3.1056 \(\int \frac{(a c-b c x)^3}{(a+b x)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{8 a^3 c^3}{b (a+b x)}-\frac{12 a^2 c^3 \log (a+b x)}{b}+5 a c^3 x-\frac{1}{2} b c^3 x^2 \]

[Out]

5*a*c^3*x - (b*c^3*x^2)/2 - (8*a^3*c^3)/(b*(a + b*x)) - (12*a^2*c^3*Log[a + b*x])/b

________________________________________________________________________________________

Rubi [A]  time = 0.0313632, antiderivative size = 54, 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} \[ -\frac{8 a^3 c^3}{b (a+b x)}-\frac{12 a^2 c^3 \log (a+b x)}{b}+5 a c^3 x-\frac{1}{2} b c^3 x^2 \]

Antiderivative was successfully verified.

[In]

Int[(a*c - b*c*x)^3/(a + b*x)^2,x]

[Out]

5*a*c^3*x - (b*c^3*x^2)/2 - (8*a^3*c^3)/(b*(a + b*x)) - (12*a^2*c^3*Log[a + b*x])/b

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 c-b c x)^3}{(a+b x)^2} \, dx &=\int \left (5 a c^3-b c^3 x+\frac{8 a^3 c^3}{(a+b x)^2}-\frac{12 a^2 c^3}{a+b x}\right ) \, dx\\ &=5 a c^3 x-\frac{1}{2} b c^3 x^2-\frac{8 a^3 c^3}{b (a+b x)}-\frac{12 a^2 c^3 \log (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0188758, size = 46, normalized size = 0.85 \[ c^3 \left (-\frac{8 a^3}{b (a+b x)}-\frac{12 a^2 \log (a+b x)}{b}+5 a x-\frac{b x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c - b*c*x)^3/(a + b*x)^2,x]

[Out]

c^3*(5*a*x - (b*x^2)/2 - (8*a^3)/(b*(a + b*x)) - (12*a^2*Log[a + b*x])/b)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 53, normalized size = 1. \begin{align*} 5\,a{c}^{3}x-{\frac{b{c}^{3}{x}^{2}}{2}}-8\,{\frac{{a}^{3}{c}^{3}}{b \left ( bx+a \right ) }}-12\,{\frac{{a}^{2}{c}^{3}\ln \left ( bx+a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*a*c^3*x-1/2*b*c^3*x^2-8*a^3*c^3/b/(b*x+a)-12*a^2*c^3*ln(b*x+a)/b

________________________________________________________________________________________

Maxima [A]  time = 1.01918, size = 72, normalized size = 1.33 \begin{align*} -\frac{1}{2} \, b c^{3} x^{2} - \frac{8 \, a^{3} c^{3}}{b^{2} x + a b} + 5 \, a c^{3} x - \frac{12 \, a^{2} c^{3} \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*c*x+a*c)^3/(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/2*b*c^3*x^2 - 8*a^3*c^3/(b^2*x + a*b) + 5*a*c^3*x - 12*a^2*c^3*log(b*x + a)/b

________________________________________________________________________________________

Fricas [A]  time = 1.51771, size = 167, normalized size = 3.09 \begin{align*} -\frac{b^{3} c^{3} x^{3} - 9 \, a b^{2} c^{3} x^{2} - 10 \, a^{2} b c^{3} x + 16 \, a^{3} c^{3} + 24 \,{\left (a^{2} b c^{3} x + a^{3} c^{3}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{2} x + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*c*x+a*c)^3/(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/2*(b^3*c^3*x^3 - 9*a*b^2*c^3*x^2 - 10*a^2*b*c^3*x + 16*a^3*c^3 + 24*(a^2*b*c^3*x + a^3*c^3)*log(b*x + a))/(
b^2*x + a*b)

________________________________________________________________________________________

Sympy [A]  time = 0.388527, size = 51, normalized size = 0.94 \begin{align*} - \frac{8 a^{3} c^{3}}{a b + b^{2} x} - \frac{12 a^{2} c^{3} \log{\left (a + b x \right )}}{b} + 5 a c^{3} x - \frac{b c^{3} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*c*x+a*c)**3/(b*x+a)**2,x)

[Out]

-8*a**3*c**3/(a*b + b**2*x) - 12*a**2*c**3*log(a + b*x)/b + 5*a*c**3*x - b*c**3*x**2/2

________________________________________________________________________________________

Giac [A]  time = 1.05334, size = 108, normalized size = 2. \begin{align*} \frac{12 \, a^{2} c^{3} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{8 \, a^{3} c^{3}}{{\left (b x + a\right )} b} + \frac{{\left (\frac{12 \, a c^{3}}{b x + a} - c^{3}\right )}{\left (b x + a\right )}^{2}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*c*x+a*c)^3/(b*x+a)^2,x, algorithm="giac")

[Out]

12*a^2*c^3*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b - 8*a^3*c^3/((b*x + a)*b) + 1/2*(12*a*c^3/(b*x + a) - c^3)
*(b*x + a)^2/b