3.49 \(\int \frac{(a+b x) (a c-b c x)^6}{x^{13}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{9 a^5 b^2 c^6}{10 x^{10}}+\frac{5 a^4 b^3 c^6}{9 x^9}+\frac{5 a^3 b^4 c^6}{8 x^8}-\frac{9 a^2 b^5 c^6}{7 x^7}+\frac{5 a^6 b c^6}{11 x^{11}}-\frac{a^7 c^6}{12 x^{12}}+\frac{5 a b^6 c^6}{6 x^6}-\frac{b^7 c^6}{5 x^5} \]

[Out]

-(a^7*c^6)/(12*x^12) + (5*a^6*b*c^6)/(11*x^11) - (9*a^5*b^2*c^6)/(10*x^10) + (5*a^4*b^3*c^6)/(9*x^9) + (5*a^3*
b^4*c^6)/(8*x^8) - (9*a^2*b^5*c^6)/(7*x^7) + (5*a*b^6*c^6)/(6*x^6) - (b^7*c^6)/(5*x^5)

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Rubi [A]  time = 0.0552477, antiderivative size = 119, 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} \[ -\frac{9 a^5 b^2 c^6}{10 x^{10}}+\frac{5 a^4 b^3 c^6}{9 x^9}+\frac{5 a^3 b^4 c^6}{8 x^8}-\frac{9 a^2 b^5 c^6}{7 x^7}+\frac{5 a^6 b c^6}{11 x^{11}}-\frac{a^7 c^6}{12 x^{12}}+\frac{5 a b^6 c^6}{6 x^6}-\frac{b^7 c^6}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^13,x]

[Out]

-(a^7*c^6)/(12*x^12) + (5*a^6*b*c^6)/(11*x^11) - (9*a^5*b^2*c^6)/(10*x^10) + (5*a^4*b^3*c^6)/(9*x^9) + (5*a^3*
b^4*c^6)/(8*x^8) - (9*a^2*b^5*c^6)/(7*x^7) + (5*a*b^6*c^6)/(6*x^6) - (b^7*c^6)/(5*x^5)

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^{13}} \, dx &=\int \left (\frac{a^7 c^6}{x^{13}}-\frac{5 a^6 b c^6}{x^{12}}+\frac{9 a^5 b^2 c^6}{x^{11}}-\frac{5 a^4 b^3 c^6}{x^{10}}-\frac{5 a^3 b^4 c^6}{x^9}+\frac{9 a^2 b^5 c^6}{x^8}-\frac{5 a b^6 c^6}{x^7}+\frac{b^7 c^6}{x^6}\right ) \, dx\\ &=-\frac{a^7 c^6}{12 x^{12}}+\frac{5 a^6 b c^6}{11 x^{11}}-\frac{9 a^5 b^2 c^6}{10 x^{10}}+\frac{5 a^4 b^3 c^6}{9 x^9}+\frac{5 a^3 b^4 c^6}{8 x^8}-\frac{9 a^2 b^5 c^6}{7 x^7}+\frac{5 a b^6 c^6}{6 x^6}-\frac{b^7 c^6}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0078419, size = 119, normalized size = 1. \[ -\frac{9 a^5 b^2 c^6}{10 x^{10}}+\frac{5 a^4 b^3 c^6}{9 x^9}+\frac{5 a^3 b^4 c^6}{8 x^8}-\frac{9 a^2 b^5 c^6}{7 x^7}+\frac{5 a^6 b c^6}{11 x^{11}}-\frac{a^7 c^6}{12 x^{12}}+\frac{5 a b^6 c^6}{6 x^6}-\frac{b^7 c^6}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^13,x]

[Out]

-(a^7*c^6)/(12*x^12) + (5*a^6*b*c^6)/(11*x^11) - (9*a^5*b^2*c^6)/(10*x^10) + (5*a^4*b^3*c^6)/(9*x^9) + (5*a^3*
b^4*c^6)/(8*x^8) - (9*a^2*b^5*c^6)/(7*x^7) + (5*a*b^6*c^6)/(6*x^6) - (b^7*c^6)/(5*x^5)

