3.551 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^7} \, dx\)

Optimal. Leaf size=132 \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{3 x^3}-\frac{5 a^2 b^3 (4 a B+3 A b)}{2 x^2}-\frac{a^5 (a B+6 A b)}{5 x^5}-\frac{3 a^4 b (2 a B+5 A b)}{4 x^4}-\frac{a^6 A}{6 x^6}-\frac{3 a b^4 (5 a B+2 A b)}{x}+b^5 \log (x) (6 a B+A b)+b^6 B x \]

[Out]

-(a^6*A)/(6*x^6) - (a^5*(6*A*b + a*B))/(5*x^5) - (3*a^4*b*(5*A*b + 2*a*B))/(4*x^4) - (5*a^3*b^2*(4*A*b + 3*a*B
))/(3*x^3) - (5*a^2*b^3*(3*A*b + 4*a*B))/(2*x^2) - (3*a*b^4*(2*A*b + 5*a*B))/x + b^6*B*x + b^5*(A*b + 6*a*B)*L
og[x]

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Rubi [A]  time = 0.0762762, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{3 x^3}-\frac{5 a^2 b^3 (4 a B+3 A b)}{2 x^2}-\frac{a^5 (a B+6 A b)}{5 x^5}-\frac{3 a^4 b (2 a B+5 A b)}{4 x^4}-\frac{a^6 A}{6 x^6}-\frac{3 a b^4 (5 a B+2 A b)}{x}+b^5 \log (x) (6 a B+A b)+b^6 B x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^7,x]

[Out]

-(a^6*A)/(6*x^6) - (a^5*(6*A*b + a*B))/(5*x^5) - (3*a^4*b*(5*A*b + 2*a*B))/(4*x^4) - (5*a^3*b^2*(4*A*b + 3*a*B
))/(3*x^3) - (5*a^2*b^3*(3*A*b + 4*a*B))/(2*x^2) - (3*a*b^4*(2*A*b + 5*a*B))/x + b^6*B*x + b^5*(A*b + 6*a*B)*L
og[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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0774836, size = 125, normalized size = 0.95 \[ b^5 \log (x) (6 a B+A b)-\frac{450 a^2 b^4 x^4 (A+2 B x)+200 a^3 b^3 x^3 (2 A+3 B x)+75 a^4 b^2 x^2 (3 A+4 B x)+18 a^5 b x (4 A+5 B x)+2 a^6 (5 A+6 B x)+360 a A b^5 x^5-60 b^6 B x^7}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^7,x]

[Out]

-(360*a*A*b^5*x^5 - 60*b^6*B*x^7 + 450*a^2*b^4*x^4*(A + 2*B*x) + 200*a^3*b^3*x^3*(2*A + 3*B*x) + 75*a^4*b^2*x^
2*(3*A + 4*B*x) + 18*a^5*b*x*(4*A + 5*B*x) + 2*a^6*(5*A + 6*B*x))/(60*x^6) + b^5*(A*b + 6*a*B)*Log[x]

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Maple [A]  time = 0.009, size = 144, normalized size = 1.1 \begin{align*}{b}^{6}Bx+A\ln \left ( x \right ){b}^{6}+6\,B\ln \left ( x \right ) a{b}^{5}-{\frac{20\,A{a}^{3}{b}^{3}}{3\,{x}^{3}}}-5\,{\frac{B{a}^{4}{b}^{2}}{{x}^{3}}}-{\frac{15\,A{a}^{2}{b}^{4}}{2\,{x}^{2}}}-10\,{\frac{B{a}^{3}{b}^{3}}{{x}^{2}}}-6\,{\frac{Aa{b}^{5}}{x}}-15\,{\frac{B{a}^{2}{b}^{4}}{x}}-{\frac{15\,A{a}^{4}{b}^{2}}{4\,{x}^{4}}}-{\frac{3\,B{a}^{5}b}{2\,{x}^{4}}}-{\frac{6\,A{a}^{5}b}{5\,{x}^{5}}}-{\frac{B{a}^{6}}{5\,{x}^{5}}}-{\frac{A{a}^{6}}{6\,{x}^{6}}} \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^7,x)

[Out]

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

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Maxima [A]  time = 0.977652, size = 193, normalized size = 1.46 \begin{align*} B b^{6} x +{\left (6 \, B a b^{5} + A b^{6}\right )} \log \left (x\right ) - \frac{10 \, A a^{6} + 180 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 150 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 12 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

B*b^6*x + (6*B*a*b^5 + A*b^6)*log(x) - 1/60*(10*A*a^6 + 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 150*(4*B*a^3*b^3 +
3*A*a^2*b^4)*x^4 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 12*(B*a^6 + 6*A*a
^5*b)*x)/x^6

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Fricas [A]  time = 1.19679, size = 332, normalized size = 2.52 \begin{align*} \frac{60 \, B b^{6} x^{7} + 60 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} \log \left (x\right ) - 10 \, A a^{6} - 180 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 150 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 100 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 12 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

1/60*(60*B*b^6*x^7 + 60*(6*B*a*b^5 + A*b^6)*x^6*log(x) - 10*A*a^6 - 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 - 150*(4
*B*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 12*(B
*a^6 + 6*A*a^5*b)*x)/x^6

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Sympy [A]  time = 4.13453, size = 141, normalized size = 1.07 \begin{align*} B b^{6} x + b^{5} \left (A b + 6 B a\right ) \log{\left (x \right )} - \frac{10 A a^{6} + x^{5} \left (360 A a b^{5} + 900 B a^{2} b^{4}\right ) + x^{4} \left (450 A a^{2} b^{4} + 600 B a^{3} b^{3}\right ) + x^{3} \left (400 A a^{3} b^{3} + 300 B a^{4} b^{2}\right ) + x^{2} \left (225 A a^{4} b^{2} + 90 B a^{5} b\right ) + x \left (72 A a^{5} b + 12 B a^{6}\right )}{60 x^{6}} \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**7,x)

[Out]

B*b**6*x + b**5*(A*b + 6*B*a)*log(x) - (10*A*a**6 + x**5*(360*A*a*b**5 + 900*B*a**2*b**4) + x**4*(450*A*a**2*b
**4 + 600*B*a**3*b**3) + x**3*(400*A*a**3*b**3 + 300*B*a**4*b**2) + x**2*(225*A*a**4*b**2 + 90*B*a**5*b) + x*(
72*A*a**5*b + 12*B*a**6))/(60*x**6)

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Giac [A]  time = 1.13134, size = 194, normalized size = 1.47 \begin{align*} B b^{6} x +{\left (6 \, B a b^{5} + A b^{6}\right )} \log \left ({\left | x \right |}\right ) - \frac{10 \, A a^{6} + 180 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 150 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 100 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 12 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{60 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

B*b^6*x + (6*B*a*b^5 + A*b^6)*log(abs(x)) - 1/60*(10*A*a^6 + 180*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5 + 150*(4*B*a^3*
b^3 + 3*A*a^2*b^4)*x^4 + 100*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 45*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 12*(B*a^6 +
6*A*a^5*b)*x)/x^6