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

Optimal. Leaf size=177 \[ \frac {3 b \log (x) (7 A b-2 a B)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8}+\frac {6 A b-a B}{a^7 x}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}-\frac {A}{2 a^6 x^2}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 177, 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, 77} \begin {gather*} \frac {6 A b-a B}{a^7 x}+\frac {5 b (3 A b-a B)}{a^7 (a+b x)}+\frac {b (5 A b-2 a B)}{a^6 (a+b x)^2}+\frac {b (2 A b-a B)}{a^5 (a+b x)^3}+\frac {b (3 A b-2 a B)}{4 a^4 (a+b x)^4}+\frac {b (A b-a B)}{5 a^3 (a+b x)^5}+\frac {3 b \log (x) (7 A b-2 a B)}{a^8}-\frac {3 b (7 A b-2 a B) \log (a+b x)}{a^8}-\frac {A}{2 a^6 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(2*a^6*x^2) + (6*A*b - a*B)/(a^7*x) + (b*(A*b - a*B))/(5*a^3*(a + b*x)^5) + (b*(3*A*b - 2*a*B))/(4*a^4*(a +
b*x)^4) + (b*(2*A*b - a*B))/(a^5*(a + b*x)^3) + (b*(5*A*b - 2*a*B))/(a^6*(a + b*x)^2) + (5*b*(3*A*b - a*B))/(
a^7*(a + b*x)) + (3*b*(7*A*b - 2*a*B)*Log[x])/a^8 - (3*b*(7*A*b - 2*a*B)*Log[a + b*x])/a^8

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 162, normalized size = 0.92 \begin {gather*} \frac {\frac {a \left (-10 a^6 (A+2 B x)+2 a^5 b x (35 A-137 B x)+7 a^4 b^2 x^2 (137 A-110 B x)+5 a^3 b^3 x^3 (539 A-188 B x)+10 a^2 b^4 x^4 (329 A-54 B x)+30 a b^5 x^5 (63 A-4 B x)+420 A b^6 x^6\right )}{x^2 (a+b x)^5}+60 b \log (x) (7 A b-2 a B)+60 b (2 a B-7 A b) \log (a+b x)}{20 a^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(420*A*b^6*x^6 + 5*a^3*b^3*x^3*(539*A - 188*B*x) + 2*a^5*b*x*(35*A - 137*B*x) + 7*a^4*b^2*x^2*(137*A - 110
*B*x) + 10*a^2*b^4*x^4*(329*A - 54*B*x) + 30*a*b^5*x^5*(63*A - 4*B*x) - 10*a^6*(A + 2*B*x)))/(x^2*(a + b*x)^5)
+ 60*b*(7*A*b - 2*a*B)*Log[x] + 60*b*(-7*A*b + 2*a*B)*Log[a + b*x])/(20*a^8)

