3.1.38 \(\int \frac {(a+b x) (a c-b c x)^5}{x^7} \, dx\)

Optimal. Leaf size=82 \[ -\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \]

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Rubi [A]  time = 0.04, antiderivative size = 82, 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}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}+\frac {4 a^5 b c^5}{5 x^5}-\frac {a^6 c^5}{6 x^6}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

-(a^6*c^5)/(6*x^6) + (4*a^5*b*c^5)/(5*x^5) - (5*a^4*b^2*c^5)/(4*x^4) + (5*a^2*b^4*c^5)/(2*x^2) - (4*a*b^5*c^5)
/x - b^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^7} \, dx &=\int \left (\frac {a^6 c^5}{x^7}-\frac {4 a^5 b c^5}{x^6}+\frac {5 a^4 b^2 c^5}{x^5}-\frac {5 a^2 b^4 c^5}{x^3}+\frac {4 a b^5 c^5}{x^2}-\frac {b^6 c^5}{x}\right ) \, dx\\ &=-\frac {a^6 c^5}{6 x^6}+\frac {4 a^5 b c^5}{5 x^5}-\frac {5 a^4 b^2 c^5}{4 x^4}+\frac {5 a^2 b^4 c^5}{2 x^2}-\frac {4 a b^5 c^5}{x}-b^6 c^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 68, normalized size = 0.83 \begin {gather*} c^5 \left (-\frac {a^6}{6 x^6}+\frac {4 a^5 b}{5 x^5}-\frac {5 a^4 b^2}{4 x^4}+\frac {5 a^2 b^4}{2 x^2}-\frac {4 a b^5}{x}-b^6 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

c^5*(-1/6*a^6/x^6 + (4*a^5*b)/(5*x^5) - (5*a^4*b^2)/(4*x^4) + (5*a^2*b^4)/(2*x^2) - (4*a*b^5)/x - b^6*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^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^5)/x^7,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^5)/x^7, x]

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fricas [A]  time = 1.59, size = 77, normalized size = 0.94 \begin {gather*} -\frac {60 \, b^{6} c^{5} x^{6} \log \relax (x) + 240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{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^7,x, algorithm="fricas")

[Out]

-1/60*(60*b^6*c^5*x^6*log(x) + 240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x +
10*a^6*c^5)/x^6

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giac [A]  time = 1.07, size = 76, normalized size = 0.93 \begin {gather*} -b^{6} c^{5} \log \left ({\left | x \right |}\right ) - \frac {240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{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^7,x, algorithm="giac")

[Out]

-b^6*c^5*log(abs(x)) - 1/60*(240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x + 1
0*a^6*c^5)/x^6

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maple [A]  time = 0.01, size = 75, normalized size = 0.91 \begin {gather*} -b^{6} c^{5} \ln \relax (x )-\frac {4 a \,b^{5} c^{5}}{x}+\frac {5 a^{2} b^{4} c^{5}}{2 x^{2}}-\frac {5 a^{4} b^{2} c^{5}}{4 x^{4}}+\frac {4 a^{5} b \,c^{5}}{5 x^{5}}-\frac {a^{6} c^{5}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^7,x)

[Out]

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

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maxima [A]  time = 1.09, size = 75, normalized size = 0.91 \begin {gather*} -b^{6} c^{5} \log \relax (x) - \frac {240 \, a b^{5} c^{5} x^{5} - 150 \, a^{2} b^{4} c^{5} x^{4} + 75 \, a^{4} b^{2} c^{5} x^{2} - 48 \, a^{5} b c^{5} x + 10 \, a^{6} c^{5}}{60 \, x^{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^7,x, algorithm="maxima")

[Out]

-b^6*c^5*log(x) - 1/60*(240*a*b^5*c^5*x^5 - 150*a^2*b^4*c^5*x^4 + 75*a^4*b^2*c^5*x^2 - 48*a^5*b*c^5*x + 10*a^6
*c^5)/x^6

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mupad [B]  time = 0.06, size = 62, normalized size = 0.76 \begin {gather*} -\frac {c^5\,\left (10\,a^6+240\,a\,b^5\,x^5+75\,a^4\,b^2\,x^2-150\,a^2\,b^4\,x^4+60\,b^6\,x^6\,\ln \relax (x)-48\,a^5\,b\,x\right )}{60\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^7,x)

[Out]

-(c^5*(10*a^6 + 240*a*b^5*x^5 + 75*a^4*b^2*x^2 - 150*a^2*b^4*x^4 + 60*b^6*x^6*log(x) - 48*a^5*b*x))/(60*x^6)

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sympy [A]  time = 0.43, size = 80, normalized size = 0.98 \begin {gather*} - b^{6} c^{5} \log {\relax (x )} - \frac {10 a^{6} c^{5} - 48 a^{5} b c^{5} x + 75 a^{4} b^{2} c^{5} x^{2} - 150 a^{2} b^{4} c^{5} x^{4} + 240 a b^{5} c^{5} x^{5}}{60 x^{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**7,x)

[Out]

-b**6*c**5*log(x) - (10*a**6*c**5 - 48*a**5*b*c**5*x + 75*a**4*b**2*c**5*x**2 - 150*a**2*b**4*c**5*x**4 + 240*
a*b**5*c**5*x**5)/(60*x**6)

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