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

Optimal. Leaf size=26 \[ x (a B+A b)-\frac {a A}{x}+\frac {1}{3} b B x^3 \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {448} \begin {gather*} x (a B+A b)-\frac {a A}{x}+\frac {1}{3} b B x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(A + B*x^2))/x^2,x]

[Out]

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

Rule 448

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} x (a B+A b)-\frac {a A}{x}+\frac {1}{3} b B x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 28, normalized size = 1.08 \begin {gather*} \frac {B b x^{4} + 3 \, {\left (B a + A b\right )} x^{2} - 3 \, A a}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.28, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, B b x^{3} + B a x + A b x - \frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*b*x^3 + B*a*x + A*b*x - A*a/x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 24, normalized size = 0.92 \begin {gather*} \frac {B b \,x^{3}}{3}+A b x +B a x -\frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*B*x^3+A*b*x+B*a*x-a*A/x

________________________________________________________________________________________

maxima [A]  time = 1.39, size = 24, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, B b x^{3} + {\left (B a + A b\right )} x - \frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*b*x^3 + (B*a + A*b)*x - A*a/x

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 24, normalized size = 0.92 \begin {gather*} x\,\left (A\,b+B\,a\right )-\frac {A\,a}{x}+\frac {B\,b\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2))/x^2,x)

[Out]

x*(A*b + B*a) - (A*a)/x + (B*b*x^3)/3

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 20, normalized size = 0.77 \begin {gather*} - \frac {A a}{x} + \frac {B b x^{3}}{3} + x \left (A b + B a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a/x + B*b*x**3/3 + x*(A*b + B*a)

________________________________________________________________________________________