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 484, normalized size = 2.73 \begin {gather*} -\frac {10 \, A a^{7} + 60 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x - 60 \, {\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \left (b x + a\right ) + 60 \, {\left ({\left (2 \, B a b^{6} - 7 \, A b^{7}\right )} x^{7} + 5 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 10 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 10 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 5 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2}\right )} \log \relax (x)}{20 \, {\left (a^{8} b^{5} x^{7} + 5 \, a^{9} b^{4} x^{6} + 10 \, a^{10} b^{3} x^{5} + 10 \, a^{11} b^{2} x^{4} + 5 \, a^{12} b x^{3} + a^{13} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(10*A*a^7 + 60*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 270*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 470*(2*B*a^4*b^3 -
7*A*a^3*b^4)*x^4 + 385*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + 137*(2*B*a^6*b - 7*A*a^5*b^2)*x^2 + 10*(2*B*a^7 - 7*A
*a^6*b)*x - 60*((2*B*a*b^6 - 7*A*b^7)*x^7 + 5*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 10*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x
^5 + 10*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 + 5*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + (2*B*a^6*b - 7*A*a^5*b^2)*x^2)*l
og(b*x + a) + 60*((2*B*a*b^6 - 7*A*b^7)*x^7 + 5*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 10*(2*B*a^3*b^4 - 7*A*a^2*b^5)
*x^5 + 10*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 + 5*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + (2*B*a^6*b - 7*A*a^5*b^2)*x^2)
*log(x))/(a^8*b^5*x^7 + 5*a^9*b^4*x^6 + 10*a^10*b^3*x^5 + 10*a^11*b^2*x^4 + 5*a^12*b*x^3 + a^13*x^2)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 205, normalized size = 1.16 \begin {gather*} -\frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {3 \, {\left (2 \, B a b^{2} - 7 \, A b^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{8} b} - \frac {10 \, A a^{7} + 60 \, {\left (2 \, B a^{2} b^{5} - 7 \, A a b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{4} b^{3} - 7 \, A a^{3} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{6} b - 7 \, A a^{5} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{7} - 7 \, A a^{6} b\right )} x}{20 \, {\left (b x + a\right )}^{5} a^{8} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(2*B*a*b - 7*A*b^2)*log(abs(x))/a^8 + 3*(2*B*a*b^2 - 7*A*b^3)*log(abs(b*x + a))/(a^8*b) - 1/20*(10*A*a^7 +
60*(2*B*a^2*b^5 - 7*A*a*b^6)*x^6 + 270*(2*B*a^3*b^4 - 7*A*a^2*b^5)*x^5 + 470*(2*B*a^4*b^3 - 7*A*a^3*b^4)*x^4 +
385*(2*B*a^5*b^2 - 7*A*a^4*b^3)*x^3 + 137*(2*B*a^6*b - 7*A*a^5*b^2)*x^2 + 10*(2*B*a^7 - 7*A*a^6*b)*x)/((b*x +
a)^5*a^8*x^2)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 228, normalized size = 1.29 \begin {gather*} \frac {A \,b^{2}}{5 \left (b x +a \right )^{5} a^{3}}-\frac {B b}{5 \left (b x +a \right )^{5} a^{2}}+\frac {3 A \,b^{2}}{4 \left (b x +a \right )^{4} a^{4}}-\frac {B b}{2 \left (b x +a \right )^{4} a^{3}}+\frac {2 A \,b^{2}}{\left (b x +a \right )^{3} a^{5}}-\frac {B b}{\left (b x +a \right )^{3} a^{4}}+\frac {5 A \,b^{2}}{\left (b x +a \right )^{2} a^{6}}-\frac {2 B b}{\left (b x +a \right )^{2} a^{5}}+\frac {15 A \,b^{2}}{\left (b x +a \right ) a^{7}}+\frac {21 A \,b^{2} \ln \relax (x )}{a^{8}}-\frac {21 A \,b^{2} \ln \left (b x +a \right )}{a^{8}}-\frac {5 B b}{\left (b x +a \right ) a^{6}}-\frac {6 B b \ln \relax (x )}{a^{7}}+\frac {6 B b \ln \left (b x +a \right )}{a^{7}}+\frac {6 A b}{a^{7} x}-\frac {B}{a^{6} x}-\frac {A}{2 a^{6} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^2/a^5/(b*x+a)^3*A-b/a^4/(b*x+a)^3*B+3/4*b^2/a^4/(b*x+a)^4*A-1/2*b/a^3/(b*x+a)^4*B+15*b^2/a^7/(b*x+a)*A-5*b
/a^6/(b*x+a)*B+5*b^2/a^6/(b*x+a)^2*A-2*b/a^5/(b*x+a)^2*B-21*b^2/a^8*ln(b*x+a)*A+6*b/a^7*ln(b*x+a)*B+1/5*b^2/a^
3/(b*x+a)^5*A-1/5*b/a^2/(b*x+a)^5*B-1/2*A/a^6/x^2+6/a^7/x*A*b-1/a^6/x*B+21*b^2/a^8*ln(x)*A-6*b/a^7*ln(x)*B

________________________________________________________________________________________

maxima [A]  time = 0.78, size = 242, normalized size = 1.37 \begin {gather*} -\frac {10 \, A a^{6} + 60 \, {\left (2 \, B a b^{5} - 7 \, A b^{6}\right )} x^{6} + 270 \, {\left (2 \, B a^{2} b^{4} - 7 \, A a b^{5}\right )} x^{5} + 470 \, {\left (2 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 385 \, {\left (2 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 137 \, {\left (2 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} + 10 \, {\left (2 \, B a^{6} - 7 \, A a^{5} b\right )} x}{20 \, {\left (a^{7} b^{5} x^{7} + 5 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{5} + 10 \, a^{10} b^{2} x^{4} + 5 \, a^{11} b x^{3} + a^{12} x^{2}\right )}} + \frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \left (b x + a\right )}{a^{8}} - \frac {3 \, {\left (2 \, B a b - 7 \, A b^{2}\right )} \log \relax (x)}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(10*A*a^6 + 60*(2*B*a*b^5 - 7*A*b^6)*x^6 + 270*(2*B*a^2*b^4 - 7*A*a*b^5)*x^5 + 470*(2*B*a^3*b^3 - 7*A*a^
2*b^4)*x^4 + 385*(2*B*a^4*b^2 - 7*A*a^3*b^3)*x^3 + 137*(2*B*a^5*b - 7*A*a^4*b^2)*x^2 + 10*(2*B*a^6 - 7*A*a^5*b
)*x)/(a^7*b^5*x^7 + 5*a^8*b^4*x^6 + 10*a^9*b^3*x^5 + 10*a^10*b^2*x^4 + 5*a^11*b*x^3 + a^12*x^2) + 3*(2*B*a*b -
7*A*b^2)*log(b*x + a)/a^8 - 3*(2*B*a*b - 7*A*b^2)*log(x)/a^8

