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

Optimal. Leaf size=86 \[ -\frac {a^4 A}{x}+a^3 \log (x) (a B+4 A b)+2 a^2 b x (2 a B+3 A b)+\frac {1}{3} b^3 x^3 (4 a B+A b)+a b^2 x^2 (3 a B+2 A b)+\frac {1}{4} b^4 B x^4 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, 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} \[ 2 a^2 b x (2 a B+3 A b)+a^3 \log (x) (a B+4 A b)-\frac {a^4 A}{x}+a b^2 x^2 (3 a B+2 A b)+\frac {1}{3} b^3 x^3 (4 a B+A b)+\frac {1}{4} b^4 B x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 85, normalized size = 0.99 \[ -\frac {a^4 A}{x}+a^3 \log (x) (a B+4 A b)+4 a^3 b B x+3 a^2 b^2 x (2 A+B x)+\frac {2}{3} a b^3 x^2 (3 A+2 B x)+\frac {1}{12} b^4 x^3 (4 A+3 B x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^4*A)/x) + 4*a^3*b*B*x + 3*a^2*b^2*x*(2*A + B*x) + (2*a*b^3*x^2*(3*A + 2*B*x))/3 + (b^4*x^3*(4*A + 3*B*x))
/12 + a^3*(4*A*b + a*B)*Log[x]

________________________________________________________________________________________

fricas [A]  time = 1.21, size = 101, normalized size = 1.17 \[ \frac {3 \, B b^{4} x^{5} - 12 \, A a^{4} + 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 12 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 24 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x \log \relax (x)}{12 \, x} \]

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)^2/x^2,x, algorithm="fricas")

[Out]

1/12*(3*B*b^4*x^5 - 12*A*a^4 + 4*(4*B*a*b^3 + A*b^4)*x^4 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 24*(2*B*a^3*b +
3*A*a^2*b^2)*x^2 + 12*(B*a^4 + 4*A*a^3*b)*x*log(x))/x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 95, normalized size = 1.10 \[ \frac {1}{4} \, B b^{4} x^{4} + \frac {4}{3} \, B a b^{3} x^{3} + \frac {1}{3} \, A b^{4} x^{3} + 3 \, B a^{2} b^{2} x^{2} + 2 \, A a b^{3} x^{2} + 4 \, B a^{3} b x + 6 \, A a^{2} b^{2} x - \frac {A a^{4}}{x} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \log \left ({\left | x \right |}\right ) \]

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)^2/x^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 95, normalized size = 1.10 \[ \frac {B \,b^{4} x^{4}}{4}+\frac {A \,b^{4} x^{3}}{3}+\frac {4 B a \,b^{3} x^{3}}{3}+2 A a \,b^{3} x^{2}+3 B \,a^{2} b^{2} x^{2}+4 A \,a^{3} b \ln \relax (x )+6 A \,a^{2} b^{2} x +B \,a^{4} \ln \relax (x )+4 B \,a^{3} b x -\frac {A \,a^{4}}{x} \]

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)^2/x^2,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 94, normalized size = 1.09 \[ \frac {1}{4} \, B b^{4} x^{4} - \frac {A a^{4}}{x} + \frac {1}{3} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{3} + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{2} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x + {\left (B a^{4} + 4 \, A a^{3} b\right )} \log \relax (x) \]

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)^2/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 86, normalized size = 1.00 \[ x^3\,\left (\frac {A\,b^4}{3}+\frac {4\,B\,a\,b^3}{3}\right )+\ln \relax (x)\,\left (B\,a^4+4\,A\,b\,a^3\right )-\frac {A\,a^4}{x}+\frac {B\,b^4\,x^4}{4}+2\,a^2\,b\,x\,\left (3\,A\,b+2\,B\,a\right )+a\,b^2\,x^2\,\left (2\,A\,b+3\,B\,a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/x^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 94, normalized size = 1.09 \[ - \frac {A a^{4}}{x} + \frac {B b^{4} x^{4}}{4} + a^{3} \left (4 A b + B a\right ) \log {\relax (x )} + x^{3} \left (\frac {A b^{4}}{3} + \frac {4 B a b^{3}}{3}\right ) + x^{2} \left (2 A a b^{3} + 3 B a^{2} b^{2}\right ) + x \left (6 A a^{2} b^{2} + 4 B a^{3} b\right ) \]

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)**2/x**2,x)

[Out]

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

________________________________________________________________________________________