3.5.62 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^8} \, dx\)

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

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

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^8,x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 88, normalized size = 0.89 \begin {gather*} -\frac {5 a^4 (6 A+7 B x)+28 a^3 b x (5 A+6 B x)+63 a^2 b^2 x^2 (4 A+5 B x)+70 a b^3 x^3 (3 A+4 B x)+35 b^4 x^4 (2 A+3 B x)}{210 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^8,x]

[Out]

-1/210*(35*b^4*x^4*(2*A + 3*B*x) + 70*a*b^3*x^3*(3*A + 4*B*x) + 63*a^2*b^2*x^2*(4*A + 5*B*x) + 28*a^3*b*x*(5*A
+ 6*B*x) + 5*a^4*(6*A + 7*B*x))/x^7

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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^8} \, 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^8,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^8, x]

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fricas [A]  time = 0.39, size = 99, normalized size = 1.00 \begin {gather*} -\frac {105 \, B b^{4} x^{5} + 30 \, A a^{4} + 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 105 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 84 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{210 \, x^{7}} \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^8,x, algorithm="fricas")

[Out]

-1/210*(105*B*b^4*x^5 + 30*A*a^4 + 70*(4*B*a*b^3 + A*b^4)*x^4 + 105*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 84*(2*B*a^
3*b + 3*A*a^2*b^2)*x^2 + 35*(B*a^4 + 4*A*a^3*b)*x)/x^7

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giac [A]  time = 0.16, size = 99, normalized size = 1.00 \begin {gather*} -\frac {105 \, B b^{4} x^{5} + 280 \, B a b^{3} x^{4} + 70 \, A b^{4} x^{4} + 315 \, B a^{2} b^{2} x^{3} + 210 \, A a b^{3} x^{3} + 168 \, B a^{3} b x^{2} + 252 \, A a^{2} b^{2} x^{2} + 35 \, B a^{4} x + 140 \, A a^{3} b x + 30 \, A a^{4}}{210 \, x^{7}} \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^8,x, algorithm="giac")

[Out]

-1/210*(105*B*b^4*x^5 + 280*B*a*b^3*x^4 + 70*A*b^4*x^4 + 315*B*a^2*b^2*x^3 + 210*A*a*b^3*x^3 + 168*B*a^3*b*x^2
+ 252*A*a^2*b^2*x^2 + 35*B*a^4*x + 140*A*a^3*b*x + 30*A*a^4)/x^7

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maple [A]  time = 0.05, size = 88, normalized size = 0.89 \begin {gather*} -\frac {B \,b^{4}}{2 x^{2}}-\frac {\left (A b +4 B a \right ) b^{3}}{3 x^{3}}-\frac {\left (2 A b +3 B a \right ) a \,b^{2}}{2 x^{4}}-\frac {A \,a^{4}}{7 x^{7}}-\frac {2 \left (3 A b +2 B a \right ) a^{2} b}{5 x^{5}}-\frac {\left (4 A b +B a \right ) a^{3}}{6 x^{6}} \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^8,x)

[Out]

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

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maxima [A]  time = 0.59, size = 99, normalized size = 1.00 \begin {gather*} -\frac {105 \, B b^{4} x^{5} + 30 \, A a^{4} + 70 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 105 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 84 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 35 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{210 \, x^{7}} \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^8,x, algorithm="maxima")

[Out]

-1/210*(105*B*b^4*x^5 + 30*A*a^4 + 70*(4*B*a*b^3 + A*b^4)*x^4 + 105*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 84*(2*B*a^
3*b + 3*A*a^2*b^2)*x^2 + 35*(B*a^4 + 4*A*a^3*b)*x)/x^7

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mupad [B]  time = 1.08, size = 96, normalized size = 0.97 \begin {gather*} -\frac {x\,\left (\frac {B\,a^4}{6}+\frac {2\,A\,b\,a^3}{3}\right )+\frac {A\,a^4}{7}+x^3\,\left (\frac {3\,B\,a^2\,b^2}{2}+A\,a\,b^3\right )+x^2\,\left (\frac {4\,B\,a^3\,b}{5}+\frac {6\,A\,a^2\,b^2}{5}\right )+x^4\,\left (\frac {A\,b^4}{3}+\frac {4\,B\,a\,b^3}{3}\right )+\frac {B\,b^4\,x^5}{2}}{x^7} \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^8,x)

[Out]

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

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sympy [A]  time = 2.85, size = 107, normalized size = 1.08 \begin {gather*} \frac {- 30 A a^{4} - 105 B b^{4} x^{5} + x^{4} \left (- 70 A b^{4} - 280 B a b^{3}\right ) + x^{3} \left (- 210 A a b^{3} - 315 B a^{2} b^{2}\right ) + x^{2} \left (- 252 A a^{2} b^{2} - 168 B a^{3} b\right ) + x \left (- 140 A a^{3} b - 35 B a^{4}\right )}{210 x^{7}} \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**8,x)

[Out]

(-30*A*a**4 - 105*B*b**4*x**5 + x**4*(-70*A*b**4 - 280*B*a*b**3) + x**3*(-210*A*a*b**3 - 315*B*a**2*b**2) + x*
*2*(-252*A*a**2*b**2 - 168*B*a**3*b) + x*(-140*A*a**3*b - 35*B*a**4))/(210*x**7)

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