3.6.70 \(\int \frac {x^6 (A+B x)}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=171 \[ -\frac {a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac {a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac {a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac {5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac {5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}-\frac {3 a (2 A b-7 a B) \log (a+b x)}{b^8}+\frac {x (A b-6 a B)}{b^7}+\frac {B x^2}{2 b^6} \]

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Rubi [A]  time = 0.22, antiderivative size = 171, 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 {a^6 (A b-a B)}{5 b^8 (a+b x)^5}+\frac {a^5 (6 A b-7 a B)}{4 b^8 (a+b x)^4}-\frac {a^4 (5 A b-7 a B)}{b^8 (a+b x)^3}+\frac {5 a^3 (4 A b-7 a B)}{2 b^8 (a+b x)^2}-\frac {5 a^2 (3 A b-7 a B)}{b^8 (a+b x)}+\frac {x (A b-6 a B)}{b^7}-\frac {3 a (2 A b-7 a B) \log (a+b x)}{b^8}+\frac {B x^2}{2 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^6*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

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

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Mathematica [A]  time = 0.10, size = 151, normalized size = 0.88 \begin {gather*} \frac {\frac {4 a^6 (a B-A b)}{(a+b x)^5}+\frac {5 a^5 (6 A b-7 a B)}{(a+b x)^4}+\frac {20 a^4 (7 a B-5 A b)}{(a+b x)^3}-\frac {50 a^3 (7 a B-4 A b)}{(a+b x)^2}+\frac {100 a^2 (7 a B-3 A b)}{a+b x}+20 b x (A b-6 a B)+60 a (7 a B-2 A b) \log (a+b x)+10 b^2 B x^2}{20 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^6*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(20*b*(A*b - 6*a*B)*x + 10*b^2*B*x^2 + (4*a^6*(-(A*b) + a*B))/(a + b*x)^5 + (5*a^5*(6*A*b - 7*a*B))/(a + b*x)^
4 + (20*a^4*(-5*A*b + 7*a*B))/(a + b*x)^3 - (50*a^3*(-4*A*b + 7*a*B))/(a + b*x)^2 + (100*a^2*(-3*A*b + 7*a*B))
/(a + b*x) + 60*a*(-2*A*b + 7*a*B)*Log[a + b*x])/(20*b^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 (A+B x)}{\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[(x^6*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(x^6*(A + B*x))/(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.40, size = 349, normalized size = 2.04 \begin {gather*} \frac {10 \, B b^{7} x^{7} + 459 \, B a^{7} - 174 \, A a^{6} b - 10 \, {\left (7 \, B a b^{6} - 2 \, A b^{7}\right )} x^{6} - 100 \, {\left (5 \, B a^{2} b^{5} - A a b^{6}\right )} x^{5} - 100 \, {\left (4 \, B a^{3} b^{4} + A a^{2} b^{5}\right )} x^{4} + 100 \, {\left (13 \, B a^{4} b^{3} - 8 \, A a^{3} b^{4}\right )} x^{3} + 300 \, {\left (9 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3}\right )} x^{2} + 375 \, {\left (5 \, B a^{6} b - 2 \, A a^{5} b^{2}\right )} x + 60 \, {\left (7 \, B a^{7} - 2 \, A a^{6} b + {\left (7 \, B a^{2} b^{5} - 2 \, A a b^{6}\right )} x^{5} + 5 \, {\left (7 \, B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} x^{4} + 10 \, {\left (7 \, B a^{4} b^{3} - 2 \, A a^{3} b^{4}\right )} x^{3} + 10 \, {\left (7 \, B a^{5} b^{2} - 2 \, A a^{4} b^{3}\right )} x^{2} + 5 \, {\left (7 \, B a^{6} b - 2 \, A a^{5} b^{2}\right )} x\right )} \log \left (b x + a\right )}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

1/20*(10*B*b^7*x^7 + 459*B*a^7 - 174*A*a^6*b - 10*(7*B*a*b^6 - 2*A*b^7)*x^6 - 100*(5*B*a^2*b^5 - A*a*b^6)*x^5
- 100*(4*B*a^3*b^4 + A*a^2*b^5)*x^4 + 100*(13*B*a^4*b^3 - 8*A*a^3*b^4)*x^3 + 300*(9*B*a^5*b^2 - 4*A*a^4*b^3)*x
^2 + 375*(5*B*a^6*b - 2*A*a^5*b^2)*x + 60*(7*B*a^7 - 2*A*a^6*b + (7*B*a^2*b^5 - 2*A*a*b^6)*x^5 + 5*(7*B*a^3*b^
4 - 2*A*a^2*b^5)*x^4 + 10*(7*B*a^4*b^3 - 2*A*a^3*b^4)*x^3 + 10*(7*B*a^5*b^2 - 2*A*a^4*b^3)*x^2 + 5*(7*B*a^6*b
- 2*A*a^5*b^2)*x)*log(b*x + a))/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a
^5*b^8)

