3.5.83 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^{12}} \, dx\)

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

________________________________________________________________________________________

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} \begin {gather*} -\frac {5 a^3 b^2 (3 a B+4 A b)}{8 x^8}-\frac {5 a^2 b^3 (4 a B+3 A b)}{7 x^7}-\frac {a^5 (a B+6 A b)}{10 x^{10}}-\frac {a^4 b (2 a B+5 A b)}{3 x^9}-\frac {a^6 A}{11 x^{11}}-\frac {a b^4 (5 a B+2 A b)}{2 x^6}-\frac {b^5 (6 a B+A b)}{5 x^5}-\frac {b^6 B}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 126, normalized size = 0.88 \begin {gather*} -\frac {84 a^6 (10 A+11 B x)+616 a^5 b x (9 A+10 B x)+1925 a^4 b^2 x^2 (8 A+9 B x)+3300 a^3 b^3 x^3 (7 A+8 B x)+3300 a^2 b^4 x^4 (6 A+7 B x)+1848 a b^5 x^5 (5 A+6 B x)+462 b^6 x^6 (4 A+5 B x)}{9240 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]

[Out]

-1/9240*(462*b^6*x^6*(4*A + 5*B*x) + 1848*a*b^5*x^5*(5*A + 6*B*x) + 3300*a^2*b^4*x^4*(6*A + 7*B*x) + 3300*a^3*
b^3*x^3*(7*A + 8*B*x) + 1925*a^4*b^2*x^2*(8*A + 9*B*x) + 616*a^5*b*x*(9*A + 10*B*x) + 84*a^6*(10*A + 11*B*x))/
x^11

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^12, x]

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 147, normalized size = 1.03 \begin {gather*} -\frac {2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \end {gather*}

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^12,x, algorithm="fricas")

[Out]

-1/9240*(2310*B*b^6*x^7 + 840*A*a^6 + 1848*(6*B*a*b^5 + A*b^6)*x^6 + 4620*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 6600
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 5775*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3080*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 +
924*(B*a^6 + 6*A*a^5*b)*x)/x^11

________________________________________________________________________________________

giac [A]  time = 0.19, size = 147, normalized size = 1.03 \begin {gather*} -\frac {2310 \, B b^{6} x^{7} + 11088 \, B a b^{5} x^{6} + 1848 \, A b^{6} x^{6} + 23100 \, B a^{2} b^{4} x^{5} + 9240 \, A a b^{5} x^{5} + 26400 \, B a^{3} b^{3} x^{4} + 19800 \, A a^{2} b^{4} x^{4} + 17325 \, B a^{4} b^{2} x^{3} + 23100 \, A a^{3} b^{3} x^{3} + 6160 \, B a^{5} b x^{2} + 15400 \, A a^{4} b^{2} x^{2} + 924 \, B a^{6} x + 5544 \, A a^{5} b x + 840 \, A a^{6}}{9240 \, x^{11}} \end {gather*}

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^12,x, algorithm="giac")

[Out]

-1/9240*(2310*B*b^6*x^7 + 11088*B*a*b^5*x^6 + 1848*A*b^6*x^6 + 23100*B*a^2*b^4*x^5 + 9240*A*a*b^5*x^5 + 26400*
B*a^3*b^3*x^4 + 19800*A*a^2*b^4*x^4 + 17325*B*a^4*b^2*x^3 + 23100*A*a^3*b^3*x^3 + 6160*B*a^5*b*x^2 + 15400*A*a
^4*b^2*x^2 + 924*B*a^6*x + 5544*A*a^5*b*x + 840*A*a^6)/x^11

________________________________________________________________________________________

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

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^12,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 147, normalized size = 1.03 \begin {gather*} -\frac {2310 \, B b^{6} x^{7} + 840 \, A a^{6} + 1848 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 4620 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 6600 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 5775 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 3080 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 924 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{9240 \, x^{11}} \end {gather*}

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^12,x, algorithm="maxima")

[Out]

-1/9240*(2310*B*b^6*x^7 + 840*A*a^6 + 1848*(6*B*a*b^5 + A*b^6)*x^6 + 4620*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 6600
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 5775*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 3080*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 +
924*(B*a^6 + 6*A*a^5*b)*x)/x^11

________________________________________________________________________________________

mupad [B]  time = 1.08, size = 142, normalized size = 0.99 \begin {gather*} -\frac {x\,\left (\frac {B\,a^6}{10}+\frac {3\,A\,b\,a^5}{5}\right )+\frac {A\,a^6}{11}+x^5\,\left (\frac {5\,B\,a^2\,b^4}{2}+A\,a\,b^5\right )+x^2\,\left (\frac {2\,B\,a^5\,b}{3}+\frac {5\,A\,a^4\,b^2}{3}\right )+x^6\,\left (\frac {A\,b^6}{5}+\frac {6\,B\,a\,b^5}{5}\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{8}+\frac {5\,A\,a^3\,b^3}{2}\right )+x^4\,\left (\frac {20\,B\,a^3\,b^3}{7}+\frac {15\,A\,a^2\,b^4}{7}\right )+\frac {B\,b^6\,x^7}{4}}{x^{11}} \end {gather*}

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^12,x)

[Out]

-(x*((B*a^6)/10 + (3*A*a^5*b)/5) + (A*a^6)/11 + x^5*((5*B*a^2*b^4)/2 + A*a*b^5) + x^2*((5*A*a^4*b^2)/3 + (2*B*
a^5*b)/3) + x^6*((A*b^6)/5 + (6*B*a*b^5)/5) + x^3*((5*A*a^3*b^3)/2 + (15*B*a^4*b^2)/8) + x^4*((15*A*a^2*b^4)/7
+ (20*B*a^3*b^3)/7) + (B*b^6*x^7)/4)/x^11

________________________________________________________________________________________

sympy [A]  time = 11.11, size = 158, normalized size = 1.10 \begin {gather*} \frac {- 840 A a^{6} - 2310 B b^{6} x^{7} + x^{6} \left (- 1848 A b^{6} - 11088 B a b^{5}\right ) + x^{5} \left (- 9240 A a b^{5} - 23100 B a^{2} b^{4}\right ) + x^{4} \left (- 19800 A a^{2} b^{4} - 26400 B a^{3} b^{3}\right ) + x^{3} \left (- 23100 A a^{3} b^{3} - 17325 B a^{4} b^{2}\right ) + x^{2} \left (- 15400 A a^{4} b^{2} - 6160 B a^{5} b\right ) + x \left (- 5544 A a^{5} b - 924 B a^{6}\right )}{9240 x^{11}} \end {gather*}

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**12,x)

[Out]

(-840*A*a**6 - 2310*B*b**6*x**7 + x**6*(-1848*A*b**6 - 11088*B*a*b**5) + x**5*(-9240*A*a*b**5 - 23100*B*a**2*b
**4) + x**4*(-19800*A*a**2*b**4 - 26400*B*a**3*b**3) + x**3*(-23100*A*a**3*b**3 - 17325*B*a**4*b**2) + x**2*(-
15400*A*a**4*b**2 - 6160*B*a**5*b) + x*(-5544*A*a**5*b - 924*B*a**6))/(9240*x**11)

________________________________________________________________________________________