________________________________________________________________________________________

mupad [B]  time = 1.21, size = 230, normalized size = 1.30 \begin {gather*} \frac {\frac {x\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{2\,a}+\frac {77\,b^2\,x^3\,\left (7\,A\,b-2\,B\,a\right )}{4\,a^4}+\frac {47\,b^3\,x^4\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^5}+\frac {27\,b^4\,x^5\,\left (7\,A\,b-2\,B\,a\right )}{2\,a^6}+\frac {3\,b^5\,x^6\,\left (7\,A\,b-2\,B\,a\right )}{a^7}+\frac {137\,b\,x^2\,\left (7\,A\,b-2\,B\,a\right )}{20\,a^3}}{a^5\,x^2+5\,a^4\,b\,x^3+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^5+5\,a\,b^4\,x^6+b^5\,x^7}-\frac {6\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (7\,A\,b-2\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (21\,A\,b^2-6\,B\,a\,b\right )}\right )\,\left (7\,A\,b-2\,B\,a\right )}{a^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(7*A*b - 2*B*a))/(2*a^2) - A/(2*a) + (77*b^2*x^3*(7*A*b - 2*B*a))/(4*a^4) + (47*b^3*x^4*(7*A*b - 2*B*a))/(
2*a^5) + (27*b^4*x^5*(7*A*b - 2*B*a))/(2*a^6) + (3*b^5*x^6*(7*A*b - 2*B*a))/a^7 + (137*b*x^2*(7*A*b - 2*B*a))/
(20*a^3))/(a^5*x^2 + b^5*x^7 + 5*a^4*b*x^3 + 5*a*b^4*x^6 + 10*a^3*b^2*x^4 + 10*a^2*b^3*x^5) - (6*b*atanh((3*b*
(7*A*b - 2*B*a)*(a + 2*b*x))/(a*(21*A*b^2 - 6*B*a*b)))*(7*A*b - 2*B*a))/a^8

________________________________________________________________________________________

sympy [A]  time = 1.19, size = 335, normalized size = 1.89 \begin {gather*} \frac {- 10 A a^{6} + x^{6} \left (420 A b^{6} - 120 B a b^{5}\right ) + x^{5} \left (1890 A a b^{5} - 540 B a^{2} b^{4}\right ) + x^{4} \left (3290 A a^{2} b^{4} - 940 B a^{3} b^{3}\right ) + x^{3} \left (2695 A a^{3} b^{3} - 770 B a^{4} b^{2}\right ) + x^{2} \left (959 A a^{4} b^{2} - 274 B a^{5} b\right ) + x \left (70 A a^{5} b - 20 B a^{6}\right )}{20 a^{12} x^{2} + 100 a^{11} b x^{3} + 200 a^{10} b^{2} x^{4} + 200 a^{9} b^{3} x^{5} + 100 a^{8} b^{4} x^{6} + 20 a^{7} b^{5} x^{7}} - \frac {3 b \left (- 7 A b + 2 B a\right ) \log {\left (x + \frac {- 21 A a b^{2} + 6 B a^{2} b - 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} + \frac {3 b \left (- 7 A b + 2 B a\right ) \log {\left (x + \frac {- 21 A a b^{2} + 6 B a^{2} b + 3 a b \left (- 7 A b + 2 B a\right )}{- 42 A b^{3} + 12 B a b^{2}} \right )}}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-10*A*a**6 + x**6*(420*A*b**6 - 120*B*a*b**5) + x**5*(1890*A*a*b**5 - 540*B*a**2*b**4) + x**4*(3290*A*a**2*b*
*4 - 940*B*a**3*b**3) + x**3*(2695*A*a**3*b**3 - 770*B*a**4*b**2) + x**2*(959*A*a**4*b**2 - 274*B*a**5*b) + x*
(70*A*a**5*b - 20*B*a**6))/(20*a**12*x**2 + 100*a**11*b*x**3 + 200*a**10*b**2*x**4 + 200*a**9*b**3*x**5 + 100*
a**8*b**4*x**6 + 20*a**7*b**5*x**7) - 3*b*(-7*A*b + 2*B*a)*log(x + (-21*A*a*b**2 + 6*B*a**2*b - 3*a*b*(-7*A*b
+ 2*B*a))/(-42*A*b**3 + 12*B*a*b**2))/a**8 + 3*b*(-7*A*b + 2*B*a)*log(x + (-21*A*a*b**2 + 6*B*a**2*b + 3*a*b*(
-7*A*b + 2*B*a))/(-42*A*b**3 + 12*B*a*b**2))/a**8

________________________________________________________________________________________