3.552 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^8} \, dx\)

Optimal. Leaf size=101 \[ -\frac{15 a^2 b^4 B}{2 x^2}-\frac{20 a^3 b^3 B}{3 x^3}-\frac{15 a^4 b^2 B}{4 x^4}-\frac{6 a^5 b B}{5 x^5}-\frac{a^6 B}{6 x^6}-\frac{A (a+b x)^7}{7 a x^7}-\frac{6 a b^5 B}{x}+b^6 B \log (x) \]

[Out]

-(a^6*B)/(6*x^6) - (6*a^5*b*B)/(5*x^5) - (15*a^4*b^2*B)/(4*x^4) - (20*a^3*b^3*B)/(3*x^3) - (15*a^2*b^4*B)/(2*x
^2) - (6*a*b^5*B)/x - (A*(a + b*x)^7)/(7*a*x^7) + b^6*B*Log[x]

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Rubi [A]  time = 0.0443396, antiderivative size = 101, 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} \[ -\frac{15 a^2 b^4 B}{2 x^2}-\frac{20 a^3 b^3 B}{3 x^3}-\frac{15 a^4 b^2 B}{4 x^4}-\frac{6 a^5 b B}{5 x^5}-\frac{a^6 B}{6 x^6}-\frac{A (a+b x)^7}{7 a x^7}-\frac{6 a b^5 B}{x}+b^6 B \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^8,x]

[Out]

-(a^6*B)/(6*x^6) - (6*a^5*b*B)/(5*x^5) - (15*a^4*b^2*B)/(4*x^4) - (20*a^3*b^3*B)/(3*x^3) - (15*a^2*b^4*B)/(2*x
^2) - (6*a*b^5*B)/x - (A*(a + b*x)^7)/(7*a*x^7) + b^6*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 )^3}{x^8} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^8} \, dx\\ &=-\frac{A (a+b x)^7}{7 a x^7}+B \int \frac{(a+b x)^6}{x^7} \, dx\\ &=-\frac{A (a+b x)^7}{7 a x^7}+B \int \left (\frac{a^6}{x^7}+\frac{6 a^5 b}{x^6}+\frac{15 a^4 b^2}{x^5}+\frac{20 a^3 b^3}{x^4}+\frac{15 a^2 b^4}{x^3}+\frac{6 a b^5}{x^2}+\frac{b^6}{x}\right ) \, dx\\ &=-\frac{a^6 B}{6 x^6}-\frac{6 a^5 b B}{5 x^5}-\frac{15 a^4 b^2 B}{4 x^4}-\frac{20 a^3 b^3 B}{3 x^3}-\frac{15 a^2 b^4 B}{2 x^2}-\frac{6 a b^5 B}{x}-\frac{A (a+b x)^7}{7 a x^7}+b^6 B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0574593, size = 132, normalized size = 1.31 \[ -\frac{3 a^4 b^2 (4 A+5 B x)}{4 x^5}-\frac{5 a^3 b^3 (3 A+4 B x)}{3 x^4}-\frac{5 a^2 b^4 (2 A+3 B x)}{2 x^3}-\frac{a^5 b (5 A+6 B x)}{5 x^6}-\frac{a^6 (6 A+7 B x)}{42 x^7}-\frac{3 a b^5 (A+2 B x)}{x^2}-\frac{A b^6}{x}+b^6 B \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^8,x]

[Out]

-((A*b^6)/x) - (3*a*b^5*(A + 2*B*x))/x^2 - (5*a^2*b^4*(2*A + 3*B*x))/(2*x^3) - (5*a^3*b^3*(3*A + 4*B*x))/(3*x^
4) - (3*a^4*b^2*(4*A + 5*B*x))/(4*x^5) - (a^5*b*(5*A + 6*B*x))/(5*x^6) - (a^6*(6*A + 7*B*x))/(42*x^7) + b^6*B*
Log[x]

