3.5.65 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^{11}} \, dx\)

Optimal. Leaf size=99 \[ -\frac {a^4 A}{10 x^{10}}-\frac {a^3 (a B+4 A b)}{9 x^9}-\frac {a^2 b (2 a B+3 A b)}{4 x^8}-\frac {b^3 (4 a B+A b)}{6 x^6}-\frac {2 a b^2 (3 a B+2 A b)}{7 x^7}-\frac {b^4 B}{5 x^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 99, 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 {a^3 (a B+4 A b)}{9 x^9}-\frac {a^2 b (2 a B+3 A b)}{4 x^8}-\frac {a^4 A}{10 x^{10}}-\frac {2 a b^2 (3 a B+2 A b)}{7 x^7}-\frac {b^3 (4 a B+A b)}{6 x^6}-\frac {b^4 B}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^11,x]

[Out]

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

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Mathematica [A]  time = 0.03, size = 88, normalized size = 0.89 \begin {gather*} -\frac {14 a^4 (9 A+10 B x)+70 a^3 b x (8 A+9 B x)+135 a^2 b^2 x^2 (7 A+8 B x)+120 a b^3 x^3 (6 A+7 B x)+42 b^4 x^4 (5 A+6 B x)}{1260 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^11,x]

[Out]

-1/1260*(42*b^4*x^4*(5*A + 6*B*x) + 120*a*b^3*x^3*(6*A + 7*B*x) + 135*a^2*b^2*x^2*(7*A + 8*B*x) + 70*a^3*b*x*(
8*A + 9*B*x) + 14*a^4*(9*A + 10*B*x))/x^10

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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 )^2}{x^{11}} \, 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)^2)/x^11,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^11, x]

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fricas [A]  time = 0.39, size = 99, normalized size = 1.00 \begin {gather*} -\frac {252 \, B b^{4} x^{5} + 126 \, A a^{4} + 210 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 360 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 315 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 140 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{1260 \, x^{10}} \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)^2/x^11,x, algorithm="fricas")

[Out]

-1/1260*(252*B*b^4*x^5 + 126*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 360*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 315*(2*
B*a^3*b + 3*A*a^2*b^2)*x^2 + 140*(B*a^4 + 4*A*a^3*b)*x)/x^10

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giac [A]  time = 0.15, size = 99, normalized size = 1.00 \begin {gather*} -\frac {252 \, B b^{4} x^{5} + 840 \, B a b^{3} x^{4} + 210 \, A b^{4} x^{4} + 1080 \, B a^{2} b^{2} x^{3} + 720 \, A a b^{3} x^{3} + 630 \, B a^{3} b x^{2} + 945 \, A a^{2} b^{2} x^{2} + 140 \, B a^{4} x + 560 \, A a^{3} b x + 126 \, A a^{4}}{1260 \, x^{10}} \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)^2/x^11,x, algorithm="giac")

[Out]

-1/1260*(252*B*b^4*x^5 + 840*B*a*b^3*x^4 + 210*A*b^4*x^4 + 1080*B*a^2*b^2*x^3 + 720*A*a*b^3*x^3 + 630*B*a^3*b*
x^2 + 945*A*a^2*b^2*x^2 + 140*B*a^4*x + 560*A*a^3*b*x + 126*A*a^4)/x^10

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maple [A]  time = 0.06, size = 88, normalized size = 0.89 \begin {gather*} -\frac {B \,b^{4}}{5 x^{5}}-\frac {\left (A b +4 B a \right ) b^{3}}{6 x^{6}}-\frac {2 \left (2 A b +3 B a \right ) a \,b^{2}}{7 x^{7}}-\frac {A \,a^{4}}{10 x^{10}}-\frac {\left (3 A b +2 B a \right ) a^{2} b}{4 x^{8}}-\frac {\left (4 A b +B a \right ) a^{3}}{9 x^{9}} \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)^2/x^11,x)

[Out]

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

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maxima [A]  time = 0.45, size = 99, normalized size = 1.00 \begin {gather*} -\frac {252 \, B b^{4} x^{5} + 126 \, A a^{4} + 210 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 360 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 315 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 140 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{1260 \, x^{10}} \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)^2/x^11,x, algorithm="maxima")

[Out]

-1/1260*(252*B*b^4*x^5 + 126*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 360*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 315*(2*
B*a^3*b + 3*A*a^2*b^2)*x^2 + 140*(B*a^4 + 4*A*a^3*b)*x)/x^10

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mupad [B]  time = 1.06, size = 97, normalized size = 0.98 \begin {gather*} -\frac {x\,\left (\frac {B\,a^4}{9}+\frac {4\,A\,b\,a^3}{9}\right )+\frac {A\,a^4}{10}+x^2\,\left (\frac {B\,a^3\,b}{2}+\frac {3\,A\,a^2\,b^2}{4}\right )+x^3\,\left (\frac {6\,B\,a^2\,b^2}{7}+\frac {4\,A\,a\,b^3}{7}\right )+x^4\,\left (\frac {A\,b^4}{6}+\frac {2\,B\,a\,b^3}{3}\right )+\frac {B\,b^4\,x^5}{5}}{x^{10}} \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)^2)/x^11,x)

[Out]

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

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sympy [A]  time = 5.45, size = 107, normalized size = 1.08 \begin {gather*} \frac {- 126 A a^{4} - 252 B b^{4} x^{5} + x^{4} \left (- 210 A b^{4} - 840 B a b^{3}\right ) + x^{3} \left (- 720 A a b^{3} - 1080 B a^{2} b^{2}\right ) + x^{2} \left (- 945 A a^{2} b^{2} - 630 B a^{3} b\right ) + x \left (- 560 A a^{3} b - 140 B a^{4}\right )}{1260 x^{10}} \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)**2/x**11,x)

[Out]

(-126*A*a**4 - 252*B*b**4*x**5 + x**4*(-210*A*b**4 - 840*B*a*b**3) + x**3*(-720*A*a*b**3 - 1080*B*a**2*b**2) +
x**2*(-945*A*a**2*b**2 - 630*B*a**3*b) + x*(-560*A*a**3*b - 140*B*a**4))/(1260*x**10)

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