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

Optimal. Leaf size=101 \[ -\frac {a^2 A}{7 x^7}-\frac {A \left (2 a c+b^2\right )+2 a b B}{5 x^5}-\frac {2 a B c+2 A b c+b^2 B}{4 x^4}-\frac {a (a B+2 A b)}{6 x^6}-\frac {c (A c+2 b B)}{3 x^3}-\frac {B c^2}{2 x^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} -\frac {a^2 A}{7 x^7}-\frac {2 a B c+2 A b c+b^2 B}{4 x^4}-\frac {A \left (2 a c+b^2\right )+2 a b B}{5 x^5}-\frac {a (a B+2 A b)}{6 x^6}-\frac {c (A c+2 b B)}{3 x^3}-\frac {B c^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 100, normalized size = 0.99 \begin {gather*} -\frac {10 a^2 (6 A+7 B x)+14 a x (2 A (5 b+6 c x)+3 B x (4 b+5 c x))+7 x^2 \left (2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )\right )}{420 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/420*(10*a^2*(6*A + 7*B*x) + 14*a*x*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x)) + 7*x^2*(5*B*x*(3*b^2 + 8*b*c*
x + 6*c^2*x^2) + 2*A*(6*b^2 + 15*b*c*x + 10*c^2*x^2)))/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+b x+c x^2\right )^2}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 93, normalized size = 0.92 \begin {gather*} -\frac {210 \, B c^{2} x^{5} + 140 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^8,x, algorithm="fricas")

[Out]

-1/420*(210*B*c^2*x^5 + 140*(2*B*b*c + A*c^2)*x^4 + 105*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 60*A*a^2 + 84*(2*B*a*b
+ A*b^2 + 2*A*a*c)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7

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giac [A]  time = 0.15, size = 101, normalized size = 1.00 \begin {gather*} -\frac {210 \, B c^{2} x^{5} + 280 \, B b c x^{4} + 140 \, A c^{2} x^{4} + 105 \, B b^{2} x^{3} + 210 \, B a c x^{3} + 210 \, A b c x^{3} + 168 \, B a b x^{2} + 84 \, A b^{2} x^{2} + 168 \, A a c x^{2} + 70 \, B a^{2} x + 140 \, A a b x + 60 \, A a^{2}}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^8,x, algorithm="giac")

[Out]

-1/420*(210*B*c^2*x^5 + 280*B*b*c*x^4 + 140*A*c^2*x^4 + 105*B*b^2*x^3 + 210*B*a*c*x^3 + 210*A*b*c*x^3 + 168*B*
a*b*x^2 + 84*A*b^2*x^2 + 168*A*a*c*x^2 + 70*B*a^2*x + 140*A*a*b*x + 60*A*a^2)/x^7

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maple [A]  time = 0.05, size = 90, normalized size = 0.89 \begin {gather*} -\frac {B \,c^{2}}{2 x^{2}}-\frac {\left (A c +2 b B \right ) c}{3 x^{3}}-\frac {A \,a^{2}}{7 x^{7}}-\frac {2 A b c +2 a B c +b^{2} B}{4 x^{4}}-\frac {\left (2 A b +B a \right ) a}{6 x^{6}}-\frac {2 A a c +A \,b^{2}+2 B a b}{5 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)^2/x^8,x)

[Out]

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

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maxima [A]  time = 0.63, size = 93, normalized size = 0.92 \begin {gather*} -\frac {210 \, B c^{2} x^{5} + 140 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 60 \, A a^{2} + 84 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 70 \, {\left (B a^{2} + 2 \, A a b\right )} x}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)^2/x^8,x, algorithm="maxima")

[Out]

-1/420*(210*B*c^2*x^5 + 140*(2*B*b*c + A*c^2)*x^4 + 105*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 60*A*a^2 + 84*(2*B*a*b
+ A*b^2 + 2*A*a*c)*x^2 + 70*(B*a^2 + 2*A*a*b)*x)/x^7

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mupad [B]  time = 1.17, size = 93, normalized size = 0.92 \begin {gather*} -\frac {x^4\,\left (\frac {A\,c^2}{3}+\frac {2\,B\,b\,c}{3}\right )+\frac {A\,a^2}{7}+x^2\,\left (\frac {A\,b^2}{5}+\frac {2\,B\,a\,b}{5}+\frac {2\,A\,a\,c}{5}\right )+x^3\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}+\frac {B\,a\,c}{2}\right )+x\,\left (\frac {B\,a^2}{6}+\frac {A\,b\,a}{3}\right )+\frac {B\,c^2\,x^5}{2}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^2)/x^8,x)

[Out]

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

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sympy [A]  time = 24.02, size = 107, normalized size = 1.06 \begin {gather*} \frac {- 60 A a^{2} - 210 B c^{2} x^{5} + x^{4} \left (- 140 A c^{2} - 280 B b c\right ) + x^{3} \left (- 210 A b c - 210 B a c - 105 B b^{2}\right ) + x^{2} \left (- 168 A a c - 84 A b^{2} - 168 B a b\right ) + x \left (- 140 A a b - 70 B a^{2}\right )}{420 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)**2/x**8,x)

[Out]

(-60*A*a**2 - 210*B*c**2*x**5 + x**4*(-140*A*c**2 - 280*B*b*c) + x**3*(-210*A*b*c - 210*B*a*c - 105*B*b**2) +
x**2*(-168*A*a*c - 84*A*b**2 - 168*B*a*b) + x*(-140*A*a*b - 70*B*a**2))/(420*x**7)

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