3.33 \(\int \frac{(a+b x) (a c-b c x)^5}{x^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac{5}{3} a^2 b^4 c^5 x^3+5 a^4 b^2 c^5 x-4 a^5 b c^5 \log (x)-\frac{a^6 c^5}{x}+a b^5 c^5 x^4-\frac{1}{5} b^6 c^5 x^5 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0355075, antiderivative size = 75, 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} \[ -\frac{5}{3} a^2 b^4 c^5 x^3+5 a^4 b^2 c^5 x-4 a^5 b c^5 \log (x)-\frac{a^6 c^5}{x}+a b^5 c^5 x^4-\frac{1}{5} b^6 c^5 x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^2,x]

[Out]

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

Mathematica [A]  time = 0.0071807, size = 61, normalized size = 0.81 \[ c^5 \left (-\frac{5}{3} a^2 b^4 x^3+5 a^4 b^2 x-4 a^5 b \log (x)-\frac{a^6}{x}+a b^5 x^4-\frac{b^6 x^5}{5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 72, normalized size = 1. \begin{align*} -{\frac{{a}^{6}{c}^{5}}{x}}+5\,{a}^{4}{b}^{2}{c}^{5}x-{\frac{5\,{a}^{2}{b}^{4}{c}^{5}{x}^{3}}{3}}+a{b}^{5}{c}^{5}{x}^{4}-{\frac{{b}^{6}{c}^{5}{x}^{5}}{5}}-4\,{a}^{5}b{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^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08676, size = 96, normalized size = 1.28 \begin{align*} -\frac{1}{5} \, b^{6} c^{5} x^{5} + a b^{5} c^{5} x^{4} - \frac{5}{3} \, a^{2} b^{4} c^{5} x^{3} + 5 \, a^{4} b^{2} c^{5} x - 4 \, a^{5} b c^{5} \log \left (x\right ) - \frac{a^{6} c^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.94164, size = 166, normalized size = 2.21 \begin{align*} -\frac{3 \, b^{6} c^{5} x^{6} - 15 \, a b^{5} c^{5} x^{5} + 25 \, a^{2} b^{4} c^{5} x^{4} - 75 \, a^{4} b^{2} c^{5} x^{2} + 60 \, a^{5} b c^{5} x \log \left (x\right ) + 15 \, a^{6} c^{5}}{15 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.357937, size = 75, normalized size = 1. \begin{align*} - \frac{a^{6} c^{5}}{x} - 4 a^{5} b c^{5} \log{\left (x \right )} + 5 a^{4} b^{2} c^{5} x - \frac{5 a^{2} b^{4} c^{5} x^{3}}{3} + a b^{5} c^{5} x^{4} - \frac{b^{6} c^{5} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21698, size = 97, normalized size = 1.29 \begin{align*} -\frac{1}{5} \, b^{6} c^{5} x^{5} + a b^{5} c^{5} x^{4} - \frac{5}{3} \, a^{2} b^{4} c^{5} x^{3} + 5 \, a^{4} b^{2} c^{5} x - 4 \, a^{5} b c^{5} \log \left ({\left | x \right |}\right ) - \frac{a^{6} c^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^2,x, algorithm="giac")

[Out]

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