3.48 \(\int \frac{(a+b x) (a c-b c x)^6}{x^{12}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{a^5 b^2 c^6}{x^9}+\frac{5 a^4 b^3 c^6}{8 x^8}+\frac{5 a^3 b^4 c^6}{7 x^7}-\frac{3 a^2 b^5 c^6}{2 x^6}+\frac{a^6 b c^6}{2 x^{10}}-\frac{a^7 c^6}{11 x^{11}}+\frac{a b^6 c^6}{x^5}-\frac{b^7 c^6}{4 x^4} \]

[Out]

-(a^7*c^6)/(11*x^11) + (a^6*b*c^6)/(2*x^10) - (a^5*b^2*c^6)/x^9 + (5*a^4*b^3*c^6)/(8*x^8) + (5*a^3*b^4*c^6)/(7
*x^7) - (3*a^2*b^5*c^6)/(2*x^6) + (a*b^6*c^6)/x^5 - (b^7*c^6)/(4*x^4)

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Rubi [A]  time = 0.0535377, antiderivative size = 114, 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} \[ -\frac{a^5 b^2 c^6}{x^9}+\frac{5 a^4 b^3 c^6}{8 x^8}+\frac{5 a^3 b^4 c^6}{7 x^7}-\frac{3 a^2 b^5 c^6}{2 x^6}+\frac{a^6 b c^6}{2 x^{10}}-\frac{a^7 c^6}{11 x^{11}}+\frac{a b^6 c^6}{x^5}-\frac{b^7 c^6}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^12,x]

[Out]

-(a^7*c^6)/(11*x^11) + (a^6*b*c^6)/(2*x^10) - (a^5*b^2*c^6)/x^9 + (5*a^4*b^3*c^6)/(8*x^8) + (5*a^3*b^4*c^6)/(7
*x^7) - (3*a^2*b^5*c^6)/(2*x^6) + (a*b^6*c^6)/x^5 - (b^7*c^6)/(4*x^4)

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)^6}{x^{12}} \, dx &=\int \left (\frac{a^7 c^6}{x^{12}}-\frac{5 a^6 b c^6}{x^{11}}+\frac{9 a^5 b^2 c^6}{x^{10}}-\frac{5 a^4 b^3 c^6}{x^9}-\frac{5 a^3 b^4 c^6}{x^8}+\frac{9 a^2 b^5 c^6}{x^7}-\frac{5 a b^6 c^6}{x^6}+\frac{b^7 c^6}{x^5}\right ) \, dx\\ &=-\frac{a^7 c^6}{11 x^{11}}+\frac{a^6 b c^6}{2 x^{10}}-\frac{a^5 b^2 c^6}{x^9}+\frac{5 a^4 b^3 c^6}{8 x^8}+\frac{5 a^3 b^4 c^6}{7 x^7}-\frac{3 a^2 b^5 c^6}{2 x^6}+\frac{a b^6 c^6}{x^5}-\frac{b^7 c^6}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.007863, size = 114, normalized size = 1. \[ -\frac{a^5 b^2 c^6}{x^9}+\frac{5 a^4 b^3 c^6}{8 x^8}+\frac{5 a^3 b^4 c^6}{7 x^7}-\frac{3 a^2 b^5 c^6}{2 x^6}+\frac{a^6 b c^6}{2 x^{10}}-\frac{a^7 c^6}{11 x^{11}}+\frac{a b^6 c^6}{x^5}-\frac{b^7 c^6}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^12,x]

[Out]

-(a^7*c^6)/(11*x^11) + (a^6*b*c^6)/(2*x^10) - (a^5*b^2*c^6)/x^9 + (5*a^4*b^3*c^6)/(8*x^8) + (5*a^3*b^4*c^6)/(7
*x^7) - (3*a^2*b^5*c^6)/(2*x^6) + (a*b^6*c^6)/x^5 - (b^7*c^6)/(4*x^4)

