Optimal. Leaf size=55 \[ -\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{3/2}}-\frac {c^2}{a x}+\frac {d^2 x}{b} \]
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Rubi [A] time = 0.05, antiderivative size = 55, 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 = {461, 205} \[ -\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{3/2}}-\frac {c^2}{a x}+\frac {d^2 x}{b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 461
Rubi steps
\begin {align*} \int \frac {\left (c+d x^2\right )^2}{x^2 \left (a+b x^2\right )} \, dx &=\int \left (\frac {d^2}{b}+\frac {c^2}{a x^2}-\frac {(-b c+a d)^2}{a b \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac {c^2}{a x}+\frac {d^2 x}{b}-\frac {(b c-a d)^2 \int \frac {1}{a+b x^2} \, dx}{a b}\\ &=-\frac {c^2}{a x}+\frac {d^2 x}{b}-\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 1.00 \[ -\frac {(a d-b c)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} b^{3/2}}-\frac {c^2}{a x}+\frac {d^2 x}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 164, normalized size = 2.98 \[ \left [\frac {2 \, a^{2} b d^{2} x^{2} - 2 \, a b^{2} c^{2} - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a b} x \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a^{2} b^{2} x}, \frac {a^{2} b d^{2} x^{2} - a b^{2} c^{2} - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b} x \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a^{2} b^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 63, normalized size = 1.15 \[ \frac {d^{2} x}{b} - \frac {c^{2}}{a x} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 1.55 \[ -\frac {a \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {b \,c^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {2 c d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {d^{2} x}{b}-\frac {c^{2}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.43, size = 63, normalized size = 1.15 \[ \frac {d^{2} x}{b} - \frac {c^{2}}{a x} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 80, normalized size = 1.45 \[ \frac {d^2\,x}{b}-\frac {c^2}{a\,x}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2}{\sqrt {a}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{3/2}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 165, normalized size = 3.00 \[ \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log {\left (- \frac {a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (a d - b c\right )^{2} \log {\left (\frac {a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {d^{2} x}{b} - \frac {c^{2}}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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