Optimal. Leaf size=98 \[ -\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+b x (2 a C+A b)-\frac {a (a C+2 A b)}{x}+\frac {1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac {1}{3} b^2 C x^3+\frac {1}{4} b^2 D x^4 \]
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Rubi [A] time = 0.09, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1802} \begin {gather*} -\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}+b x (2 a C+A b)-\frac {a (a C+2 A b)}{x}+\frac {1}{2} b x^2 (2 a D+b B)+a \log (x) (a D+2 b B)+\frac {1}{3} b^2 C x^3+\frac {1}{4} b^2 D x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 1802
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x^4} \, dx &=\int \left (b (A b+2 a C)+\frac {a^2 A}{x^4}+\frac {a^2 B}{x^3}+\frac {a (2 A b+a C)}{x^2}+\frac {a (2 b B+a D)}{x}+b (b B+2 a D) x+b^2 C x^2+b^2 D x^3\right ) \, dx\\ &=-\frac {a^2 A}{3 x^3}-\frac {a^2 B}{2 x^2}-\frac {a (2 A b+a C)}{x}+b (A b+2 a C) x+\frac {1}{2} b (b B+2 a D) x^2+\frac {1}{3} b^2 C x^3+\frac {1}{4} b^2 D x^4+a (2 b B+a D) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.85 \begin {gather*} -\frac {a^2 (2 A+3 x (B+2 C x))}{6 x^3}-\frac {2 a A b}{x}+a \log (x) (a D+2 b B)+a b x (2 C+D x)+\frac {1}{12} b^2 x \left (12 A+x \left (6 B+4 C x+3 D x^2\right )\right ) \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 (A+B x+C x^2+D x^3\right )}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.88, size = 103, normalized size = 1.05 \begin {gather*} \frac {3 \, D b^{2} x^{7} + 4 \, C b^{2} x^{6} + 6 \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + 12 \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + 12 \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} \log \relax (x) - 6 \, B a^{2} x - 4 \, A a^{2} - 12 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 97, normalized size = 0.99 \begin {gather*} \frac {1}{4} \, D b^{2} x^{4} + \frac {1}{3} \, C b^{2} x^{3} + D a b x^{2} + \frac {1}{2} \, B b^{2} x^{2} + 2 \, C a b x + A b^{2} x + {\left (D a^{2} + 2 \, B a b\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, B a^{2} x + 2 \, A a^{2} + 6 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 97, normalized size = 0.99 \begin {gather*} \frac {D b^{2} x^{4}}{4}+\frac {C \,b^{2} x^{3}}{3}+\frac {B \,b^{2} x^{2}}{2}+D a b \,x^{2}+A \,b^{2} x +2 B a b \ln \relax (x )+2 C a b x +D a^{2} \ln \relax (x )-\frac {2 A a b}{x}-\frac {C \,a^{2}}{x}-\frac {B \,a^{2}}{2 x^{2}}-\frac {A \,a^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 97, normalized size = 0.99 \begin {gather*} \frac {1}{4} \, D b^{2} x^{4} + \frac {1}{3} \, C b^{2} x^{3} + \frac {1}{2} \, {\left (2 \, D a b + B b^{2}\right )} x^{2} + {\left (2 \, C a b + A b^{2}\right )} x + {\left (D a^{2} + 2 \, B a b\right )} \log \relax (x) - \frac {3 \, B a^{2} x + 2 \, A a^{2} + 6 \, {\left (C a^{2} + 2 \, A a b\right )} x^{2}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 106, normalized size = 1.08 \begin {gather*} \frac {b^2\,x^4\,D}{4}+\frac {a^2\,\ln \left (x^2\right )\,D}{2}-\frac {A\,\left (a^2+6\,a\,b\,x^2-3\,b^2\,x^4\right )}{3\,x^3}+\frac {B\,\left (b^2\,x^4-a^2+4\,a\,b\,x^2\,\ln \relax (x)\right )}{2\,x^2}+\frac {C\,\left (-3\,a^2+6\,a\,b\,x^2+b^2\,x^4\right )}{3\,x}+a\,b\,x^2\,D \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.46, size = 100, normalized size = 1.02 \begin {gather*} \frac {C b^{2} x^{3}}{3} + \frac {D b^{2} x^{4}}{4} + a \left (2 B b + D a\right ) \log {\relax (x )} + x^{2} \left (\frac {B b^{2}}{2} + D a b\right ) + x \left (A b^{2} + 2 C a b\right ) + \frac {- 2 A a^{2} - 3 B a^{2} x + x^{2} \left (- 12 A a b - 6 C a^{2}\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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