3.5.59 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^2}{x^5} \, dx\)

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

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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} \begin {gather*} -\frac {a^3 (a B+4 A b)}{3 x^3}-\frac {a^2 b (2 a B+3 A b)}{x^2}-\frac {a^4 A}{4 x^4}-\frac {2 a b^2 (3 a B+2 A b)}{x}+b^3 \log (x) (4 a B+A b)+b^4 B x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^5,x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^5,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^5,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/x^5, x]

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fricas [A]  time = 0.41, size = 101, normalized size = 1.17 \begin {gather*} \frac {12 \, B b^{4} x^{5} + 12 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} \log \relax (x) - 3 \, A a^{4} - 24 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} - 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{12 \, x^{4}} \end {gather*}

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^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 96, normalized size = 1.12 \begin {gather*} B b^{4} x + {\left (4 \, B a b^{3} + A b^{4}\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{4} + 24 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{12 \, x^{4}} \end {gather*}

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^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 96, normalized size = 1.12 \begin {gather*} A \,b^{4} \ln \relax (x )+4 B a \,b^{3} \ln \relax (x )+B \,b^{4} x -\frac {4 A a \,b^{3}}{x}-\frac {6 B \,a^{2} b^{2}}{x}-\frac {3 A \,a^{2} b^{2}}{x^{2}}-\frac {2 B \,a^{3} b}{x^{2}}-\frac {4 A \,a^{3} b}{3 x^{3}}-\frac {B \,a^{4}}{3 x^{3}}-\frac {A \,a^{4}}{4 x^{4}} \end {gather*}

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^5,x)

[Out]

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

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maxima [A]  time = 0.46, size = 95, normalized size = 1.10 \begin {gather*} B b^{4} x + {\left (4 \, B a b^{3} + A b^{4}\right )} \log \relax (x) - \frac {3 \, A a^{4} + 24 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 12 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 4 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x}{12 \, x^{4}} \end {gather*}

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^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.09, size = 93, normalized size = 1.08 \begin {gather*} \ln \relax (x)\,\left (A\,b^4+4\,B\,a\,b^3\right )-\frac {x\,\left (\frac {B\,a^4}{3}+\frac {4\,A\,b\,a^3}{3}\right )+\frac {A\,a^4}{4}+x^2\,\left (2\,B\,a^3\,b+3\,A\,a^2\,b^2\right )+x^3\,\left (6\,B\,a^2\,b^2+4\,A\,a\,b^3\right )}{x^4}+B\,b^4\,x \end {gather*}

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^5,x)

[Out]

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

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sympy [A]  time = 1.22, size = 99, normalized size = 1.15 \begin {gather*} B b^{4} x + b^{3} \left (A b + 4 B a\right ) \log {\relax (x )} + \frac {- 3 A a^{4} + x^{3} \left (- 48 A a b^{3} - 72 B a^{2} b^{2}\right ) + x^{2} \left (- 36 A a^{2} b^{2} - 24 B a^{3} b\right ) + x \left (- 16 A a^{3} b - 4 B a^{4}\right )}{12 x^{4}} \end {gather*}

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**5,x)

[Out]

B*b**4*x + b**3*(A*b + 4*B*a)*log(x) + (-3*A*a**4 + x**3*(-48*A*a*b**3 - 72*B*a**2*b**2) + x**2*(-36*A*a**2*b*
*2 - 24*B*a**3*b) + x*(-16*A*a**3*b - 4*B*a**4))/(12*x**4)

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