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

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

[Out]

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

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Rubi [A]  time = 0.0798859, 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} \[ \frac{5}{2} a^2 b^3 x^2 (4 a B+3 A b)+5 a^3 b^2 x (3 a B+4 A b)-\frac{a^5 (a B+6 A b)}{x}+3 a^4 b \log (x) (2 a B+5 A b)-\frac{a^6 A}{2 x^2}+a b^4 x^3 (5 a B+2 A b)+\frac{1}{4} b^5 x^4 (6 a B+A b)+\frac{1}{5} b^6 B x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0576652, size = 128, normalized size = 0.98 \[ \frac{5}{2} a^2 b^4 x^2 (3 A+2 B x)+10 a^3 b^3 x (2 A+B x)+3 a^4 b \log (x) (2 a B+5 A b)-\frac{6 a^5 A b}{x}-\frac{a^6 (A+2 B x)}{2 x^2}+15 a^4 b^2 B x+\frac{1}{2} a b^5 x^3 (4 A+3 B x)+\frac{1}{20} b^6 x^4 (5 A+4 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 144, normalized size = 1.1 \begin{align*}{\frac{{b}^{6}B{x}^{5}}{5}}+{\frac{A{x}^{4}{b}^{6}}{4}}+{\frac{3\,B{x}^{4}a{b}^{5}}{2}}+2\,A{x}^{3}a{b}^{5}+5\,B{x}^{3}{a}^{2}{b}^{4}+{\frac{15\,A{x}^{2}{a}^{2}{b}^{4}}{2}}+10\,B{x}^{2}{a}^{3}{b}^{3}+20\,A{a}^{3}{b}^{3}x+15\,B{a}^{4}{b}^{2}x+15\,A\ln \left ( x \right ){a}^{4}{b}^{2}+6\,B\ln \left ( x \right ){a}^{5}b-{\frac{A{a}^{6}}{2\,{x}^{2}}}-6\,{\frac{A{a}^{5}b}{x}}-{\frac{B{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^3,x)

[Out]

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

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Maxima [A]  time = 1.02242, size = 193, normalized size = 1.47 \begin{align*} \frac{1}{5} \, B b^{6} x^{5} + \frac{1}{4} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{4} +{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{3} + \frac{5}{2} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{2} + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x + 3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \log \left (x\right ) - \frac{A a^{6} + 2 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.56457, size = 327, normalized size = 2.5 \begin{align*} \frac{4 \, B b^{6} x^{7} - 10 \, A a^{6} + 5 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 20 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 50 \,{\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} + 60 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} \log \left (x\right ) - 20 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{20 \, x^{2}} \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^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.677956, size = 146, normalized size = 1.11 \begin{align*} \frac{B b^{6} x^{5}}{5} + 3 a^{4} b \left (5 A b + 2 B a\right ) \log{\left (x \right )} + x^{4} \left (\frac{A b^{6}}{4} + \frac{3 B a b^{5}}{2}\right ) + x^{3} \left (2 A a b^{5} + 5 B a^{2} b^{4}\right ) + x^{2} \left (\frac{15 A a^{2} b^{4}}{2} + 10 B a^{3} b^{3}\right ) + x \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right ) - \frac{A a^{6} + x \left (12 A a^{5} b + 2 B a^{6}\right )}{2 x^{2}} \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**3,x)

[Out]

B*b**6*x**5/5 + 3*a**4*b*(5*A*b + 2*B*a)*log(x) + x**4*(A*b**6/4 + 3*B*a*b**5/2) + x**3*(2*A*a*b**5 + 5*B*a**2
*b**4) + x**2*(15*A*a**2*b**4/2 + 10*B*a**3*b**3) + x*(20*A*a**3*b**3 + 15*B*a**4*b**2) - (A*a**6 + x*(12*A*a*
*5*b + 2*B*a**6))/(2*x**2)

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Giac [A]  time = 1.14491, size = 194, normalized size = 1.48 \begin{align*} \frac{1}{5} \, B b^{6} x^{5} + \frac{3}{2} \, B a b^{5} x^{4} + \frac{1}{4} \, A b^{6} x^{4} + 5 \, B a^{2} b^{4} x^{3} + 2 \, A a b^{5} x^{3} + 10 \, B a^{3} b^{3} x^{2} + \frac{15}{2} \, A a^{2} b^{4} x^{2} + 15 \, B a^{4} b^{2} x + 20 \, A a^{3} b^{3} x + 3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} \log \left ({\left | x \right |}\right ) - \frac{A a^{6} + 2 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{2 \, x^{2}} \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^3,x, algorithm="giac")

[Out]

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