3.7 \(\int \frac{(a+b x^2) (A+B x^2)}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.015107, antiderivative size = 26, normalized size of antiderivative = 1., 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} \[ -\frac{a B+A b}{x}-\frac{a A}{3 x^3}+b B x \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(A + B*x^2))/x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0124445, size = 27, normalized size = 1.04 \[ \frac{-a B-A b}{x}-\frac{a A}{3 x^3}+b B x \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)*(A + B*x^2))/x^4,x]

[Out]

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

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Maple [A]  time = 0.004, size = 25, normalized size = 1. \begin{align*} bBx-{\frac{Aa}{3\,{x}^{3}}}-{\frac{Ab+Ba}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)*(B*x^2+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 0.986408, size = 35, normalized size = 1.35 \begin{align*} B b x - \frac{3 \,{\left (B a + A b\right )} x^{2} + A a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.43557, size = 63, normalized size = 2.42 \begin{align*} \frac{3 \, B b x^{4} - 3 \,{\left (B a + A b\right )} x^{2} - A a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.332068, size = 26, normalized size = 1. \begin{align*} B b x - \frac{A a + x^{2} \left (3 A b + 3 B a\right )}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)*(B*x**2+A)/x**4,x)

[Out]

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

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Giac [A]  time = 1.09227, size = 38, normalized size = 1.46 \begin{align*} B b x - \frac{3 \, B a x^{2} + 3 \, A b x^{2} + A a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)*(B*x^2+A)/x^4,x, algorithm="giac")

[Out]

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