Optimal. Leaf size=28 \[ A b x+\frac {1}{3} (b B+A c) x^3+\frac {1}{5} B c x^5 \]
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Rubi [A]
time = 0.01, antiderivative size = 28, 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 = {1598, 380}
\begin {gather*} \frac {1}{3} x^3 (A c+b B)+A b x+\frac {1}{5} B c x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^2} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b+(b B+A c) x^2+B c x^4\right ) \, dx\\ &=A b x+\frac {1}{3} (b B+A c) x^3+\frac {1}{5} B c x^5\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} A b x+\frac {1}{3} (b B+A c) x^3+\frac {1}{5} B c x^5 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 25, normalized size = 0.89
method | result | size |
default | \(A b x +\frac {\left (A c +B b \right ) x^{3}}{3}+\frac {B c \,x^{5}}{5}\) | \(25\) |
risch | \(A b x +\frac {1}{3} x^{3} A c +\frac {1}{3} b B \,x^{3}+\frac {1}{5} B c \,x^{5}\) | \(27\) |
gosper | \(\frac {x \left (3 B c \,x^{4}+5 A c \,x^{2}+5 b B \,x^{2}+15 A b \right )}{15}\) | \(30\) |
norman | \(\frac {\left (\frac {A c}{3}+\frac {B b}{3}\right ) x^{4}+A b \,x^{2}+\frac {B c \,x^{6}}{5}}{x}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, B c x^{5} + \frac {1}{3} \, {\left (B b + A c\right )} x^{3} + A b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.43, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{5} \, B c x^{5} + \frac {1}{3} \, {\left (B b + A c\right )} x^{3} + A b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 26, normalized size = 0.93 \begin {gather*} A b x + \frac {B c x^{5}}{5} + x^{3} \left (\frac {A c}{3} + \frac {B b}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 26, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, B c x^{5} + \frac {1}{3} \, B b x^{3} + \frac {1}{3} \, A c x^{3} + A b x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 25, normalized size = 0.89 \begin {gather*} \frac {B\,c\,x^5}{5}+\left (\frac {A\,c}{3}+\frac {B\,b}{3}\right )\,x^3+A\,b\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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