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Maple [A]  time = 0.006, size = 83, normalized size = 0.7 \begin{align*}{c}^{6} \left ({\frac{a{b}^{6}}{{x}^{5}}}-{\frac{{a}^{7}}{11\,{x}^{11}}}-{\frac{{b}^{7}}{4\,{x}^{4}}}+{\frac{5\,{a}^{4}{b}^{3}}{8\,{x}^{8}}}-{\frac{3\,{a}^{2}{b}^{5}}{2\,{x}^{6}}}+{\frac{5\,{a}^{3}{b}^{4}}{7\,{x}^{7}}}-{\frac{{a}^{5}{b}^{2}}{{x}^{9}}}+{\frac{{a}^{6}b}{2\,{x}^{10}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^6/x^12,x)

[Out]

c^6*(a*b^6/x^5-1/11*a^7/x^11-1/4*b^7/x^4+5/8*a^4*b^3/x^8-3/2*a^2*b^5/x^6+5/7*a^3*b^4/x^7-a^5*b^2/x^9+1/2*a^6*b
/x^10)

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Maxima [A]  time = 1.06414, size = 139, normalized size = 1.22 \begin{align*} -\frac{154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^12,x, algorithm="maxima")

[Out]

-1/616*(154*b^7*c^6*x^7 - 616*a*b^6*c^6*x^6 + 924*a^2*b^5*c^6*x^5 - 440*a^3*b^4*c^6*x^4 - 385*a^4*b^3*c^6*x^3
+ 616*a^5*b^2*c^6*x^2 - 308*a^6*b*c^6*x + 56*a^7*c^6)/x^11

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Fricas [A]  time = 1.91935, size = 230, normalized size = 2.02 \begin{align*} -\frac{154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^12,x, algorithm="fricas")

[Out]

-1/616*(154*b^7*c^6*x^7 - 616*a*b^6*c^6*x^6 + 924*a^2*b^5*c^6*x^5 - 440*a^3*b^4*c^6*x^4 - 385*a^4*b^3*c^6*x^3
+ 616*a^5*b^2*c^6*x^2 - 308*a^6*b*c^6*x + 56*a^7*c^6)/x^11

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Sympy [A]  time = 0.972669, size = 112, normalized size = 0.98 \begin{align*} - \frac{56 a^{7} c^{6} - 308 a^{6} b c^{6} x + 616 a^{5} b^{2} c^{6} x^{2} - 385 a^{4} b^{3} c^{6} x^{3} - 440 a^{3} b^{4} c^{6} x^{4} + 924 a^{2} b^{5} c^{6} x^{5} - 616 a b^{6} c^{6} x^{6} + 154 b^{7} c^{6} x^{7}}{616 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**12,x)

[Out]

-(56*a**7*c**6 - 308*a**6*b*c**6*x + 616*a**5*b**2*c**6*x**2 - 385*a**4*b**3*c**6*x**3 - 440*a**3*b**4*c**6*x*
*4 + 924*a**2*b**5*c**6*x**5 - 616*a*b**6*c**6*x**6 + 154*b**7*c**6*x**7)/(616*x**11)

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Giac [A]  time = 1.18843, size = 139, normalized size = 1.22 \begin{align*} -\frac{154 \, b^{7} c^{6} x^{7} - 616 \, a b^{6} c^{6} x^{6} + 924 \, a^{2} b^{5} c^{6} x^{5} - 440 \, a^{3} b^{4} c^{6} x^{4} - 385 \, a^{4} b^{3} c^{6} x^{3} + 616 \, a^{5} b^{2} c^{6} x^{2} - 308 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{616 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^12,x, algorithm="giac")

[Out]

-1/616*(154*b^7*c^6*x^7 - 616*a*b^6*c^6*x^6 + 924*a^2*b^5*c^6*x^5 - 440*a^3*b^4*c^6*x^4 - 385*a^4*b^3*c^6*x^3
+ 616*a^5*b^2*c^6*x^2 - 308*a^6*b*c^6*x + 56*a^7*c^6)/x^11