3.5.63 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^9} \, dx\)

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

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

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9,x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 88, normalized size = 0.89 \begin {gather*} -\frac {15 a^4 (7 A+8 B x)+80 a^3 b x (6 A+7 B x)+168 a^2 b^2 x^2 (5 A+6 B x)+168 a b^3 x^3 (4 A+5 B x)+70 b^4 x^4 (3 A+4 B x)}{840 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9,x]

[Out]

-1/840*(70*b^4*x^4*(3*A + 4*B*x) + 168*a*b^3*x^3*(4*A + 5*B*x) + 168*a^2*b^2*x^2*(5*A + 6*B*x) + 80*a^3*b*x*(6
*A + 7*B*x) + 15*a^4*(7*A + 8*B*x))/x^8

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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^9} \, 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^9,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^9, x]

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fricas [A]  time = 0.39, size = 99, normalized size = 1.00 \begin {gather*} -\frac {280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \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^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*b^4*x^5 + 105*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 336*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 280*(2*B
*a^3*b + 3*A*a^2*b^2)*x^2 + 120*(B*a^4 + 4*A*a^3*b)*x)/x^8

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giac [A]  time = 0.16, size = 99, normalized size = 1.00 \begin {gather*} -\frac {280 \, B b^{4} x^{5} + 840 \, B a b^{3} x^{4} + 210 \, A b^{4} x^{4} + 1008 \, B a^{2} b^{2} x^{3} + 672 \, A a b^{3} x^{3} + 560 \, B a^{3} b x^{2} + 840 \, A a^{2} b^{2} x^{2} + 120 \, B a^{4} x + 480 \, A a^{3} b x + 105 \, A a^{4}}{840 \, x^{8}} \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^9,x, algorithm="giac")

[Out]

-1/840*(280*B*b^4*x^5 + 840*B*a*b^3*x^4 + 210*A*b^4*x^4 + 1008*B*a^2*b^2*x^3 + 672*A*a*b^3*x^3 + 560*B*a^3*b*x
^2 + 840*A*a^2*b^2*x^2 + 120*B*a^4*x + 480*A*a^3*b*x + 105*A*a^4)/x^8

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maple [A]  time = 0.05, size = 88, normalized size = 0.89 \begin {gather*} -\frac {B \,b^{4}}{3 x^{3}}-\frac {\left (A b +4 B a \right ) b^{3}}{4 x^{4}}-\frac {2 \left (2 A b +3 B a \right ) a \,b^{2}}{5 x^{5}}-\frac {A \,a^{4}}{8 x^{8}}-\frac {\left (3 A b +2 B a \right ) a^{2} b}{3 x^{6}}-\frac {\left (4 A b +B a \right ) a^{3}}{7 x^{7}} \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^9,x)

[Out]

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

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maxima [A]  time = 0.52, size = 99, normalized size = 1.00 \begin {gather*} -\frac {280 \, B b^{4} x^{5} + 105 \, A a^{4} + 210 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 336 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 280 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 120 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{840 \, x^{8}} \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^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*b^4*x^5 + 105*A*a^4 + 210*(4*B*a*b^3 + A*b^4)*x^4 + 336*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 280*(2*B
*a^3*b + 3*A*a^2*b^2)*x^2 + 120*(B*a^4 + 4*A*a^3*b)*x)/x^8

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mupad [B]  time = 1.07, size = 95, normalized size = 0.96 \begin {gather*} -\frac {x\,\left (\frac {B\,a^4}{7}+\frac {4\,A\,b\,a^3}{7}\right )+\frac {A\,a^4}{8}+x^2\,\left (\frac {2\,B\,a^3\,b}{3}+A\,a^2\,b^2\right )+x^3\,\left (\frac {6\,B\,a^2\,b^2}{5}+\frac {4\,A\,a\,b^3}{5}\right )+x^4\,\left (\frac {A\,b^4}{4}+B\,a\,b^3\right )+\frac {B\,b^4\,x^5}{3}}{x^8} \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^9,x)

[Out]

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

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sympy [A]  time = 3.71, size = 107, normalized size = 1.08 \begin {gather*} \frac {- 105 A a^{4} - 280 B b^{4} x^{5} + x^{4} \left (- 210 A b^{4} - 840 B a b^{3}\right ) + x^{3} \left (- 672 A a b^{3} - 1008 B a^{2} b^{2}\right ) + x^{2} \left (- 840 A a^{2} b^{2} - 560 B a^{3} b\right ) + x \left (- 480 A a^{3} b - 120 B a^{4}\right )}{840 x^{8}} \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**9,x)

[Out]

(-105*A*a**4 - 280*B*b**4*x**5 + x**4*(-210*A*b**4 - 840*B*a*b**3) + x**3*(-672*A*a*b**3 - 1008*B*a**2*b**2) +
x**2*(-840*A*a**2*b**2 - 560*B*a**3*b) + x*(-480*A*a**3*b - 120*B*a**4))/(840*x**8)

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