3.533 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^6} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 a^2 b^2 B}{x^2}-\frac{4 a^3 b B}{3 x^3}-\frac{a^4 B}{4 x^4}-\frac{A (a+b x)^5}{5 a x^5}-\frac{4 a b^3 B}{x}+b^4 B \log (x) \]

[Out]

-(a^4*B)/(4*x^4) - (4*a^3*b*B)/(3*x^3) - (3*a^2*b^2*B)/x^2 - (4*a*b^3*B)/x - (A*(a + b*x)^5)/(5*a*x^5) + b^4*B
*Log[x]

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Rubi [A]  time = 0.0293582, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 78, 43} \[ -\frac{3 a^2 b^2 B}{x^2}-\frac{4 a^3 b B}{3 x^3}-\frac{a^4 B}{4 x^4}-\frac{A (a+b x)^5}{5 a x^5}-\frac{4 a b^3 B}{x}+b^4 B \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^6,x]

[Out]

-(a^4*B)/(4*x^4) - (4*a^3*b*B)/(3*x^3) - (3*a^2*b^2*B)/x^2 - (4*a*b^3*B)/x - (A*(a + b*x)^5)/(5*a*x^5) + b^4*B
*Log[x]

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0459834, size = 87, normalized size = 1.23 \[ b^4 B \log (x)-\frac{60 a^2 b^2 x^2 (2 A+3 B x)+20 a^3 b x (3 A+4 B x)+3 a^4 (4 A+5 B x)+120 a b^3 x^3 (A+2 B x)+60 A b^4 x^4}{60 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^6,x]

[Out]

-(60*A*b^4*x^4 + 120*a*b^3*x^3*(A + 2*B*x) + 60*a^2*b^2*x^2*(2*A + 3*B*x) + 20*a^3*b*x*(3*A + 4*B*x) + 3*a^4*(
4*A + 5*B*x))/(60*x^5) + b^4*B*Log[x]

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Maple [A]  time = 0.006, size = 100, normalized size = 1.4 \begin{align*}{b}^{4}B\ln \left ( x \right ) -2\,{\frac{A{a}^{2}{b}^{2}}{{x}^{3}}}-{\frac{4\,B{a}^{3}b}{3\,{x}^{3}}}-{\frac{A{a}^{4}}{5\,{x}^{5}}}-2\,{\frac{Aa{b}^{3}}{{x}^{2}}}-3\,{\frac{B{a}^{2}{b}^{2}}{{x}^{2}}}-{\frac{A{b}^{4}}{x}}-4\,{\frac{Ba{b}^{3}}{x}}-{\frac{A{a}^{3}b}{{x}^{4}}}-{\frac{B{a}^{4}}{4\,{x}^{4}}} \end{align*}

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^6,x)

[Out]

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

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Maxima [A]  time = 0.999264, size = 132, normalized size = 1.86 \begin{align*} B b^{4} \log \left (x\right ) - \frac{12 \, A a^{4} + 60 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 60 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 40 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \end{align*}

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^6,x, algorithm="maxima")

[Out]

B*b^4*log(x) - 1/60*(12*A*a^4 + 60*(4*B*a*b^3 + A*b^4)*x^4 + 60*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 40*(2*B*a^3*b
+ 3*A*a^2*b^2)*x^2 + 15*(B*a^4 + 4*A*a^3*b)*x)/x^5

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Fricas [A]  time = 1.33469, size = 228, normalized size = 3.21 \begin{align*} \frac{60 \, B b^{4} x^{5} \log \left (x\right ) - 12 \, A a^{4} - 60 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} - 60 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 40 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \end{align*}

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^6,x, algorithm="fricas")

[Out]

1/60*(60*B*b^4*x^5*log(x) - 12*A*a^4 - 60*(4*B*a*b^3 + A*b^4)*x^4 - 60*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 - 40*(2*B
*a^3*b + 3*A*a^2*b^2)*x^2 - 15*(B*a^4 + 4*A*a^3*b)*x)/x^5

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Sympy [A]  time = 2.19416, size = 99, normalized size = 1.39 \begin{align*} B b^{4} \log{\left (x \right )} - \frac{12 A a^{4} + x^{4} \left (60 A b^{4} + 240 B a b^{3}\right ) + x^{3} \left (120 A a b^{3} + 180 B a^{2} b^{2}\right ) + x^{2} \left (120 A a^{2} b^{2} + 80 B a^{3} b\right ) + x \left (60 A a^{3} b + 15 B a^{4}\right )}{60 x^{5}} \end{align*}

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**6,x)

[Out]

B*b**4*log(x) - (12*A*a**4 + x**4*(60*A*b**4 + 240*B*a*b**3) + x**3*(120*A*a*b**3 + 180*B*a**2*b**2) + x**2*(1
20*A*a**2*b**2 + 80*B*a**3*b) + x*(60*A*a**3*b + 15*B*a**4))/(60*x**5)

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Giac [A]  time = 1.14849, size = 134, normalized size = 1.89 \begin{align*} B b^{4} \log \left ({\left | x \right |}\right ) - \frac{12 \, A a^{4} + 60 \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 60 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 40 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 15 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x}{60 \, x^{5}} \end{align*}

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^6,x, algorithm="giac")

[Out]

B*b^4*log(abs(x)) - 1/60*(12*A*a^4 + 60*(4*B*a*b^3 + A*b^4)*x^4 + 60*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 40*(2*B*a
^3*b + 3*A*a^2*b^2)*x^2 + 15*(B*a^4 + 4*A*a^3*b)*x)/x^5