3.557 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{13}} \, dx\)

Optimal. Leaf size=143 \[ -\frac {a^6 A}{12 x^{12}}-\frac {a^5 (a B+6 A b)}{11 x^{11}}-\frac {3 a^4 b (2 a B+5 A b)}{10 x^{10}}-\frac {5 a^3 b^2 (3 a B+4 A b)}{9 x^9}-\frac {5 a^2 b^3 (4 a B+3 A b)}{8 x^8}-\frac {b^5 (6 a B+A b)}{6 x^6}-\frac {3 a b^4 (5 a B+2 A b)}{7 x^7}-\frac {b^6 B}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac {5 a^3 b^2 (3 a B+4 A b)}{9 x^9}-\frac {5 a^2 b^3 (4 a B+3 A b)}{8 x^8}-\frac {a^5 (a B+6 A b)}{11 x^{11}}-\frac {3 a^4 b (2 a B+5 A b)}{10 x^{10}}-\frac {a^6 A}{12 x^{12}}-\frac {3 a b^4 (5 a B+2 A b)}{7 x^7}-\frac {b^5 (6 a B+A b)}{6 x^6}-\frac {b^6 B}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^13,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

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] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 126, normalized size = 0.88 \[ -\frac {210 a^6 (11 A+12 B x)+1512 a^5 b x (10 A+11 B x)+4620 a^4 b^2 x^2 (9 A+10 B x)+7700 a^3 b^3 x^3 (8 A+9 B x)+7425 a^2 b^4 x^4 (7 A+8 B x)+3960 a b^5 x^5 (6 A+7 B x)+924 b^6 x^6 (5 A+6 B x)}{27720 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^13,x]

[Out]

-1/27720*(924*b^6*x^6*(5*A + 6*B*x) + 3960*a*b^5*x^5*(6*A + 7*B*x) + 7425*a^2*b^4*x^4*(7*A + 8*B*x) + 7700*a^3
*b^3*x^3*(8*A + 9*B*x) + 4620*a^4*b^2*x^2*(9*A + 10*B*x) + 1512*a^5*b*x*(10*A + 11*B*x) + 210*a^6*(11*A + 12*B
*x))/x^12

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fricas [A]  time = 0.85, size = 147, normalized size = 1.03 \[ -\frac {5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{27720 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^13,x, algorithm="fricas")

[Out]

-1/27720*(5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 11880*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 1
7325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b + 5*A*a^4*b^2)*
x^2 + 2520*(B*a^6 + 6*A*a^5*b)*x)/x^12

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giac [A]  time = 0.16, size = 147, normalized size = 1.03 \[ -\frac {5544 \, B b^{6} x^{7} + 27720 \, B a b^{5} x^{6} + 4620 \, A b^{6} x^{6} + 59400 \, B a^{2} b^{4} x^{5} + 23760 \, A a b^{5} x^{5} + 69300 \, B a^{3} b^{3} x^{4} + 51975 \, A a^{2} b^{4} x^{4} + 46200 \, B a^{4} b^{2} x^{3} + 61600 \, A a^{3} b^{3} x^{3} + 16632 \, B a^{5} b x^{2} + 41580 \, A a^{4} b^{2} x^{2} + 2520 \, B a^{6} x + 15120 \, A a^{5} b x + 2310 \, A a^{6}}{27720 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^13,x, algorithm="giac")

[Out]

-1/27720*(5544*B*b^6*x^7 + 27720*B*a*b^5*x^6 + 4620*A*b^6*x^6 + 59400*B*a^2*b^4*x^5 + 23760*A*a*b^5*x^5 + 6930
0*B*a^3*b^3*x^4 + 51975*A*a^2*b^4*x^4 + 46200*B*a^4*b^2*x^3 + 61600*A*a^3*b^3*x^3 + 16632*B*a^5*b*x^2 + 41580*
A*a^4*b^2*x^2 + 2520*B*a^6*x + 15120*A*a^5*b*x + 2310*A*a^6)/x^12

