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.
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Rule 448
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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