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

Optimal. Leaf size=79 \[ a^6 c^5 \log (x)-4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6 \]

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Rubi [A]  time = 0.03, antiderivative size = 79, 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}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4-4 a^5 b c^5 x+a^6 c^5 \log (x)+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x,x]

[Out]

-4*a^5*b*c^5*x + (5*a^4*b^2*c^5*x^2)/2 - (5*a^2*b^4*c^5*x^4)/4 + (4*a*b^5*c^5*x^5)/5 - (b^6*c^5*x^6)/6 + a^6*c
^5*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)^5}{x} \, dx &=\int \left (-4 a^5 b c^5+\frac {a^6 c^5}{x}+5 a^4 b^2 c^5 x-5 a^2 b^4 c^5 x^3+4 a b^5 c^5 x^4-b^6 c^5 x^5\right ) \, dx\\ &=-4 a^5 b c^5 x+\frac {5}{2} a^4 b^2 c^5 x^2-\frac {5}{4} a^2 b^4 c^5 x^4+\frac {4}{5} a b^5 c^5 x^5-\frac {1}{6} b^6 c^5 x^6+a^6 c^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 75, normalized size = 0.95 \begin {gather*} c^5 \left (a^6 \log (-b x)+\frac {127 a^6}{60}-4 a^5 b x+\frac {5}{2} a^4 b^2 x^2-\frac {5}{4} a^2 b^4 x^4+\frac {4}{5} a b^5 x^5-\frac {b^6 x^6}{6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

c^5*((127*a^6)/60 - 4*a^5*b*x + (5*a^4*b^2*x^2)/2 - (5*a^2*b^4*x^4)/4 + (4*a*b^5*x^5)/5 - (b^6*x^6)/6 + a^6*Lo
g[-(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)^5}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.61, size = 71, normalized size = 0.90 \begin {gather*} -\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \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)^5/x,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.12, size = 72, normalized size = 0.91 \begin {gather*} -\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \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)^5/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 72, normalized size = 0.91 \begin {gather*} -\frac {b^{6} c^{5} x^{6}}{6}+\frac {4 a \,b^{5} c^{5} x^{5}}{5}-\frac {5 a^{2} b^{4} c^{5} x^{4}}{4}+\frac {5 a^{4} b^{2} c^{5} x^{2}}{2}+a^{6} c^{5} \ln \relax (x )-4 a^{5} b \,c^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.01, size = 71, normalized size = 0.90 \begin {gather*} -\frac {1}{6} \, b^{6} c^{5} x^{6} + \frac {4}{5} \, a b^{5} c^{5} x^{5} - \frac {5}{4} \, a^{2} b^{4} c^{5} x^{4} + \frac {5}{2} \, a^{4} b^{2} c^{5} x^{2} - 4 \, a^{5} b c^{5} x + a^{6} c^{5} \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)^5/x,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 71, normalized size = 0.90 \begin {gather*} a^6\,c^5\,\ln \relax (x)-\frac {b^6\,c^5\,x^6}{6}+\frac {4\,a\,b^5\,c^5\,x^5}{5}+\frac {5\,a^4\,b^2\,c^5\,x^2}{2}-\frac {5\,a^2\,b^4\,c^5\,x^4}{4}-4\,a^5\,b\,c^5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x,x)

[Out]

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

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sympy [A]  time = 0.16, size = 82, normalized size = 1.04 \begin {gather*} a^{6} c^{5} \log {\relax (x )} - 4 a^{5} b c^{5} x + \frac {5 a^{4} b^{2} c^{5} x^{2}}{2} - \frac {5 a^{2} b^{4} c^{5} x^{4}}{4} + \frac {4 a b^{5} c^{5} x^{5}}{5} - \frac {b^{6} c^{5} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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