3.550 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^3}{x^6} \, dx\)

Optimal. Leaf size=131 \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{2 x^2}-\frac{5 a^2 b^3 (4 a B+3 A b)}{x}-\frac{a^5 (a B+6 A b)}{4 x^4}-\frac{a^4 b (2 a B+5 A b)}{x^3}-\frac{a^6 A}{5 x^5}+b^5 x (6 a B+A b)+3 a b^4 \log (x) (5 a B+2 A b)+\frac{1}{2} b^6 B x^2 \]

[Out]

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

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Rubi [A]  time = 0.0810445, antiderivative size = 131, 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} \[ -\frac{5 a^3 b^2 (3 a B+4 A b)}{2 x^2}-\frac{5 a^2 b^3 (4 a B+3 A b)}{x}-\frac{a^5 (a B+6 A b)}{4 x^4}-\frac{a^4 b (2 a B+5 A b)}{x^3}-\frac{a^6 A}{5 x^5}+b^5 x (6 a B+A b)+3 a b^4 \log (x) (5 a B+2 A b)+\frac{1}{2} b^6 B x^2 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.051135, size = 128, normalized size = 0.98 \[ -\frac{5 a^4 b^2 (2 A+3 B x)}{2 x^3}-\frac{10 a^3 b^3 (A+2 B x)}{x^2}-\frac{15 a^2 A b^4}{x}-\frac{a^5 b (3 A+4 B x)}{2 x^4}-\frac{a^6 (4 A+5 B x)}{20 x^5}+3 a b^4 \log (x) (5 a B+2 A b)+6 a b^5 B x+\frac{1}{2} b^6 x (2 A+B x) \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3)/x^6,x]

[Out]

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

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Maple [A]  time = 0.011, size = 143, normalized size = 1.1 \begin{align*}{\frac{{b}^{6}B{x}^{2}}{2}}+A{b}^{6}x+6\,Ba{b}^{5}x+6\,A\ln \left ( x \right ) a{b}^{5}+15\,B\ln \left ( x \right ){a}^{2}{b}^{4}-5\,{\frac{A{a}^{4}{b}^{2}}{{x}^{3}}}-2\,{\frac{B{a}^{5}b}{{x}^{3}}}-{\frac{A{a}^{6}}{5\,{x}^{5}}}-10\,{\frac{A{a}^{3}{b}^{3}}{{x}^{2}}}-{\frac{15\,B{a}^{4}{b}^{2}}{2\,{x}^{2}}}-15\,{\frac{A{a}^{2}{b}^{4}}{x}}-20\,{\frac{B{a}^{3}{b}^{3}}{x}}-{\frac{3\,A{a}^{5}b}{2\,{x}^{4}}}-{\frac{B{a}^{6}}{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^6,x)

[Out]

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

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Maxima [A]  time = 1.01047, size = 194, normalized size = 1.48 \begin{align*} \frac{1}{2} \, B b^{6} x^{2} +{\left (6 \, B a b^{5} + A b^{6}\right )} x + 3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \log \left (x\right ) - \frac{4 \, A a^{6} + 100 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 50 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 20 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, 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)^3/x^6,x, algorithm="maxima")

[Out]

1/2*B*b^6*x^2 + (6*B*a*b^5 + A*b^6)*x + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*log(x) - 1/20*(4*A*a^6 + 100*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*x^4 + 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 20*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 5*(B*a^6 + 6*A*a^
5*b)*x)/x^5

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Fricas [A]  time = 1.17694, size = 327, normalized size = 2.5 \begin{align*} \frac{10 \, B b^{6} x^{7} - 4 \, A a^{6} + 20 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 60 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} \log \left (x\right ) - 100 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 50 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 20 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 5 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, 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)^3/x^6,x, algorithm="fricas")

[Out]

1/20*(10*B*b^6*x^7 - 4*A*a^6 + 20*(6*B*a*b^5 + A*b^6)*x^6 + 60*(5*B*a^2*b^4 + 2*A*a*b^5)*x^5*log(x) - 100*(4*B
*a^3*b^3 + 3*A*a^2*b^4)*x^4 - 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 - 20*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 - 5*(B*a^6
+ 6*A*a^5*b)*x)/x^5

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Sympy [A]  time = 2.52407, size = 143, normalized size = 1.09 \begin{align*} \frac{B b^{6} x^{2}}{2} + 3 a b^{4} \left (2 A b + 5 B a\right ) \log{\left (x \right )} + x \left (A b^{6} + 6 B a b^{5}\right ) - \frac{4 A a^{6} + x^{4} \left (300 A a^{2} b^{4} + 400 B a^{3} b^{3}\right ) + x^{3} \left (200 A a^{3} b^{3} + 150 B a^{4} b^{2}\right ) + x^{2} \left (100 A a^{4} b^{2} + 40 B a^{5} b\right ) + x \left (30 A a^{5} b + 5 B a^{6}\right )}{20 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)**3/x**6,x)

[Out]

B*b**6*x**2/2 + 3*a*b**4*(2*A*b + 5*B*a)*log(x) + x*(A*b**6 + 6*B*a*b**5) - (4*A*a**6 + x**4*(300*A*a**2*b**4
+ 400*B*a**3*b**3) + x**3*(200*A*a**3*b**3 + 150*B*a**4*b**2) + x**2*(100*A*a**4*b**2 + 40*B*a**5*b) + x*(30*A
*a**5*b + 5*B*a**6))/(20*x**5)

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Giac [A]  time = 1.13282, size = 194, normalized size = 1.48 \begin{align*} \frac{1}{2} \, B b^{6} x^{2} + 6 \, B a b^{5} x + A b^{6} x + 3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \log \left ({\left | x \right |}\right ) - \frac{4 \, A a^{6} + 100 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 50 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 20 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, 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)^3/x^6,x, algorithm="giac")

[Out]

1/2*B*b^6*x^2 + 6*B*a*b^5*x + A*b^6*x + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*log(abs(x)) - 1/20*(4*A*a^6 + 100*(4*B*a^3
*b^3 + 3*A*a^2*b^4)*x^4 + 50*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^3 + 20*(2*B*a^5*b + 5*A*a^4*b^2)*x^2 + 5*(B*a^6 + 6
*A*a^5*b)*x)/x^5