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

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

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1584, 448} \begin {gather*} x (A c+b B)-\frac {A b}{x}+\frac {1}{3} B c x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b)/x) + (b*B + A*c)*x + (B*c*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]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.36, size = 28, normalized size = 1.08 \begin {gather*} \frac {B c x^{4} + 3 \, {\left (B b + A c\right )} x^{2} - 3 \, A b}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, B c x^{3} + B b x + A c x - \frac {A b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*c*x^3 + B*b*x + A*c*x - A*b/x

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maple [A]  time = 0.06, size = 24, normalized size = 0.92 \begin {gather*} \frac {B c \,x^{3}}{3}+A c x +B b x -\frac {A b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 24, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, B c x^{3} + {\left (B b + A c\right )} x - \frac {A b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*c*x^3 + (B*b + A*c)*x - A*b/x

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mupad [B]  time = 0.07, size = 24, normalized size = 0.92 \begin {gather*} x\,\left (A\,c+B\,b\right )-\frac {A\,b}{x}+\frac {B\,c\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(A*c + B*b) - (A*b)/x + (B*c*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*b/x + B*c*x**3/3 + x*(A*c + B*b)

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