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

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

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Rubi [A]  time = 0.06, antiderivative size = 119, 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 {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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 119, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(a^7*c^6)/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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (a c-b c x)^6}{x^{13}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.23, size = 103, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.01, size = 103, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.01, size = 84, normalized size = 0.71 \begin {gather*} \left (-\frac {b^{7}}{5 x^{5}}+\frac {5 a \,b^{6}}{6 x^{6}}-\frac {9 a^{2} b^{5}}{7 x^{7}}+\frac {5 a^{3} b^{4}}{8 x^{8}}+\frac {5 a^{4} b^{3}}{9 x^{9}}-\frac {9 a^{5} b^{2}}{10 x^{10}}+\frac {5 a^{6} b}{11 x^{11}}-\frac {a^{7}}{12 x^{12}}\right ) c^{6} \end {gather*}

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

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maxima [A]  time = 1.11, size = 103, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.07, size = 103, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^7\,c^6}{12}-\frac {5\,a^6\,b\,c^6\,x}{11}+\frac {9\,a^5\,b^2\,c^6\,x^2}{10}-\frac {5\,a^4\,b^3\,c^6\,x^3}{9}-\frac {5\,a^3\,b^4\,c^6\,x^4}{8}+\frac {9\,a^2\,b^5\,c^6\,x^5}{7}-\frac {5\,a\,b^6\,c^6\,x^6}{6}+\frac {b^7\,c^6\,x^7}{5}}{x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.77, size = 110, normalized size = 0.92 \begin {gather*} \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 {gather*}

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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