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maple [A]  time = 0.05, size = 128, normalized size = 0.90 \[ -\frac {B \,b^{6}}{5 x^{5}}-\frac {\left (A b +6 B a \right ) b^{5}}{6 x^{6}}-\frac {3 \left (2 A b +5 B a \right ) a \,b^{4}}{7 x^{7}}-\frac {5 \left (3 A b +4 B a \right ) a^{2} b^{3}}{8 x^{8}}-\frac {5 \left (4 A b +3 B a \right ) a^{3} b^{2}}{9 x^{9}}-\frac {A \,a^{6}}{12 x^{12}}-\frac {3 \left (5 A b +2 B a \right ) a^{4} b}{10 x^{10}}-\frac {\left (6 A b +B a \right ) a^{5}}{11 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^13,x)

[Out]

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

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maxima [A]  time = 0.58, size = 147, normalized size = 1.03 \[ -\frac {5544 \, B b^{6} x^{7} + 2310 \, A a^{6} + 4620 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 11880 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 17325 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 15400 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 8316 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 2520 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{27720 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3/x^13,x, algorithm="maxima")

[Out]

-1/27720*(5544*B*b^6*x^7 + 2310*A*a^6 + 4620*(6*B*a*b^5 + A*b^6)*x^6 + 11880*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 1
7325*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 15400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 8316*(2*B*a^5*b + 5*A*a^4*b^2)*
x^2 + 2520*(B*a^6 + 6*A*a^5*b)*x)/x^12

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mupad [B]  time = 0.07, size = 142, normalized size = 0.99 \[ -\frac {x\,\left (\frac {B\,a^6}{11}+\frac {6\,A\,b\,a^5}{11}\right )+\frac {A\,a^6}{12}+x^2\,\left (\frac {3\,B\,a^5\,b}{5}+\frac {3\,A\,a^4\,b^2}{2}\right )+x^5\,\left (\frac {15\,B\,a^2\,b^4}{7}+\frac {6\,A\,a\,b^5}{7}\right )+x^6\,\left (\frac {A\,b^6}{6}+B\,a\,b^5\right )+x^4\,\left (\frac {5\,B\,a^3\,b^3}{2}+\frac {15\,A\,a^2\,b^4}{8}\right )+x^3\,\left (\frac {5\,B\,a^4\,b^2}{3}+\frac {20\,A\,a^3\,b^3}{9}\right )+\frac {B\,b^6\,x^7}{5}}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^3)/x^13,x)

[Out]

-(x*((B*a^6)/11 + (6*A*a^5*b)/11) + (A*a^6)/12 + x^2*((3*A*a^4*b^2)/2 + (3*B*a^5*b)/5) + x^5*((15*B*a^2*b^4)/7
+ (6*A*a*b^5)/7) + x^6*((A*b^6)/6 + B*a*b^5) + x^4*((15*A*a^2*b^4)/8 + (5*B*a^3*b^3)/2) + x^3*((20*A*a^3*b^3)
/9 + (5*B*a^4*b^2)/3) + (B*b^6*x^7)/5)/x^12

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sympy [A]  time = 13.84, size = 158, normalized size = 1.10 \[ \frac {- 2310 A a^{6} - 5544 B b^{6} x^{7} + x^{6} \left (- 4620 A b^{6} - 27720 B a b^{5}\right ) + x^{5} \left (- 23760 A a b^{5} - 59400 B a^{2} b^{4}\right ) + x^{4} \left (- 51975 A a^{2} b^{4} - 69300 B a^{3} b^{3}\right ) + x^{3} \left (- 61600 A a^{3} b^{3} - 46200 B a^{4} b^{2}\right ) + x^{2} \left (- 41580 A a^{4} b^{2} - 16632 B a^{5} b\right ) + x \left (- 15120 A a^{5} b - 2520 B a^{6}\right )}{27720 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3/x**13,x)

[Out]

(-2310*A*a**6 - 5544*B*b**6*x**7 + x**6*(-4620*A*b**6 - 27720*B*a*b**5) + x**5*(-23760*A*a*b**5 - 59400*B*a**2
*b**4) + x**4*(-51975*A*a**2*b**4 - 69300*B*a**3*b**3) + x**3*(-61600*A*a**3*b**3 - 46200*B*a**4*b**2) + x**2*
(-41580*A*a**4*b**2 - 16632*B*a**5*b) + x*(-15120*A*a**5*b - 2520*B*a**6))/(27720*x**12)

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