3.1.37 \(\int \frac {(a+b x) (a c-b c x)^5}{x^6} \, dx\)

Optimal. Leaf size=75 \[ -\frac {a^6 c^5}{5 x^5}+\frac {a^5 b c^5}{x^4}-\frac {5 a^4 b^2 c^5}{3 x^3}+\frac {5 a^2 b^4 c^5}{x}+4 a b^5 c^5 \log (x)-b^6 c^5 x \]

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Rubi [A]  time = 0.03, antiderivative size = 75, 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*} -\frac {5 a^4 b^2 c^5}{3 x^3}+\frac {5 a^2 b^4 c^5}{x}+\frac {a^5 b c^5}{x^4}-\frac {a^6 c^5}{5 x^5}+4 a b^5 c^5 \log (x)-b^6 c^5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^6,x]

[Out]

-(a^6*c^5)/(5*x^5) + (a^5*b*c^5)/x^4 - (5*a^4*b^2*c^5)/(3*x^3) + (5*a^2*b^4*c^5)/x - b^6*c^5*x + 4*a*b^5*c^5*L
og[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)^5}{x^6} \, dx &=\int \left (-b^6 c^5+\frac {a^6 c^5}{x^6}-\frac {4 a^5 b c^5}{x^5}+\frac {5 a^4 b^2 c^5}{x^4}-\frac {5 a^2 b^4 c^5}{x^2}+\frac {4 a b^5 c^5}{x}\right ) \, dx\\ &=-\frac {a^6 c^5}{5 x^5}+\frac {a^5 b c^5}{x^4}-\frac {5 a^4 b^2 c^5}{3 x^3}+\frac {5 a^2 b^4 c^5}{x}-b^6 c^5 x+4 a b^5 c^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 61, normalized size = 0.81 \begin {gather*} c^5 \left (-\frac {a^6}{5 x^5}+\frac {a^5 b}{x^4}-\frac {5 a^4 b^2}{3 x^3}+\frac {5 a^2 b^4}{x}+4 a b^5 \log (x)-b^6 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^6,x]

[Out]

c^5*(-1/5*a^6/x^5 + (a^5*b)/x^4 - (5*a^4*b^2)/(3*x^3) + (5*a^2*b^4)/x - b^6*x + 4*a*b^5*Log[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)^5}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^5)/x^6,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^5)/x^6, x]

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fricas [A]  time = 2.37, size = 77, normalized size = 1.03 \begin {gather*} -\frac {15 \, b^{6} c^{5} x^{6} - 60 \, a b^{5} c^{5} x^{5} \log \relax (x) - 75 \, a^{2} b^{4} c^{5} x^{4} + 25 \, a^{4} b^{2} c^{5} x^{2} - 15 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^6,x, algorithm="fricas")

[Out]

-1/15*(15*b^6*c^5*x^6 - 60*a*b^5*c^5*x^5*log(x) - 75*a^2*b^4*c^5*x^4 + 25*a^4*b^2*c^5*x^2 - 15*a^5*b*c^5*x + 3
*a^6*c^5)/x^5

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giac [A]  time = 0.96, size = 74, normalized size = 0.99 \begin {gather*} -b^{6} c^{5} x + 4 \, a b^{5} c^{5} \log \left ({\left | x \right |}\right ) + \frac {75 \, a^{2} b^{4} c^{5} x^{4} - 25 \, a^{4} b^{2} c^{5} x^{2} + 15 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^6,x, algorithm="giac")

[Out]

-b^6*c^5*x + 4*a*b^5*c^5*log(abs(x)) + 1/15*(75*a^2*b^4*c^5*x^4 - 25*a^4*b^2*c^5*x^2 + 15*a^5*b*c^5*x - 3*a^6*
c^5)/x^5

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maple [A]  time = 0.01, size = 72, normalized size = 0.96 \begin {gather*} 4 a \,b^{5} c^{5} \ln \relax (x )-b^{6} c^{5} x +\frac {5 a^{2} b^{4} c^{5}}{x}-\frac {5 a^{4} b^{2} c^{5}}{3 x^{3}}+\frac {a^{5} b \,c^{5}}{x^{4}}-\frac {a^{6} c^{5}}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^6,x)

[Out]

-1/5*a^6*c^5/x^5+a^5*b*c^5/x^4-5/3*a^4*b^2*c^5/x^3+5*a^2*b^4*c^5/x-b^6*c^5*x+4*a*b^5*c^5*ln(x)

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maxima [A]  time = 1.06, size = 73, normalized size = 0.97 \begin {gather*} -b^{6} c^{5} x + 4 \, a b^{5} c^{5} \log \relax (x) + \frac {75 \, a^{2} b^{4} c^{5} x^{4} - 25 \, a^{4} b^{2} c^{5} x^{2} + 15 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{15 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^6,x, algorithm="maxima")

[Out]

-b^6*c^5*x + 4*a*b^5*c^5*log(x) + 1/15*(75*a^2*b^4*c^5*x^4 - 25*a^4*b^2*c^5*x^2 + 15*a^5*b*c^5*x - 3*a^6*c^5)/
x^5

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mupad [B]  time = 0.05, size = 61, normalized size = 0.81 \begin {gather*} -\frac {c^5\,\left (\frac {a^6}{5}+b^6\,x^6+\frac {5\,a^4\,b^2\,x^2}{3}-5\,a^2\,b^4\,x^4-a^5\,b\,x-4\,a\,b^5\,x^5\,\ln \relax (x)\right )}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^6,x)

[Out]

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

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sympy [A]  time = 0.36, size = 76, normalized size = 1.01 \begin {gather*} 4 a b^{5} c^{5} \log {\relax (x )} - b^{6} c^{5} x - \frac {3 a^{6} c^{5} - 15 a^{5} b c^{5} x + 25 a^{4} b^{2} c^{5} x^{2} - 75 a^{2} b^{4} c^{5} x^{4}}{15 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**6,x)

[Out]

4*a*b**5*c**5*log(x) - b**6*c**5*x - (3*a**6*c**5 - 15*a**5*b*c**5*x + 25*a**4*b**2*c**5*x**2 - 75*a**2*b**4*c
**5*x**4)/(15*x**5)

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