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

Optimal. Leaf size=29 \[ \log (x) (a B+A b)-\frac {a A}{2 x^2}+\frac {1}{2} b B x^2 \]

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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*} \log (x) (a B+A b)-\frac {a A}{2 x^2}+\frac {1}{2} b B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(A + B*x^2))/x^3,x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \begin {gather*} \log (x) (a B+A b)-\frac {a A}{2 x^2}+\frac {1}{2} b B x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*A)/x^2 + (b*B*x^2)/2 + (A*b + a*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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 30, normalized size = 1.03 \begin {gather*} \frac {B b x^{4} + 2 \, {\left (B a + A b\right )} x^{2} \log \relax (x) - A a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.41, size = 42, normalized size = 1.45 \begin {gather*} \frac {1}{2} \, B b x^{2} + \frac {1}{2} \, {\left (B a + A b\right )} \log \left (x^{2}\right ) - \frac {B a x^{2} + A b x^{2} + A a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 26, normalized size = 0.90 \begin {gather*} \frac {B b \,x^{2}}{2}+A b \ln \relax (x )+B a \ln \relax (x )-\frac {A a}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(B*x^2+A)/x^3,x)

[Out]

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

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maxima [A]  time = 1.35, size = 28, normalized size = 0.97 \begin {gather*} \frac {1}{2} \, B b x^{2} + \frac {1}{2} \, {\left (B a + A b\right )} \log \left (x^{2}\right ) - \frac {A a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 25, normalized size = 0.86 \begin {gather*} \ln \relax (x)\,\left (A\,b+B\,a\right )-\frac {A\,a}{2\,x^2}+\frac {B\,b\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2))/x^3,x)

[Out]

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

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sympy [A]  time = 0.18, size = 26, normalized size = 0.90 \begin {gather*} - \frac {A a}{2 x^{2}} + \frac {B b x^{2}}{2} + \left (A b + B a\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(B*x**2+A)/x**3,x)

[Out]

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

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