3.9 \(\int \frac {(a+b x) (a c-b c x)^3}{x^5} \, dx\)

Optimal. Leaf size=50 \[ -\frac {a^4 c^3}{4 x^4}+\frac {2 a^3 b c^3}{3 x^3}-\frac {2 a b^3 c^3}{x}-b^4 c^3 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 50, 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} \[ \frac {2 a^3 b c^3}{3 x^3}-\frac {a^4 c^3}{4 x^4}-\frac {2 a b^3 c^3}{x}-b^4 c^3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^5,x]

[Out]

-(a^4*c^3)/(4*x^4) + (2*a^3*b*c^3)/(3*x^3) - (2*a*b^3*c^3)/x - b^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^5} \, dx &=\int \left (\frac {a^4 c^3}{x^5}-\frac {2 a^3 b c^3}{x^4}+\frac {2 a b^3 c^3}{x^2}-\frac {b^4 c^3}{x}\right ) \, dx\\ &=-\frac {a^4 c^3}{4 x^4}+\frac {2 a^3 b c^3}{3 x^3}-\frac {2 a b^3 c^3}{x}-b^4 c^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 0.84 \[ c^3 \left (-\frac {a^4}{4 x^4}+\frac {2 a^3 b}{3 x^3}-\frac {2 a b^3}{x}-b^4 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^5,x]

[Out]

c^3*(-1/4*a^4/x^4 + (2*a^3*b)/(3*x^3) - (2*a*b^3)/x - b^4*Log[x])

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fricas [A]  time = 0.62, size = 49, normalized size = 0.98 \[ -\frac {12 \, b^{4} c^{3} x^{4} \log \relax (x) + 24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^5,x, algorithm="fricas")

[Out]

-1/12*(12*b^4*c^3*x^4*log(x) + 24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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giac [A]  time = 0.95, size = 48, normalized size = 0.96 \[ -b^{4} c^{3} \log \left ({\left | x \right |}\right ) - \frac {24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^5,x, algorithm="giac")

[Out]

-b^4*c^3*log(abs(x)) - 1/12*(24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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maple [A]  time = 0.01, size = 47, normalized size = 0.94 \[ -b^{4} c^{3} \ln \relax (x )-\frac {2 a \,b^{3} c^{3}}{x}+\frac {2 a^{3} b \,c^{3}}{3 x^{3}}-\frac {a^{4} c^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^5,x)

[Out]

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

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maxima [A]  time = 1.12, size = 47, normalized size = 0.94 \[ -b^{4} c^{3} \log \relax (x) - \frac {24 \, a b^{3} c^{3} x^{3} - 8 \, a^{3} b c^{3} x + 3 \, a^{4} c^{3}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^5,x, algorithm="maxima")

[Out]

-b^4*c^3*log(x) - 1/12*(24*a*b^3*c^3*x^3 - 8*a^3*b*c^3*x + 3*a^4*c^3)/x^4

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mupad [B]  time = 0.06, size = 40, normalized size = 0.80 \[ -\frac {c^3\,\left (3\,a^4+24\,a\,b^3\,x^3+12\,b^4\,x^4\,\ln \relax (x)-8\,a^3\,b\,x\right )}{12\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^5,x)

[Out]

-(c^3*(3*a^4 + 24*a*b^3*x^3 + 12*b^4*x^4*log(x) - 8*a^3*b*x))/(12*x^4)

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sympy [A]  time = 0.26, size = 49, normalized size = 0.98 \[ - b^{4} c^{3} \log {\relax (x )} - \frac {3 a^{4} c^{3} - 8 a^{3} b c^{3} x + 24 a b^{3} c^{3} x^{3}}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**5,x)

[Out]

-b**4*c**3*log(x) - (3*a**4*c**3 - 8*a**3*b*c**3*x + 24*a*b**3*c**3*x**3)/(12*x**4)

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