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 = 0.0536301, 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 \[ \frac{8 a^3 c^3 \log (a+b x)}{b}-4 a^2 c^3 x+\frac{c^3 (a-b x)^3}{3 b}+\frac{a c^3 (a-b x)^2}{b} \]

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

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Rubi in Sympy [A]  time = 14.9552, size = 53, normalized size = 0.87 \[ \frac{8 a^{3} c^{3} \log{\left (a + b x \right )}}{b} - 4 a^{2} c^{3} x + \frac{a c^{3} \left (a - b x\right )^{2}}{b} + \frac{c^{3} \left (a - b x\right )^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*c*x+a*c)**3/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.010585, 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 = 0.004, size = 49, normalized size = 0.8 \[ -{\frac{{c}^{3}{b}^{2}{x}^{3}}{3}}+2\,{c}^{3}b{x}^{2}a-7\,{a}^{2}{c}^{3}x+8\,{\frac{{a}^{3}{c}^{3}\ln \left ( bx+a \right ) }{b}} \]

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*b^2*x^3+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 = 1.3436, size = 65, normalized size = 1.07 \[ -\frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{3} x^{2} - 7 \, a^{2} c^{3} x + \frac{8 \, a^{3} c^{3} \log \left (b x + a\right )}{b} \]

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 = 0.217258, size = 70, normalized size = 1.15 \[ -\frac{b^{3} c^{3} x^{3} - 6 \, a b^{2} c^{3} x^{2} + 21 \, a^{2} b c^{3} x - 24 \, a^{3} c^{3} \log \left (b x + a\right )}{3 \, b} \]

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 = 1.28158, size = 49, normalized size = 0.8 \[ \frac{8 a^{3} c^{3} \log{\left (a + b x \right )}}{b} - 7 a^{2} c^{3} x + 2 a b c^{3} x^{2} - \frac{b^{2} c^{3} x^{3}}{3} \]

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/XCAS [A]  time = 0.205888, size = 80, normalized size = 1.31 \[ \frac{8 \, a^{3} c^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b} - \frac{b^{5} c^{3} x^{3} - 6 \, a b^{4} c^{3} x^{2} + 21 \, a^{2} b^{3} c^{3} x}{3 \, b^{3}} \]

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*ln(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