Optimal. Leaf size=48 \[ -\frac {a^2 c}{x}+\frac {1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac {1}{5} b^2 d x^5 \]
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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} \[ -\frac {a^2 c}{x}+\frac {1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac {1}{5} b^2 d x^5 \]
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^2} \, dx &=\int \left (a (2 b c+a d)+\frac {a^2 c}{x^2}+b (b c+2 a d) x^2+b^2 d x^4\right ) \, dx\\ &=-\frac {a^2 c}{x}+a (2 b c+a d) x+\frac {1}{3} b (b c+2 a d) x^3+\frac {1}{5} b^2 d x^5\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.00 \[ -\frac {a^2 c}{x}+\frac {1}{3} b x^3 (2 a d+b c)+a x (a d+2 b c)+\frac {1}{5} b^2 d x^5 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 53, normalized size = 1.10 \[ \frac {3 \, b^{2} d x^{6} + 5 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} - 15 \, a^{2} c + 15 \, {\left (2 \, a b c + a^{2} d\right )} x^{2}}{15 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 48, normalized size = 1.00 \[ \frac {1}{5} \, b^{2} d x^{5} + \frac {1}{3} \, b^{2} c x^{3} + \frac {2}{3} \, a b d x^{3} + 2 \, a b c x + a^{2} d x - \frac {a^{2} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 49, normalized size = 1.02 \[ \frac {b^{2} d \,x^{5}}{5}+\frac {2 a b d \,x^{3}}{3}+\frac {b^{2} c \,x^{3}}{3}+a^{2} d x +2 a b c x -\frac {a^{2} c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 48, normalized size = 1.00 \[ \frac {1}{5} \, b^{2} d x^{5} + \frac {1}{3} \, {\left (b^{2} c + 2 \, a b d\right )} x^{3} - \frac {a^{2} c}{x} + {\left (2 \, a b c + a^{2} d\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 48, normalized size = 1.00 \[ x\,\left (d\,a^2+2\,b\,c\,a\right )+x^3\,\left (\frac {c\,b^2}{3}+\frac {2\,a\,d\,b}{3}\right )-\frac {a^2\,c}{x}+\frac {b^2\,d\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 48, normalized size = 1.00 \[ - \frac {a^{2} c}{x} + \frac {b^{2} d x^{5}}{5} + x^{3} \left (\frac {2 a b d}{3} + \frac {b^{2} c}{3}\right ) + x \left (a^{2} d + 2 a b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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