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

Optimal. Leaf size=133 \[ \frac{5}{2} a^3 b^2 x^2 (3 a B+4 A b)+\frac{5}{3} a^2 b^3 x^3 (4 a B+3 A b)+3 a^4 b x (2 a B+5 A b)+a^5 \log (x) (a B+6 A b)-\frac{a^6 A}{x}+\frac{3}{4} a b^4 x^4 (5 a B+2 A b)+\frac{1}{5} b^5 x^5 (6 a B+A b)+\frac{1}{6} b^6 B x^6 \]

[Out]

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

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Rubi [A]  time = 0.0766646, antiderivative size = 133, 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} \[ \frac{5}{2} a^3 b^2 x^2 (3 a B+4 A b)+\frac{5}{3} a^2 b^3 x^3 (4 a B+3 A b)+3 a^4 b x (2 a B+5 A b)+a^5 \log (x) (a B+6 A b)-\frac{a^6 A}{x}+\frac{3}{4} a b^4 x^4 (5 a B+2 A b)+\frac{1}{5} b^5 x^5 (6 a B+A b)+\frac{1}{6} b^6 B x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^6*A)/x) + 3*a^4*b*(5*A*b + 2*a*B)*x + (5*a^3*b^2*(4*A*b + 3*a*B)*x^2)/2 + (5*a^2*b^3*(3*A*b + 4*a*B)*x^3)
/3 + (3*a*b^4*(2*A*b + 5*a*B)*x^4)/4 + (b^5*(A*b + 6*a*B)*x^5)/5 + (b^6*B*x^6)/6 + a^5*(6*A*b + a*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 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^2} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^2} \, dx\\ &=\int \left (3 a^4 b (5 A b+2 a B)+\frac{a^6 A}{x^2}+\frac{a^5 (6 A b+a B)}{x}+5 a^3 b^2 (4 A b+3 a B) x+5 a^2 b^3 (3 A b+4 a B) x^2+3 a b^4 (2 A b+5 a B) x^3+b^5 (A b+6 a B) x^4+b^6 B x^5\right ) \, dx\\ &=-\frac{a^6 A}{x}+3 a^4 b (5 A b+2 a B) x+\frac{5}{2} a^3 b^2 (4 A b+3 a B) x^2+\frac{5}{3} a^2 b^3 (3 A b+4 a B) x^3+\frac{3}{4} a b^4 (2 A b+5 a B) x^4+\frac{1}{5} b^5 (A b+6 a B) x^5+\frac{1}{6} b^6 B x^6+a^5 (6 A b+a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0527824, size = 129, normalized size = 0.97 \[ \frac{10}{3} a^3 b^3 x^2 (3 A+2 B x)+\frac{5}{4} a^2 b^4 x^3 (4 A+3 B x)+\frac{15}{2} a^4 b^2 x (2 A+B x)+a^5 \log (x) (a B+6 A b)-\frac{a^6 A}{x}+6 a^5 b B x+\frac{3}{10} a b^5 x^4 (5 A+4 B x)+\frac{1}{30} b^6 x^5 (6 A+5 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^6*A)/x) + 6*a^5*b*B*x + (15*a^4*b^2*x*(2*A + B*x))/2 + (10*a^3*b^3*x^2*(3*A + 2*B*x))/3 + (5*a^2*b^4*x^3*
(4*A + 3*B*x))/4 + (3*a*b^5*x^4*(5*A + 4*B*x))/10 + (b^6*x^5*(6*A + 5*B*x))/30 + a^5*(6*A*b + a*B)*Log[x]

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Maple [A]  time = 0.006, size = 143, normalized size = 1.1 \begin{align*}{\frac{{b}^{6}B{x}^{6}}{6}}+{\frac{A{x}^{5}{b}^{6}}{5}}+{\frac{6\,B{x}^{5}a{b}^{5}}{5}}+{\frac{3\,A{x}^{4}a{b}^{5}}{2}}+{\frac{15\,B{x}^{4}{a}^{2}{b}^{4}}{4}}+5\,A{x}^{3}{a}^{2}{b}^{4}+{\frac{20\,B{x}^{3}{a}^{3}{b}^{3}}{3}}+10\,A{x}^{2}{a}^{3}{b}^{3}+{\frac{15\,B{x}^{2}{a}^{4}{b}^{2}}{2}}+15\,A{a}^{4}{b}^{2}x+6\,B{a}^{5}bx+6\,A\ln \left ( x \right ){a}^{5}b+B\ln \left ( x \right ){a}^{6}-{\frac{A{a}^{6}}{x}} \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^2,x)

[Out]

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

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Maxima [A]  time = 1.01248, size = 193, normalized size = 1.45 \begin{align*} \frac{1}{6} \, B b^{6} x^{6} - \frac{A a^{6}}{x} + \frac{1}{5} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{5} + \frac{3}{4} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{4} + \frac{5}{3} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{3} + \frac{5}{2} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{2} + 3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x +{\left (B a^{6} + 6 \, A a^{5} b\right )} \log \left (x\right ) \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.62346, size = 329, normalized size = 2.47 \begin{align*} \frac{10 \, B b^{6} x^{7} - 60 \, A a^{6} + 12 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 45 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 100 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 150 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 180 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 60 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x \log \left (x\right )}{60 \, x} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.508757, size = 148, normalized size = 1.11 \begin{align*} - \frac{A a^{6}}{x} + \frac{B b^{6} x^{6}}{6} + a^{5} \left (6 A b + B a\right ) \log{\left (x \right )} + x^{5} \left (\frac{A b^{6}}{5} + \frac{6 B a b^{5}}{5}\right ) + x^{4} \left (\frac{3 A a b^{5}}{2} + \frac{15 B a^{2} b^{4}}{4}\right ) + x^{3} \left (5 A a^{2} b^{4} + \frac{20 B a^{3} b^{3}}{3}\right ) + x^{2} \left (10 A a^{3} b^{3} + \frac{15 B a^{4} b^{2}}{2}\right ) + x \left (15 A a^{4} b^{2} + 6 B a^{5} b\right ) \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**2,x)

[Out]

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

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Giac [A]  time = 1.1364, size = 193, normalized size = 1.45 \begin{align*} \frac{1}{6} \, B b^{6} x^{6} + \frac{6}{5} \, B a b^{5} x^{5} + \frac{1}{5} \, A b^{6} x^{5} + \frac{15}{4} \, B a^{2} b^{4} x^{4} + \frac{3}{2} \, A a b^{5} x^{4} + \frac{20}{3} \, B a^{3} b^{3} x^{3} + 5 \, A a^{2} b^{4} x^{3} + \frac{15}{2} \, B a^{4} b^{2} x^{2} + 10 \, A a^{3} b^{3} x^{2} + 6 \, B a^{5} b x + 15 \, A a^{4} b^{2} x - \frac{A a^{6}}{x} +{\left (B a^{6} + 6 \, A a^{5} b\right )} \log \left ({\left | x \right |}\right ) \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^2,x, algorithm="giac")

[Out]

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