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giac [A]  time = 0.16, size = 172, normalized size = 1.01 \begin {gather*} \frac {3 \, {\left (7 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} + \frac {B b^{6} x^{2} - 12 \, B a b^{5} x + 2 \, A b^{6} x}{2 \, b^{12}} + \frac {459 \, B a^{7} - 174 \, A a^{6} b + 100 \, {\left (7 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 50 \, {\left (49 \, B a^{4} b^{3} - 20 \, A a^{3} b^{4}\right )} x^{3} + 10 \, {\left (329 \, B a^{5} b^{2} - 130 \, A a^{4} b^{3}\right )} x^{2} + 35 \, {\left (57 \, B a^{6} b - 22 \, A a^{5} b^{2}\right )} x}{20 \, {\left (b x + a\right )}^{5} b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

3*(7*B*a^2 - 2*A*a*b)*log(abs(b*x + a))/b^8 + 1/2*(B*b^6*x^2 - 12*B*a*b^5*x + 2*A*b^6*x)/b^12 + 1/20*(459*B*a^
7 - 174*A*a^6*b + 100*(7*B*a^3*b^4 - 3*A*a^2*b^5)*x^4 + 50*(49*B*a^4*b^3 - 20*A*a^3*b^4)*x^3 + 10*(329*B*a^5*b
^2 - 130*A*a^4*b^3)*x^2 + 35*(57*B*a^6*b - 22*A*a^5*b^2)*x)/((b*x + a)^5*b^8)

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maple [A]  time = 0.07, size = 213, normalized size = 1.25 \begin {gather*} -\frac {A \,a^{6}}{5 \left (b x +a \right )^{5} b^{7}}+\frac {B \,a^{7}}{5 \left (b x +a \right )^{5} b^{8}}+\frac {3 A \,a^{5}}{2 \left (b x +a \right )^{4} b^{7}}-\frac {7 B \,a^{6}}{4 \left (b x +a \right )^{4} b^{8}}-\frac {5 A \,a^{4}}{\left (b x +a \right )^{3} b^{7}}+\frac {7 B \,a^{5}}{\left (b x +a \right )^{3} b^{8}}+\frac {10 A \,a^{3}}{\left (b x +a \right )^{2} b^{7}}-\frac {35 B \,a^{4}}{2 \left (b x +a \right )^{2} b^{8}}+\frac {B \,x^{2}}{2 b^{6}}-\frac {15 A \,a^{2}}{\left (b x +a \right ) b^{7}}-\frac {6 A a \ln \left (b x +a \right )}{b^{7}}+\frac {A x}{b^{6}}+\frac {35 B \,a^{3}}{\left (b x +a \right ) b^{8}}+\frac {21 B \,a^{2} \ln \left (b x +a \right )}{b^{8}}-\frac {6 B a x}{b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/2*B*x^2/b^6+1/b^6*A*x-6/b^7*B*a*x-5*a^4/b^7/(b*x+a)^3*A+7*a^5/b^8/(b*x+a)^3*B-1/5*a^6/b^7/(b*x+a)^5*A+1/5*a^
7/b^8/(b*x+a)^5*B+10*a^3/b^7/(b*x+a)^2*A-35/2*a^4/b^8/(b*x+a)^2*B+3/2*a^5/b^7/(b*x+a)^4*A-7/4*a^6/b^8/(b*x+a)^
4*B-6*a/b^7*ln(b*x+a)*A+21*a^2/b^8*ln(b*x+a)*B-15*a^2/b^7/(b*x+a)*A+35*a^3/b^8/(b*x+a)*B

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maxima [A]  time = 0.70, size = 213, normalized size = 1.25 \begin {gather*} \frac {459 \, B a^{7} - 174 \, A a^{6} b + 100 \, {\left (7 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5}\right )} x^{4} + 50 \, {\left (49 \, B a^{4} b^{3} - 20 \, A a^{3} b^{4}\right )} x^{3} + 10 \, {\left (329 \, B a^{5} b^{2} - 130 \, A a^{4} b^{3}\right )} x^{2} + 35 \, {\left (57 \, B a^{6} b - 22 \, A a^{5} b^{2}\right )} x}{20 \, {\left (b^{13} x^{5} + 5 \, a b^{12} x^{4} + 10 \, a^{2} b^{11} x^{3} + 10 \, a^{3} b^{10} x^{2} + 5 \, a^{4} b^{9} x + a^{5} b^{8}\right )}} + \frac {B b x^{2} - 2 \, {\left (6 \, B a - A b\right )} x}{2 \, b^{7}} + \frac {3 \, {\left (7 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(B*x+A)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

