3.864 \(\int \frac{(A+B x) (a+b x+c x^2)^2}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0536574, antiderivative size = 99, 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} \[ -\frac{a^2 A}{6 x^6}-\frac{2 a B c+2 A b c+b^2 B}{3 x^3}-\frac{A \left (2 a c+b^2\right )+2 a b B}{4 x^4}-\frac{a (a B+2 A b)}{5 x^5}-\frac{c (A c+2 b B)}{2 x^2}-\frac{B c^2}{x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/x^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0352415, size = 97, normalized size = 0.98 \[ -\frac{2 a^2 (5 A+6 B x)+2 a x (3 A (4 b+5 c x)+5 B x (3 b+4 c x))+5 x^2 \left (A \left (3 b^2+8 b c x+6 c^2 x^2\right )+4 B x \left (b^2+3 b c x+3 c^2 x^2\right )\right )}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/x^7,x]

[Out]

-(2*a^2*(5*A + 6*B*x) + 2*a*x*(5*B*x*(3*b + 4*c*x) + 3*A*(4*b + 5*c*x)) + 5*x^2*(4*B*x*(b^2 + 3*b*c*x + 3*c^2*
x^2) + A*(3*b^2 + 8*b*c*x + 6*c^2*x^2)))/(60*x^6)

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

[Out]

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

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Maxima [A]  time = 1.10291, size = 126, normalized size = 1.27 \begin{align*} -\frac{60 \, B c^{2} x^{5} + 30 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 20 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 10 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 12 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

-1/60*(60*B*c^2*x^5 + 30*(2*B*b*c + A*c^2)*x^4 + 20*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 10*A*a^2 + 15*(2*B*a*b + A
*b^2 + 2*A*a*c)*x^2 + 12*(B*a^2 + 2*A*a*b)*x)/x^6

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Fricas [A]  time = 1.24708, size = 217, normalized size = 2.19 \begin{align*} -\frac{60 \, B c^{2} x^{5} + 30 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 20 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 10 \, A a^{2} + 15 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 12 \,{\left (B a^{2} + 2 \, A a b\right )} x}{60 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

-1/60*(60*B*c^2*x^5 + 30*(2*B*b*c + A*c^2)*x^4 + 20*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 10*A*a^2 + 15*(2*B*a*b + A
*b^2 + 2*A*a*c)*x^2 + 12*(B*a^2 + 2*A*a*b)*x)/x^6

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Sympy [A]  time = 18.0439, size = 102, normalized size = 1.03 \begin{align*} - \frac{10 A a^{2} + 60 B c^{2} x^{5} + x^{4} \left (30 A c^{2} + 60 B b c\right ) + x^{3} \left (40 A b c + 40 B a c + 20 B b^{2}\right ) + x^{2} \left (30 A a c + 15 A b^{2} + 30 B a b\right ) + x \left (24 A a b + 12 B a^{2}\right )}{60 x^{6}} \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**7,x)

[Out]

-(10*A*a**2 + 60*B*c**2*x**5 + x**4*(30*A*c**2 + 60*B*b*c) + x**3*(40*A*b*c + 40*B*a*c + 20*B*b**2) + x**2*(30
*A*a*c + 15*A*b**2 + 30*B*a*b) + x*(24*A*a*b + 12*B*a**2))/(60*x**6)

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Giac [A]  time = 1.25423, size = 136, normalized size = 1.37 \begin{align*} -\frac{60 \, B c^{2} x^{5} + 60 \, B b c x^{4} + 30 \, A c^{2} x^{4} + 20 \, B b^{2} x^{3} + 40 \, B a c x^{3} + 40 \, A b c x^{3} + 30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 30 \, A a c x^{2} + 12 \, B a^{2} x + 24 \, A a b x + 10 \, A a^{2}}{60 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

-1/60*(60*B*c^2*x^5 + 60*B*b*c*x^4 + 30*A*c^2*x^4 + 20*B*b^2*x^3 + 40*B*a*c*x^3 + 40*A*b*c*x^3 + 30*B*a*b*x^2
+ 15*A*b^2*x^2 + 30*A*a*c*x^2 + 12*B*a^2*x + 24*A*a*b*x + 10*A*a^2)/x^6