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

Optimal. Leaf size=80 \[ -\frac{5 a^4 b^2 c^5}{2 x^2}-5 a^2 b^4 c^5 \log (x)+\frac{4 a^5 b c^5}{3 x^3}-\frac{a^6 c^5}{4 x^4}+4 a b^5 c^5 x-\frac{1}{2} b^6 c^5 x^2 \]

[Out]

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

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Rubi [A]  time = 0.0349633, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {75} \[ -\frac{5 a^4 b^2 c^5}{2 x^2}-5 a^2 b^4 c^5 \log (x)+\frac{4 a^5 b c^5}{3 x^3}-\frac{a^6 c^5}{4 x^4}+4 a b^5 c^5 x-\frac{1}{2} b^6 c^5 x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0067089, size = 66, normalized size = 0.82 \[ c^5 \left (-\frac{5 a^4 b^2}{2 x^2}-5 a^2 b^4 \log (x)+\frac{4 a^5 b}{3 x^3}-\frac{a^6}{4 x^4}+4 a b^5 x-\frac{b^6 x^2}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 73, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}{c}^{5}}{4\,{x}^{4}}}+{\frac{4\,{a}^{5}b{c}^{5}}{3\,{x}^{3}}}-{\frac{5\,{a}^{4}{b}^{2}{c}^{5}}{2\,{x}^{2}}}+4\,a{b}^{5}{c}^{5}x-{\frac{{b}^{6}{c}^{5}{x}^{2}}{2}}-5\,{a}^{2}{b}^{4}{c}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0364, size = 99, normalized size = 1.24 \begin{align*} -\frac{1}{2} \, b^{6} c^{5} x^{2} + 4 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{5} \log \left (x\right ) - \frac{30 \, a^{4} b^{2} c^{5} x^{2} - 16 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^6*c^5*x^2 + 4*a*b^5*c^5*x - 5*a^2*b^4*c^5*log(x) - 1/12*(30*a^4*b^2*c^5*x^2 - 16*a^5*b*c^5*x + 3*a^6*c^
5)/x^4

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Fricas [A]  time = 1.81404, size = 167, normalized size = 2.09 \begin{align*} -\frac{6 \, b^{6} c^{5} x^{6} - 48 \, a b^{5} c^{5} x^{5} + 60 \, a^{2} b^{4} c^{5} x^{4} \log \left (x\right ) + 30 \, a^{4} b^{2} c^{5} x^{2} - 16 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*b^6*c^5*x^6 - 48*a*b^5*c^5*x^5 + 60*a^2*b^4*c^5*x^4*log(x) + 30*a^4*b^2*c^5*x^2 - 16*a^5*b*c^5*x + 3*
a^6*c^5)/x^4

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Sympy [A]  time = 0.484149, size = 78, normalized size = 0.98 \begin{align*} - 5 a^{2} b^{4} c^{5} \log{\left (x \right )} + 4 a b^{5} c^{5} x - \frac{b^{6} c^{5} x^{2}}{2} - \frac{3 a^{6} c^{5} - 16 a^{5} b c^{5} x + 30 a^{4} b^{2} c^{5} x^{2}}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*a**2*b**4*c**5*log(x) + 4*a*b**5*c**5*x - b**6*c**5*x**2/2 - (3*a**6*c**5 - 16*a**5*b*c**5*x + 30*a**4*b**2
*c**5*x**2)/(12*x**4)

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Giac [A]  time = 1.19449, size = 100, normalized size = 1.25 \begin{align*} -\frac{1}{2} \, b^{6} c^{5} x^{2} + 4 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{5} \log \left ({\left | x \right |}\right ) - \frac{30 \, a^{4} b^{2} c^{5} x^{2} - 16 \, a^{5} b c^{5} x + 3 \, a^{6} c^{5}}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^6*c^5*x^2 + 4*a*b^5*c^5*x - 5*a^2*b^4*c^5*log(abs(x)) - 1/12*(30*a^4*b^2*c^5*x^2 - 16*a^5*b*c^5*x + 3*a
^6*c^5)/x^4