Optimal. Leaf size=92 \[ \frac {(a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}+\frac {x (a d+3 b c)}{8 a^2 b \left (a+b x^2\right )}+\frac {x (b c-a d)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {385, 199, 205} \begin {gather*} \frac {(a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}+\frac {x (a d+3 b c)}{8 a^2 b \left (a+b x^2\right )}+\frac {x (b c-a d)}{4 a b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 385
Rubi steps
\begin {align*} \int \frac {c+d x^2}{\left (a+b x^2\right )^3} \, dx &=\frac {(b c-a d) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 b c+a d) \int \frac {1}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(b c-a d) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 b c+a d) x}{8 a^2 b \left (a+b x^2\right )}+\frac {(3 b c+a d) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b}\\ &=\frac {(b c-a d) x}{4 a b \left (a+b x^2\right )^2}+\frac {(3 b c+a d) x}{8 a^2 b \left (a+b x^2\right )}+\frac {(3 b c+a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 84, normalized size = 0.91 \begin {gather*} \frac {(a d+3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2}}+\frac {x \left (a^2 (-d)+a b \left (5 c+d x^2\right )+3 b^2 c x^2\right )}{8 a^2 b \left (a+b x^2\right )^2} \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 {c+d x^2}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 301, normalized size = 3.27 \begin {gather*} \left [\frac {2 \, {\left (3 \, a b^{3} c + a^{2} b^{2} d\right )} x^{3} - {\left ({\left (3 \, b^{3} c + a b^{2} d\right )} x^{4} + 3 \, a^{2} b c + a^{3} d + 2 \, {\left (3 \, a b^{2} c + a^{2} b d\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (5 \, a^{2} b^{2} c - a^{3} b d\right )} x}{16 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}, \frac {{\left (3 \, a b^{3} c + a^{2} b^{2} d\right )} x^{3} + {\left ({\left (3 \, b^{3} c + a b^{2} d\right )} x^{4} + 3 \, a^{2} b c + a^{3} d + 2 \, {\left (3 \, a b^{2} c + a^{2} b d\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (5 \, a^{2} b^{2} c - a^{3} b d\right )} x}{8 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.58, size = 78, normalized size = 0.85 \begin {gather*} \frac {{\left (3 \, b c + a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b} + \frac {3 \, b^{2} c x^{3} + a b d x^{3} + 5 \, a b c x - a^{2} d x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 0.97 \begin {gather*} \frac {d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {3 c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}+\frac {\frac {\left (a d +3 b c \right ) x^{3}}{8 a^{2}}-\frac {\left (a d -5 b c \right ) x}{8 a b}}{\left (b \,x^{2}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 92, normalized size = 1.00 \begin {gather*} \frac {{\left (3 \, b^{2} c + a b d\right )} x^{3} + {\left (5 \, a b c - a^{2} d\right )} x}{8 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {{\left (3 \, b c + a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.02, size = 81, normalized size = 0.88 \begin {gather*} \frac {\frac {x^3\,\left (a\,d+3\,b\,c\right )}{8\,a^2}-\frac {x\,\left (a\,d-5\,b\,c\right )}{8\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d+3\,b\,c\right )}{8\,a^{5/2}\,b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 150, normalized size = 1.63 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d + 3 b c\right ) \log {\left (- a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} b^{3}}} \left (a d + 3 b c\right ) \log {\left (a^{3} b \sqrt {- \frac {1}{a^{5} b^{3}}} + x \right )}}{16} + \frac {x^{3} \left (a b d + 3 b^{2} c\right ) + x \left (- a^{2} d + 5 a b c\right )}{8 a^{4} b + 16 a^{3} b^{2} x^{2} + 8 a^{2} b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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