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Maple [A]  time = 0.007, size = 84, normalized size = 0.7 \begin{align*}{c}^{6} \left ( -{\frac{{b}^{7}}{5\,{x}^{5}}}-{\frac{{a}^{7}}{12\,{x}^{12}}}+{\frac{5\,{a}^{6}b}{11\,{x}^{11}}}+{\frac{5\,{a}^{3}{b}^{4}}{8\,{x}^{8}}}+{\frac{5\,a{b}^{6}}{6\,{x}^{6}}}-{\frac{9\,{a}^{2}{b}^{5}}{7\,{x}^{7}}}+{\frac{5\,{a}^{4}{b}^{3}}{9\,{x}^{9}}}-{\frac{9\,{a}^{5}{b}^{2}}{10\,{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^13,x)

[Out]

c^6*(-1/5*b^7/x^5-1/12*a^7/x^12+5/11*a^6*b/x^11+5/8*a^3*b^4/x^8+5/6*a*b^6/x^6-9/7*a^2*b^5/x^7+5/9*a^4*b^3/x^9-
9/10*a^5*b^2/x^10)

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Maxima [A]  time = 1.06568, size = 139, normalized size = 1.17 \begin{align*} -\frac{5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^13,x, algorithm="maxima")

[Out]

-1/27720*(5544*b^7*c^6*x^7 - 23100*a*b^6*c^6*x^6 + 35640*a^2*b^5*c^6*x^5 - 17325*a^3*b^4*c^6*x^4 - 15400*a^4*b
^3*c^6*x^3 + 24948*a^5*b^2*c^6*x^2 - 12600*a^6*b*c^6*x + 2310*a^7*c^6)/x^12

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Fricas [A]  time = 1.9853, size = 252, normalized size = 2.12 \begin{align*} -\frac{5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^13,x, algorithm="fricas")

[Out]

-1/27720*(5544*b^7*c^6*x^7 - 23100*a*b^6*c^6*x^6 + 35640*a^2*b^5*c^6*x^5 - 17325*a^3*b^4*c^6*x^4 - 15400*a^4*b
^3*c^6*x^3 + 24948*a^5*b^2*c^6*x^2 - 12600*a^6*b*c^6*x + 2310*a^7*c^6)/x^12

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Sympy [A]  time = 1.04905, size = 112, normalized size = 0.94 \begin{align*} - \frac{2310 a^{7} c^{6} - 12600 a^{6} b c^{6} x + 24948 a^{5} b^{2} c^{6} x^{2} - 15400 a^{4} b^{3} c^{6} x^{3} - 17325 a^{3} b^{4} c^{6} x^{4} + 35640 a^{2} b^{5} c^{6} x^{5} - 23100 a b^{6} c^{6} x^{6} + 5544 b^{7} c^{6} x^{7}}{27720 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**13,x)

[Out]

-(2310*a**7*c**6 - 12600*a**6*b*c**6*x + 24948*a**5*b**2*c**6*x**2 - 15400*a**4*b**3*c**6*x**3 - 17325*a**3*b*
*4*c**6*x**4 + 35640*a**2*b**5*c**6*x**5 - 23100*a*b**6*c**6*x**6 + 5544*b**7*c**6*x**7)/(27720*x**12)

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Giac [A]  time = 1.24482, size = 139, normalized size = 1.17 \begin{align*} -\frac{5544 \, b^{7} c^{6} x^{7} - 23100 \, a b^{6} c^{6} x^{6} + 35640 \, a^{2} b^{5} c^{6} x^{5} - 17325 \, a^{3} b^{4} c^{6} x^{4} - 15400 \, a^{4} b^{3} c^{6} x^{3} + 24948 \, a^{5} b^{2} c^{6} x^{2} - 12600 \, a^{6} b c^{6} x + 2310 \, a^{7} c^{6}}{27720 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^13,x, algorithm="giac")

[Out]

-1/27720*(5544*b^7*c^6*x^7 - 23100*a*b^6*c^6*x^6 + 35640*a^2*b^5*c^6*x^5 - 17325*a^3*b^4*c^6*x^4 - 15400*a^4*b
^3*c^6*x^3 + 24948*a^5*b^2*c^6*x^2 - 12600*a^6*b*c^6*x + 2310*a^7*c^6)/x^12