Optimal. Leaf size=54 \[ -\frac {a A}{3 x^3}-\frac {a B}{2 x^2}-\frac {A b+a C}{x}+b C x+\frac {1}{2} b D x^2+(b B+a D) \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1816}
\begin {gather*} -\frac {a C+A b}{x}-\frac {a A}{3 x^3}+\log (x) (a D+b B)-\frac {a B}{2 x^2}+b C x+\frac {1}{2} b D x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 1816
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right )}{x^4} \, dx &=\int \left (b C+\frac {a A}{x^4}+\frac {a B}{x^3}+\frac {A b+a C}{x^2}+\frac {b B+a D}{x}+b D x\right ) \, dx\\ &=-\frac {a A}{3 x^3}-\frac {a B}{2 x^2}-\frac {A b+a C}{x}+b C x+\frac {1}{2} b D x^2+(b B+a D) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 55, normalized size = 1.02 \begin {gather*} -\frac {a A}{3 x^3}-\frac {a B}{2 x^2}+\frac {-A b-a C}{x}+b C x+\frac {1}{2} b D x^2+(b B+a D) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 49, normalized size = 0.91
method | result | size |
default | \(\frac {b D x^{2}}{2}+b C x -\frac {a B}{2 x^{2}}-\frac {a A}{3 x^{3}}+\left (B b +a D\right ) \ln \left (x \right )-\frac {A b +a C}{x}\) | \(49\) |
norman | \(\frac {\left (-A b -a C \right ) x^{2}+b C \,x^{4}-\frac {A a}{3}-\frac {B a x}{2}+\frac {b D x^{5}}{2}}{x^{3}}+\left (B b +a D\right ) \ln \left (x \right )\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 49, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, D b x^{2} + C b x + {\left (D a + B b\right )} \log \left (x\right ) - \frac {3 \, B a x + 6 \, {\left (C a + A b\right )} x^{2} + 2 \, A a}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.77, size = 55, normalized size = 1.02 \begin {gather*} \frac {3 \, D b x^{5} + 6 \, C b x^{4} + 6 \, {\left (D a + B b\right )} x^{3} \log \left (x\right ) - 3 \, B a x - 6 \, {\left (C a + A b\right )} x^{2} - 2 \, A a}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.38, size = 54, normalized size = 1.00 \begin {gather*} C b x + \frac {D b x^{2}}{2} + \left (B b + D a\right ) \log {\left (x \right )} + \frac {- 2 A a - 3 B a x + x^{2} \left (- 6 A b - 6 C a\right )}{6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 50, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, D b x^{2} + C b x + {\left (D a + B b\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, B a x + 6 \, {\left (C a + A b\right )} x^{2} + 2 \, A a}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 50, normalized size = 0.93 \begin {gather*} \frac {b\,x^2\,D}{2}+a\,\ln \left (x\right )\,D+C\,b\,x-\frac {A\,a}{3\,x^3}-\frac {A\,b}{x}-\frac {B\,a}{2\,x^2}-\frac {C\,a}{x}+B\,b\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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