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

Optimal. Leaf size=48 \[ -\frac {a^2 A}{3 x^3}+b x (2 a B+A b)-\frac {a (a B+2 A b)}{x}+\frac {1}{3} b^2 B x^3 \]

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \begin {gather*} -\frac {a^2 A}{3 x^3}+b x (2 a B+A b)-\frac {a (a B+2 A b)}{x}+\frac {1}{3} b^2 B x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.02, size = 50, normalized size = 1.04 \begin {gather*} \frac {a^2 (-B)-2 a A b}{x}-\frac {a^2 A}{3 x^3}+b x (2 a B+A b)+\frac {1}{3} b^2 B x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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 )^2 \left (A+B x^2\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 52, normalized size = 1.08 \begin {gather*} \frac {B b^{2} x^{6} + 3 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - A a^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.35, size = 50, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, B b^{2} x^{3} + 2 \, B a b x + A b^{2} x - \frac {3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + A a^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 46, normalized size = 0.96 \begin {gather*} \frac {B \,b^{2} x^{3}}{3}+A \,b^{2} x +2 B a b x -\frac {A \,a^{2}}{3 x^{3}}-\frac {\left (2 A b +B a \right ) a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 50, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, B b^{2} x^{3} + {\left (2 \, B a b + A b^{2}\right )} x - \frac {A a^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 50, normalized size = 1.04 \begin {gather*} x\,\left (A\,b^2+2\,B\,a\,b\right )-\frac {x^2\,\left (B\,a^2+2\,A\,b\,a\right )+\frac {A\,a^2}{3}}{x^3}+\frac {B\,b^2\,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)^2)/x^4,x)

[Out]

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

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sympy [A]  time = 0.25, size = 51, normalized size = 1.06 \begin {gather*} \frac {B b^{2} x^{3}}{3} + x \left (A b^{2} + 2 B a b\right ) + \frac {- A a^{2} + x^{2} \left (- 6 A a b - 3 B a^{2}\right )}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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