Optimal. Leaf size=66 \[ -\frac {a^2}{3 c x^3}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}}-\frac {a (2 b c-a d)}{c^2 x} \]
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Rubi [A] time = 0.05, antiderivative size = 66, 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} \[ -\frac {a^2}{3 c x^3}-\frac {a (2 b c-a d)}{c^2 x}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 461
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )} \, dx &=\int \left (\frac {a^2}{c x^4}-\frac {a (-2 b c+a d)}{c^2 x^2}+\frac {(b c-a d)^2}{c^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {a^2}{3 c x^3}-\frac {a (2 b c-a d)}{c^2 x}+\frac {(b c-a d)^2 \int \frac {1}{c+d x^2} \, dx}{c^2}\\ &=-\frac {a^2}{3 c x^3}-\frac {a (2 b c-a d)}{c^2 x}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 64, normalized size = 0.97 \[ -\frac {a^2}{3 c x^3}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}}+\frac {a (a d-2 b c)}{c^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 192, normalized size = 2.91 \[ \left [-\frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-c d} x^{3} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 2 \, a^{2} c^{2} d + 6 \, {\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{2}}{6 \, c^{3} d x^{3}}, \frac {3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {c d} x^{3} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) - a^{2} c^{2} d - 3 \, {\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{2}}{3 \, c^{3} d x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 71, normalized size = 1.08 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{2}} - \frac {6 \, a b c x^{2} - 3 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 98, normalized size = 1.48 \[ \frac {a^{2} d^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, c^{2}}-\frac {2 a b d \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, c}+\frac {b^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}}+\frac {a^{2} d}{c^{2} x}-\frac {2 a b}{c x}-\frac {a^{2}}{3 c \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 71, normalized size = 1.08 \[ \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{2}} - \frac {a^{2} c + 3 \, {\left (2 \, a b c - a^{2} d\right )} x^{2}}{3 \, c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 90, normalized size = 1.36 \[ \frac {a^2\,d}{c^2\,x}-\frac {a^2}{3\,c\,x^3}+\frac {a^2\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{5/2}}+\frac {b^2\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{\sqrt {c}\,\sqrt {d}}-\frac {2\,a\,b}{c\,x}-\frac {2\,a\,b\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.65, size = 172, normalized size = 2.61 \[ - \frac {\sqrt {- \frac {1}{c^{5} d}} \left (a d - b c\right )^{2} \log {\left (- \frac {c^{3} \sqrt {- \frac {1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{c^{5} d}} \left (a d - b c\right )^{2} \log {\left (\frac {c^{3} \sqrt {- \frac {1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {- a^{2} c + x^{2} \left (3 a^{2} d - 6 a b c\right )}{3 c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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