Optimal. Leaf size=92 \[ \frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{3/2}}+\frac {x (3 a d+b c)}{8 c^2 d \left (c+d x^2\right )}-\frac {x (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
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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} \[ \frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{3/2}}+\frac {x (3 a d+b c)}{8 c^2 d \left (c+d x^2\right )}-\frac {x (b c-a d)}{4 c d \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 385
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\left (c+d x^2\right )^3} \, dx &=-\frac {(b c-a d) x}{4 c d \left (c+d x^2\right )^2}+\frac {(b c+3 a d) \int \frac {1}{\left (c+d x^2\right )^2} \, dx}{4 c d}\\ &=-\frac {(b c-a d) x}{4 c d \left (c+d x^2\right )^2}+\frac {(b c+3 a d) x}{8 c^2 d \left (c+d x^2\right )}+\frac {(b c+3 a d) \int \frac {1}{c+d x^2} \, dx}{8 c^2 d}\\ &=-\frac {(b c-a d) x}{4 c d \left (c+d x^2\right )^2}+\frac {(b c+3 a d) x}{8 c^2 d \left (c+d x^2\right )}+\frac {(b c+3 a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 82, normalized size = 0.89 \[ \frac {(3 a d+b c) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{8 c^{5/2} d^{3/2}}+\frac {x \left (a d \left (5 c+3 d x^2\right )+b c \left (d x^2-c\right )\right )}{8 c^2 d \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 300, normalized size = 3.26 \[ \left [\frac {2 \, {\left (b c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{3} - {\left ({\left (b c d^{2} + 3 \, a d^{3}\right )} x^{4} + b c^{3} + 3 \, a c^{2} d + 2 \, {\left (b c^{2} d + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) - 2 \, {\left (b c^{3} d - 5 \, a c^{2} d^{2}\right )} x}{16 \, {\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} d^{2}\right )}}, \frac {{\left (b c^{2} d^{2} + 3 \, a c d^{3}\right )} x^{3} + {\left ({\left (b c d^{2} + 3 \, a d^{3}\right )} x^{4} + b c^{3} + 3 \, a c^{2} d + 2 \, {\left (b c^{2} d + 3 \, a c d^{2}\right )} x^{2}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - {\left (b c^{3} d - 5 \, a c^{2} d^{2}\right )} x}{8 \, {\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} d^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 78, normalized size = 0.85 \[ \frac {{\left (b c + 3 \, a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{2} d} + \frac {b c d x^{3} + 3 \, a d^{2} x^{3} - b c^{2} x + 5 \, a c d x}{8 \, {\left (d x^{2} + c\right )}^{2} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.98 \[ \frac {3 a \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c^{2}}+\frac {b \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \sqrt {c d}\, c d}+\frac {\frac {\left (3 a d +b c \right ) x^{3}}{8 c^{2}}+\frac {\left (5 a d -b c \right ) x}{8 c d}}{\left (d \,x^{2}+c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 92, normalized size = 1.00 \[ \frac {{\left (b c d + 3 \, a d^{2}\right )} x^{3} - {\left (b c^{2} - 5 \, a c d\right )} x}{8 \, {\left (c^{2} d^{3} x^{4} + 2 \, c^{3} d^{2} x^{2} + c^{4} d\right )}} + \frac {{\left (b c + 3 \, a d\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{8 \, \sqrt {c d} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.06, size = 82, normalized size = 0.89 \[ \frac {\frac {x^3\,\left (3\,a\,d+b\,c\right )}{8\,c^2}+\frac {x\,\left (5\,a\,d-b\,c\right )}{8\,c\,d}}{c^2+2\,c\,d\,x^2+d^2\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )\,\left (3\,a\,d+b\,c\right )}{8\,c^{5/2}\,d^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 150, normalized size = 1.63 \[ - \frac {\sqrt {- \frac {1}{c^{5} d^{3}}} \left (3 a d + b c\right ) \log {\left (- c^{3} d \sqrt {- \frac {1}{c^{5} d^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{c^{5} d^{3}}} \left (3 a d + b c\right ) \log {\left (c^{3} d \sqrt {- \frac {1}{c^{5} d^{3}}} + x \right )}}{16} + \frac {x^{3} \left (3 a d^{2} + b c d\right ) + x \left (5 a c d - b c^{2}\right )}{8 c^{4} d + 16 c^{3} d^{2} x^{2} + 8 c^{2} d^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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