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Maple [A]  time = 0.008, size = 148, normalized size = 1.5 \begin{align*}{b}^{6}B\ln \left ( x \right ) -5\,{\frac{A{a}^{2}{b}^{4}}{{x}^{3}}}-{\frac{20\,B{a}^{3}{b}^{3}}{3\,{x}^{3}}}-3\,{\frac{Aa{b}^{5}}{{x}^{2}}}-{\frac{15\,B{a}^{2}{b}^{4}}{2\,{x}^{2}}}-{\frac{A{b}^{6}}{x}}-6\,{\frac{Ba{b}^{5}}{x}}-{\frac{A{a}^{6}}{7\,{x}^{7}}}-{\frac{A{a}^{5}b}{{x}^{6}}}-{\frac{B{a}^{6}}{6\,{x}^{6}}}-3\,{\frac{A{a}^{4}{b}^{2}}{{x}^{5}}}-{\frac{6\,B{a}^{5}b}{5\,{x}^{5}}}-5\,{\frac{A{a}^{3}{b}^{3}}{{x}^{4}}}-{\frac{15\,B{a}^{4}{b}^{2}}{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)^3/x^8,x)

[Out]

b^6*B*ln(x)-5*a^2*b^4/x^3*A-20/3*a^3*b^3*B/x^3-3*a*b^5/x^2*A-15/2*a^2*b^4*B/x^2-b^6/x*A-6*a*b^5*B/x-1/7*A*a^6/
x^7-a^5/x^6*A*b-1/6*a^6*B/x^6-3*a^4*b^2/x^5*A-6/5*a^5*b*B/x^5-5*a^3*b^3/x^4*A-15/4*a^4*b^2*B/x^4

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Maxima [A]  time = 1.00298, size = 197, normalized size = 1.95 \begin{align*} B b^{6} \log \left (x\right ) - \frac{60 \, A a^{6} + 420 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \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)^3/x^8,x, algorithm="maxima")

[Out]

B*b^6*log(x) - 1/420*(60*A*a^6 + 420*(6*B*a*b^5 + A*b^6)*x^6 + 630*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 700*(4*B*a^
3*b^3 + 3*A*a^2*b^4)*x^4 + 525*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 70*(B*a^6
+ 6*A*a^5*b)*x)/x^7

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Fricas [A]  time = 1.22781, size = 338, normalized size = 3.35 \begin{align*} \frac{420 \, B b^{6} x^{7} \log \left (x\right ) - 60 \, A a^{6} - 420 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} - 630 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 700 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 252 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 70 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \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)^3/x^8,x, algorithm="fricas")

[Out]

1/420*(420*B*b^6*x^7*log(x) - 60*A*a^6 - 420*(6*B*a*b^5 + A*b^6)*x^6 - 630*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 - 700
*(4*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 525*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 252*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 7
0*(B*a^6 + 6*A*a^5*b)*x)/x^7

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Sympy [A]  time = 5.93251, size = 146, normalized size = 1.45 \begin{align*} B b^{6} \log{\left (x \right )} - \frac{60 A a^{6} + x^{6} \left (420 A b^{6} + 2520 B a b^{5}\right ) + x^{5} \left (1260 A a b^{5} + 3150 B a^{2} b^{4}\right ) + x^{4} \left (2100 A a^{2} b^{4} + 2800 B a^{3} b^{3}\right ) + x^{3} \left (2100 A a^{3} b^{3} + 1575 B a^{4} b^{2}\right ) + x^{2} \left (1260 A a^{4} b^{2} + 504 B a^{5} b\right ) + x \left (420 A a^{5} b + 70 B a^{6}\right )}{420 x^{7}} \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)**3/x**8,x)

[Out]

B*b**6*log(x) - (60*A*a**6 + x**6*(420*A*b**6 + 2520*B*a*b**5) + x**5*(1260*A*a*b**5 + 3150*B*a**2*b**4) + x**
4*(2100*A*a**2*b**4 + 2800*B*a**3*b**3) + x**3*(2100*A*a**3*b**3 + 1575*B*a**4*b**2) + x**2*(1260*A*a**4*b**2
+ 504*B*a**5*b) + x*(420*A*a**5*b + 70*B*a**6))/(420*x**7)

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Giac [A]  time = 1.1623, size = 198, normalized size = 1.96 \begin{align*} B b^{6} \log \left ({\left | x \right |}\right ) - \frac{60 \, A a^{6} + 420 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \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)^3/x^8,x, algorithm="giac")

[Out]

B*b^6*log(abs(x)) - 1/420*(60*A*a^6 + 420*(6*B*a*b^5 + A*b^6)*x^6 + 630*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 700*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 + 525*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 252*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 70*(
B*a^6 + 6*A*a^5*b)*x)/x^7