Optimal. Leaf size=63 \[ \frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}+\frac {d x (2 b c-a d)}{b^2}+\frac {d^2 x^3}{3 b} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {390, 205} \[ \frac {d x (2 b c-a d)}{b^2}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}+\frac {d^2 x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 390
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx &=\int \left (\frac {d (2 b c-a d)}{b^2}+\frac {d^2 x^2}{b}+\frac {b^2 c^2-2 a b c d+a^2 d^2}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^3}{3 b}+\frac {(b c-a d)^2 \int \frac {1}{a+b x^2} \, dx}{b^2}\\ &=\frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^3}{3 b}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.94 \[ \frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}+\frac {d x \left (-3 a d+6 b c+b d x^2\right )}{3 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 181, normalized size = 2.87 \[ \left [\frac {2 \, a b^{2} d^{2} x^{3} - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x}{6 \, a b^{3}}, \frac {a b^{2} d^{2} x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x}{3 \, a b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 72, normalized size = 1.14 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} d^{2} x^{3} + 6 \, b^{2} c d x - 3 \, a b d^{2} x}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 95, normalized size = 1.51 \[ \frac {d^{2} x^{3}}{3 b}+\frac {a^{2} d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {2 a c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {a \,d^{2} x}{b^{2}}+\frac {2 c d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.40, size = 69, normalized size = 1.10 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b d^{2} x^{3} + 3 \, {\left (2 \, b c d - a d^{2}\right )} x}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 90, normalized size = 1.43 \[ \frac {d^2\,x^3}{3\,b}-x\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2}{\sqrt {a}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{\sqrt {a}\,b^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.44, size = 172, normalized size = 2.73 \[ x \left (- \frac {a d^{2}}{b^{2}} + \frac {2 c d}{b}\right ) - \frac {\sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2} \log {\left (- \frac {a b^{2} \sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2} \log {\left (\frac {a b^{2} \sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {d^{2} x^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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