1/20*(459*B*a^7 - 174*A*a^6*b + 100*(7*B*a^3*b^4 - 3*A*a^2*b^5)*x^4 + 50*(49*B*a^4*b^3 - 20*A*a^3*b^4)*x^3 + 1
0*(329*B*a^5*b^2 - 130*A*a^4*b^3)*x^2 + 35*(57*B*a^6*b - 22*A*a^5*b^2)*x)/(b^13*x^5 + 5*a*b^12*x^4 + 10*a^2*b^
11*x^3 + 10*a^3*b^10*x^2 + 5*a^4*b^9*x + a^5*b^8) + 1/2*(B*b*x^2 - 2*(6*B*a - A*b)*x)/b^7 + 3*(7*B*a^2 - 2*A*a
*b)*log(b*x + a)/b^8

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mupad [B]  time = 1.16, size = 210, normalized size = 1.23 \begin {gather*} x\,\left (\frac {A}{b^6}-\frac {6\,B\,a}{b^7}\right )-\frac {x^2\,\left (65\,A\,a^4\,b^2-\frac {329\,B\,a^5\,b}{2}\right )-x\,\left (\frac {399\,B\,a^6}{4}-\frac {77\,A\,a^5\,b}{2}\right )-\frac {3\,\left (153\,B\,a^7-58\,A\,a^6\,b\right )}{20\,b}+x^4\,\left (15\,A\,a^2\,b^4-35\,B\,a^3\,b^3\right )+x^3\,\left (50\,A\,a^3\,b^3-\frac {245\,B\,a^4\,b^2}{2}\right )}{a^5\,b^7+5\,a^4\,b^8\,x+10\,a^3\,b^9\,x^2+10\,a^2\,b^{10}\,x^3+5\,a\,b^{11}\,x^4+b^{12}\,x^5}+\frac {B\,x^2}{2\,b^6}+\frac {\ln \left (a+b\,x\right )\,\left (21\,B\,a^2-6\,A\,a\,b\right )}{b^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(A + B*x))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

x*(A/b^6 - (6*B*a)/b^7) - (x^2*(65*A*a^4*b^2 - (329*B*a^5*b)/2) - x*((399*B*a^6)/4 - (77*A*a^5*b)/2) - (3*(153
*B*a^7 - 58*A*a^6*b))/(20*b) + x^4*(15*A*a^2*b^4 - 35*B*a^3*b^3) + x^3*(50*A*a^3*b^3 - (245*B*a^4*b^2)/2))/(a^
5*b^7 + b^12*x^5 + 5*a^4*b^8*x + 5*a*b^11*x^4 + 10*a^3*b^9*x^2 + 10*a^2*b^10*x^3) + (B*x^2)/(2*b^6) + (log(a +
b*x)*(21*B*a^2 - 6*A*a*b))/b^8

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sympy [A]  time = 2.75, size = 216, normalized size = 1.26 \begin {gather*} \frac {B x^{2}}{2 b^{6}} + \frac {3 a \left (- 2 A b + 7 B a\right ) \log {\left (a + b x \right )}}{b^{8}} + x \left (\frac {A}{b^{6}} - \frac {6 B a}{b^{7}}\right ) + \frac {- 174 A a^{6} b + 459 B a^{7} + x^{4} \left (- 300 A a^{2} b^{5} + 700 B a^{3} b^{4}\right ) + x^{3} \left (- 1000 A a^{3} b^{4} + 2450 B a^{4} b^{3}\right ) + x^{2} \left (- 1300 A a^{4} b^{3} + 3290 B a^{5} b^{2}\right ) + x \left (- 770 A a^{5} b^{2} + 1995 B a^{6} b\right )}{20 a^{5} b^{8} + 100 a^{4} b^{9} x + 200 a^{3} b^{10} x^{2} + 200 a^{2} b^{11} x^{3} + 100 a b^{12} x^{4} + 20 b^{13} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

B*x**2/(2*b**6) + 3*a*(-2*A*b + 7*B*a)*log(a + b*x)/b**8 + x*(A/b**6 - 6*B*a/b**7) + (-174*A*a**6*b + 459*B*a*
*7 + x**4*(-300*A*a**2*b**5 + 700*B*a**3*b**4) + x**3*(-1000*A*a**3*b**4 + 2450*B*a**4*b**3) + x**2*(-1300*A*a
**4*b**3 + 3290*B*a**5*b**2) + x*(-770*A*a**5*b**2 + 1995*B*a**6*b))/(20*a**5*b**8 + 100*a**4*b**9*x + 200*a**
3*b**10*x**2 + 200*a**2*b**11*x**3 + 100*a*b**12*x**4 + 20*b**13*x**5)

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