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

Optimal. Leaf size=47 \[ a^4 c^3 \log (x)-2 a^3 b c^3 x+\frac {2}{3} a b^3 c^3 x^3-\frac {1}{4} b^4 c^3 x^4 \]

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \begin {gather*} -2 a^3 b c^3 x+a^4 c^3 \log (x)+\frac {2}{3} a b^3 c^3 x^3-\frac {1}{4} b^4 c^3 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x,x]

[Out]

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

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 48, normalized size = 1.02 \begin {gather*} c^3 \left (a^4 \log (-b x)+\frac {1}{12} \left (19 a^4-24 a^3 b x+8 a b^3 x^3-3 b^4 x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^3*((19*a^4 - 24*a^3*b*x + 8*a*b^3*x^3 - 3*b^4*x^4)/12 + a^4*Log[-(b*x)])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.39, size = 43, normalized size = 0.91 \begin {gather*} -\frac {1}{4} \, b^{4} c^{3} x^{4} + \frac {2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.07, size = 44, normalized size = 0.94 \begin {gather*} -\frac {1}{4} \, b^{4} c^{3} x^{4} + \frac {2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 44, normalized size = 0.94 \begin {gather*} -\frac {b^{4} c^{3} x^{4}}{4}+\frac {2 a \,b^{3} c^{3} x^{3}}{3}+a^{4} c^{3} \ln \relax (x )-2 a^{3} b \,c^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.01, size = 43, normalized size = 0.91 \begin {gather*} -\frac {1}{4} \, b^{4} c^{3} x^{4} + \frac {2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 43, normalized size = 0.91 \begin {gather*} a^4\,c^3\,\ln \relax (x)-\frac {b^4\,c^3\,x^4}{4}+\frac {2\,a\,b^3\,c^3\,x^3}{3}-2\,a^3\,b\,c^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x,x)

[Out]

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

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sympy [A]  time = 0.13, size = 48, normalized size = 1.02 \begin {gather*} a^{4} c^{3} \log {\relax (x )} - 2 a^{3} b c^{3} x + \frac {2 a b^{3} c^{3} x^{3}}{3} - \frac {b^{4} c^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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