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

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

[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)

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Rubi [A]  time = 0.0528436, antiderivative size = 101, normalized size of antiderivative = 1., 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} \[ -\frac{a^2 A}{7 x^7}-\frac{2 a B c+2 A b c+b^2 B}{4 x^4}-\frac{A \left (2 a c+b^2\right )+2 a b B}{5 x^5}-\frac{a (a B+2 A b)}{6 x^6}-\frac{c (A c+2 b B)}{3 x^3}-\frac{B c^2}{2 x^2} \]

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*}

Mathematica [A]  time = 0.0363004, size = 100, normalized size = 0.99 \[ -\frac{10 a^2 (6 A+7 B x)+14 a x (2 A (5 b+6 c x)+3 B x (4 b+5 c x))+7 x^2 \left (2 A \left (6 b^2+15 b c x+10 c^2 x^2\right )+5 B x \left (3 b^2+8 b c x+6 c^2 x^2\right )\right )}{420 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)))/(420*x^7)

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Maple [A]  time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{c \left ( Ac+2\,bB \right ) }{3\,{x}^{3}}}-{\frac{B{c}^{2}}{2\,{x}^{2}}}-{\frac{A{a}^{2}}{7\,{x}^{7}}}-{\frac{2\,aAc+A{b}^{2}+2\,abB}{5\,{x}^{5}}}-{\frac{2\,Abc+2\,aBc+{b}^{2}B}{4\,{x}^{4}}}-{\frac{a \left ( 2\,Ab+aB \right ) }{6\,{x}^{6}}} \end{align*}

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]

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

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Maxima [A]  time = 1.10712, size = 126, normalized size = 1.25 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.22555, size = 223, normalized size = 2.21 \begin{align*} -\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{align*}

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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Sympy [A]  time = 28.3867, size = 102, normalized size = 1.01 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.14005, size = 136, normalized size = 1.35 \begin{align*} -\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{align*}

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