Optimal. Leaf size=42 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1586, 635, 205, 260} \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1586
Rubi steps
\begin {align*} \int \frac {a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{\left (a+b x^2\right )^3} \, dx &=\int \frac {a c+a d x+b c x^2+b d x^3}{\left (a+b x^2\right )^2} \, dx\\ &=\int \frac {c+d x}{a+b x^2} \, dx\\ &=c \int \frac {1}{a+b x^2} \, dx+d \int \frac {x}{a+b x^2} \, dx\\ &=\frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.00 \[ \frac {c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+\frac {d \log \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 98, normalized size = 2.33 \[ \left [\frac {a d \log \left (b x^{2} + a\right ) - \sqrt {-a b} c \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {a d \log \left (b x^{2} + a\right ) + 2 \, \sqrt {a b} c \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{2 \, a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 31, normalized size = 0.74 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} + \frac {d \log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.76 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {d \ln \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 31, normalized size = 0.74 \[ \frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} + \frac {d \log \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 32, normalized size = 0.76 \[ \frac {d\,\ln \left (b\,x^2+a\right )}{2\,b}+\frac {c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 124, normalized size = 2.95 \[ \left (\frac {d}{2 b} - \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) \log {\left (x + \frac {2 a b \left (\frac {d}{2 b} - \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} + \left (\frac {d}{2 b} + \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) \log {\left (x + \frac {2 a b \left (\frac {d}{2 b} + \frac {c \sqrt {- a b^{3}}}{2 a b^{2}}\right ) - a d}{b c} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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