3.11.49 \(\int \frac {(a c-b c x)^3}{a+b x} \, dx\) [1049]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} \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} \end {gather*}

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 45

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*}

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Mathematica [A]
time = 0.00, size = 42, normalized size = 0.69 \begin {gather*} c^3 \left (-7 a^2 x+2 a b x^2-\frac {b^2 x^3}{3}+\frac {8 a^3 \log (a+b x)}{b}\right ) \end {gather*}

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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Mathics [A]
time = 1.79, size = 42, normalized size = 0.69 \begin {gather*} \frac {c^3 \left (24 a^3 \text {Log}\left [a+b x\right ]+b x \left (-21 a^2+6 a b x-b^2 x^2\right )\right )}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[1/(a + b*x)*(a*c - b*c*x)^3,x]')

[Out]

c ^ 3 (24 a ^ 3 Log[a + b x] + b x (-21 a ^ 2 + 6 a b x - b ^ 2 x ^ 2)) / (3 b)

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Maple [A]
time = 0.15, size = 41, normalized size = 0.67

method	result	size
default	\(c^{3} \left (-\frac {b^{2} x^{3}}{3}+2 a b \,x^{2}-7 a^{2} x +\frac {8 a^{3} \ln \left (b x +a \right )}{b}\right )\)	\(41\)
norman	\(-7 a^{2} c^{3} x -\frac {b^{2} c^{3} x^{3}}{3}+2 a \,c^{3} b \,x^{2}+\frac {8 a^{3} c^{3} \ln \left (b x +a \right )}{b}\)	\(49\)
risch	\(-7 a^{2} c^{3} x -\frac {b^{2} c^{3} x^{3}}{3}+2 a \,c^{3} b \,x^{2}+\frac {8 a^{3} c^{3} \ln \left (b x +a \right )}{b}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*c*x+a*c)^3/(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 48, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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 = 0.29, size = 52, normalized size = 0.85 \begin {gather*} -\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 {gather*}

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 = 0.10, size = 49, normalized size = 0.80 \begin {gather*} \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 {gather*}

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 = 0.00, size = 63, normalized size = 1.03 \begin {gather*} \frac {-\frac {1}{3} x^{3} b^{5} c^{3}+2 x^{2} b^{4} c^{3} a-7 x b^{3} c^{3} a^{2}}{b^{3}}+\frac {8 c^{3} a^{3} \ln \left |x b+a\right |}{b} \end {gather*}

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(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

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Mupad [B]
time = 0.05, size = 48, normalized size = 0.79 \begin {gather*} \frac {8\,a^3\,c^3\,\ln \left (a+b\,x\right )}{b}-\frac {b^2\,c^3\,x^3}{3}-7\,a^2\,c^3\,x+2\,a\,b\,c^3\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*c - b*c*x)^3/(a + b*x),x)

[Out]

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

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