}{{\left (b c x - a c\right )}^{2} b c^{2}}}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^5,x, algorithm="giac")

[Out]

1/6*(6*a^2/((b*c*x - a*c)^4*b) + 8*a/((b*c*x - a*c)^3*b*c) + 3/((b*c*x - a*c)^2*b*c^2))/c