Optimal. Leaf size=80 \[ \frac {1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac {1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac {1}{8} b^2 d^2 x^8 \]
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Rubi [A] time = 0.08, antiderivative size = 80, 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 = {446, 88} \begin {gather*} \frac {1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac {1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac {1}{8} b^2 d^2 x^8 \end {gather*}
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^2}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (2 a c (b c+a d)+\frac {a^2 c^2}{x}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x+2 b d (b c+a d) x^2+b^2 d^2 x^3\right ) \, dx,x,x^2\right )\\ &=a c (b c+a d) x^2+\frac {1}{4} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+\frac {1}{3} b d (b c+a d) x^6+\frac {1}{8} b^2 d^2 x^8+a^2 c^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 80, normalized size = 1.00 \begin {gather*} \frac {1}{4} x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c^2 \log (x)+\frac {1}{3} b d x^6 (a d+b c)+a c x^2 (a d+b c)+\frac {1}{8} b^2 d^2 x^8 \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 )^2}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.67, size = 82, normalized size = 1.02 \begin {gather*} \frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} \log \relax (x) + {\left (a b c^{2} + a^{2} c d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 92, normalized size = 1.15 \begin {gather*} \frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, b^{2} c d x^{6} + \frac {1}{3} \, a b d^{2} x^{6} + \frac {1}{4} \, b^{2} c^{2} x^{4} + a b c d x^{4} + \frac {1}{4} \, a^{2} d^{2} x^{4} + a b c^{2} x^{2} + a^{2} c d x^{2} + \frac {1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 1.12 \begin {gather*} \frac {b^{2} d^{2} x^{8}}{8}+\frac {a b \,d^{2} x^{6}}{3}+\frac {b^{2} c d \,x^{6}}{3}+\frac {a^{2} d^{2} x^{4}}{4}+a b c d \,x^{4}+\frac {b^{2} c^{2} x^{4}}{4}+a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+a^{2} c^{2} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 85, normalized size = 1.06 \begin {gather*} \frac {1}{8} \, b^{2} d^{2} x^{8} + \frac {1}{3} \, {\left (b^{2} c d + a b d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + \frac {1}{2} \, a^{2} c^{2} \log \left (x^{2}\right ) + {\left (a b c^{2} + a^{2} c d\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 74, normalized size = 0.92 \begin {gather*} x^4\,\left (\frac {a^2\,d^2}{4}+a\,b\,c\,d+\frac {b^2\,c^2}{4}\right )+\frac {b^2\,d^2\,x^8}{8}+a^2\,c^2\,\ln \relax (x)+a\,c\,x^2\,\left (a\,d+b\,c\right )+\frac {b\,d\,x^6\,\left (a\,d+b\,c\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 85, normalized size = 1.06 \begin {gather*} a^{2} c^{2} \log {\relax (x )} + \frac {b^{2} d^{2} x^{8}}{8} + x^{6} \left (\frac {a b d^{2}}{3} + \frac {b^{2} c d}{3}\right ) + x^{4} \left (\frac {a^{2} d^{2}}{4} + a b c d + \frac {b^{2} c^{2}}{4}\right ) + x^{2} \left (a^{2} c d + a b c^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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