Optimal. Leaf size=66 \[ -\frac {a^2}{3 c x^3}-\frac {a (2 b c-a d)}{c^2 x}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}} \]
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Rubi [A]
time = 0.04, 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 = {472, 211}
\begin {gather*} -\frac {a^2}{3 c x^3}+\frac {(b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} \sqrt {d}}-\frac {a (2 b c-a d)}{c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 472
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 = 0.04, size = 64, normalized size = 0.97 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 68, normalized size = 1.03
method | result | size |
default | \(\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{c^{2} \sqrt {c d}}-\frac {a^{2}}{3 c \,x^{3}}+\frac {a \left (a d -2 b c \right )}{c^{2} x}\) | \(68\) |
risch | \(\frac {\frac {a \left (a d -2 b c \right ) x^{2}}{c^{2}}-\frac {a^{2}}{3 c}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c^{5} d \,\textit {\_Z}^{2}+a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} c^{5} d +2 a^{4} d^{4}-8 a^{3} b c \,d^{3}+12 a^{2} b^{2} c^{2} d^{2}-8 a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x +\left (-a^{2} c^{3} d^{2}+2 a b \,c^{4} d -b^{2} c^{5}\right ) \textit {\_R} \right )\right )}{2}\) | \(190\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.68, size = 71, normalized size = 1.08 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.06, size = 192, normalized size = 2.91 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 172 vs.
\(2 (56) = 112\).
time = 0.39, size = 172, normalized size = 2.61 \begin {gather*} - \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} \cdot \left (3 a^{2} d - 6 a b c\right )}{3 c^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 71, normalized size = 1.08 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 90, normalized size = 1.36 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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