3.1049 \(\int \frac{(a c-b c x)^3}{a+b x} \, dx\)

Optimal. Leaf size=61 \[ \frac{8 a^3 c^3 \log (a+b x)}{b}-4 a^2 c^3 x+\frac{c^3 (a-b x)^3}{3 b}+\frac{a c^3 (a-b x)^2}{b} \]

[Out]

-4*a^2*c^3*x + (a*c^3*(a - b*x)^2)/b + (c^3*(a - b*x)^3)/(3*b) + (8*a^3*c^3*Log[a + b*x])/b

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Rubi [A]  time = 0.0212865, antiderivative size = 61, 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} \[ \frac{8 a^3 c^3 \log (a+b x)}{b}-4 a^2 c^3 x+\frac{c^3 (a-b x)^3}{3 b}+\frac{a c^3 (a-b x)^2}{b} \]

Antiderivative was successfully verified.

[In]

Int[(a*c - b*c*x)^3/(a + b*x),x]

[Out]

-4*a^2*c^3*x + (a*c^3*(a - b*x)^2)/b + (c^3*(a - b*x)^3)/(3*b) + (8*a^3*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} \, dx &=\int \left (-4 a^2 c^3+\frac{8 a^3 c^3}{a+b x}-2 a c^2 (a c-b c x)-c (a c-b c x)^2\right ) \, dx\\ &=-4 a^2 c^3 x+\frac{a c^3 (a-b x)^2}{b}+\frac{c^3 (a-b x)^3}{3 b}+\frac{8 a^3 c^3 \log (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0056005, size = 42, normalized size = 0.69 \[ c^3 \left (\frac{8 a^3 \log (a+b x)}{b}-7 a^2 x+2 a b x^2-\frac{b^2 x^3}{3}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*(-7*a^2*x + 2*a*b*x^2 - (b^2*x^3)/3 + (8*a^3*Log[a + b*x])/b)

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Maple [A]  time = 0.002, size = 49, normalized size = 0.8 \begin{align*} -{\frac{{c}^{3}{x}^{3}{b}^{2}}{3}}+2\,{c}^{3}b{x}^{2}a-7\,{a}^{2}{c}^{3}x+8\,{\frac{{a}^{3}{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),x)

[Out]

-1/3*c^3*x^3*b^2+2*c^3*b*x^2*a-7*a^2*c^3*x+8*a^3*c^3*ln(b*x+a)/b

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Maxima [A]  time = 1.05212, size = 65, normalized size = 1.07 \begin{align*} -\frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{3} x^{2} - 7 \, a^{2} c^{3} x + \frac{8 \, a^{3} 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),x, algorithm="maxima")

[Out]

-1/3*b^2*c^3*x^3 + 2*a*b*c^3*x^2 - 7*a^2*c^3*x + 8*a^3*c^3*log(b*x + a)/b

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Fricas [A]  time = 1.56054, size = 112, normalized size = 1.84 \begin{align*} -\frac{b^{3} c^{3} x^{3} - 6 \, a b^{2} c^{3} x^{2} + 21 \, a^{2} b c^{3} x - 24 \, a^{3} c^{3} \log \left (b x + a\right )}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b^3*c^3*x^3 - 6*a*b^2*c^3*x^2 + 21*a^2*b*c^3*x - 24*a^3*c^3*log(b*x + a))/b

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Sympy [A]  time = 0.317511, size = 49, normalized size = 0.8 \begin{align*} \frac{8 a^{3} c^{3} \log{\left (a + b x \right )}}{b} - 7 a^{2} c^{3} x + 2 a b c^{3} x^{2} - \frac{b^{2} c^{3} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a**3*c**3*log(a + b*x)/b - 7*a**2*c**3*x + 2*a*b*c**3*x**2 - b**2*c**3*x**3/3

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Giac [A]  time = 1.04772, size = 80, normalized size = 1.31 \begin{align*} \frac{8 \, a^{3} c^{3} \log \left ({\left | b x + a \right |}\right )}{b} - \frac{b^{5} c^{3} x^{3} - 6 \, a b^{4} c^{3} x^{2} + 21 \, a^{2} b^{3} c^{3} x}{3 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a^3*c^3*log(abs(b*x + a))/b - 1/3*(b^5*c^3*x^3 - 6*a*b^4*c^3*x^2 + 21*a^2*b^3*c^3*x)/b^3