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

Optimal. Leaf size=29 \[ -\frac {a B+A b}{2 x^2}-\frac {a A}{4 x^4}+b B \log (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {446, 76} \begin {gather*} -\frac {a B+A b}{2 x^2}-\frac {a A}{4 x^4}+b B \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(A + B*x^2))/x^5,x]

[Out]

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

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])

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 31, normalized size = 1.07 \begin {gather*} \frac {-a B-A b}{2 x^2}-\frac {a A}{4 x^4}+b B \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a*A)/x^4 + (-(A*b) - a*B)/(2*x^2) + b*B*Log[x]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 31, normalized size = 1.07 \begin {gather*} \frac {4 \, B b x^{4} \log \relax (x) - 2 \, {\left (B a + A b\right )} x^{2} - A a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.32, size = 39, normalized size = 1.34 \begin {gather*} \frac {1}{2} \, B b \log \left (x^{2}\right ) - \frac {3 \, B b x^{4} + 2 \, B a x^{2} + 2 \, A b x^{2} + A a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 28, normalized size = 0.97 \begin {gather*} B b \ln \relax (x )-\frac {A b}{2 x^{2}}-\frac {B a}{2 x^{2}}-\frac {A a}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(B*x^2+A)/x^5,x)

[Out]

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

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maxima [A]  time = 1.39, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{2} \, B b \log \left (x^{2}\right ) - \frac {2 \, {\left (B a + A b\right )} x^{2} + A a}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.07, size = 29, normalized size = 1.00 \begin {gather*} B\,b\,\ln \relax (x)-\frac {\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x^2+\frac {A\,a}{4}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2))/x^5,x)

[Out]

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

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sympy [A]  time = 0.36, size = 29, normalized size = 1.00 \begin {gather*} B b \log {\relax (x )} + \frac {- A a + x^{2} \left (- 2 A b - 2 B a\right )}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(B*x**2+A)/x**5,x)

[Out]

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

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