3.2.49 \(\int \frac {(a+b x^2)^2 (c+d x^2)}{x^4} \, dx\)

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

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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 c}{3 x^3}+b x (2 a d+b c)-\frac {a (a d+2 b c)}{x}+\frac {1}{3} b^2 d x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2))/x^4,x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2))/x^4,x]

[Out]

-1/3*(a^2*c)/x^3 + (-2*a*b*c - a^2*d)/x + b*(b*c + 2*a*d)*x + (b^2*d*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 (c+d x^2\right )}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^4,x]

[Out]

IntegrateAlgebraic[((a + b*x^2)^2*(c + d*x^2))/x^4, x]

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fricas [A]  time = 0.49, size = 52, normalized size = 1.08 \begin {gather*} \frac {b^{2} d x^{6} + 3 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} - a^{2} c - 3 \, {\left (2 \, a b c + a^{2} d\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*(d*x^2+c)/x^4,x, algorithm="fricas")

[Out]

1/3*(b^2*d*x^6 + 3*(b^2*c + 2*a*b*d)*x^4 - a^2*c - 3*(2*a*b*c + a^2*d)*x^2)/x^3

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giac [A]  time = 0.42, size = 50, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, b^{2} d x^{3} + b^{2} c x + 2 \, a b d x - \frac {6 \, a b c x^{2} + 3 \, a^{2} d x^{2} + a^{2} c}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*d*x^3 + b^2*c*x + 2*a*b*d*x - 1/3*(6*a*b*c*x^2 + 3*a^2*d*x^2 + a^2*c)/x^3

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maple [A]  time = 0.00, size = 46, normalized size = 0.96 \begin {gather*} \frac {b^{2} d \,x^{3}}{3}+2 a b d x +b^{2} c x -\frac {a^{2} c}{3 x^{3}}-\frac {\left (a d +2 b c \right ) a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)/x^4,x)

[Out]

1/3*b^2*d*x^3+2*a*b*d*x+b^2*c*x-a*(a*d+2*b*c)/x-1/3*a^2*c/x^3

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maxima [A]  time = 1.04, size = 50, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, b^{2} d x^{3} + {\left (b^{2} c + 2 \, a b d\right )} x - \frac {a^{2} c + 3 \, {\left (2 \, a b c + a^{2} d\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*(d*x^2+c)/x^4,x, algorithm="maxima")

[Out]

1/3*b^2*d*x^3 + (b^2*c + 2*a*b*d)*x - 1/3*(a^2*c + 3*(2*a*b*c + a^2*d)*x^2)/x^3

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mupad [B]  time = 0.05, size = 50, normalized size = 1.04 \begin {gather*} x\,\left (c\,b^2+2\,a\,d\,b\right )-\frac {\frac {a^2\,c}{3}+x^2\,\left (d\,a^2+2\,b\,c\,a\right )}{x^3}+\frac {b^2\,d\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x^2)^2*(c + d*x^2))/x^4,x)

[Out]

x*(b^2*c + 2*a*b*d) - ((a^2*c)/3 + x^2*(a^2*d + 2*a*b*c))/x^3 + (b^2*d*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)/x**4,x)

[Out]

b**2*d*x**3/3 + x*(2*a*b*d + b**2*c) + (-a**2*c + x**2*(-3*a**2*d - 6*a*b*c))/(3*x**3)

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