Optimal. Leaf size=87 \[ -\frac {a^2}{5 c x^5}-\frac {a (2 b c-a d)}{3 c^2 x^3}-\frac {(b c-a d)^2}{c^3 x}-\frac {\sqrt {d} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 87, 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}{5 c x^5}-\frac {\sqrt {d} (b c-a d)^2 \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}}-\frac {(b c-a d)^2}{c^3 x}-\frac {a (2 b c-a d)}{3 c^2 x^3} \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^6 \left (c+d x^2\right )} \, dx &=\int \left (\frac {a^2}{c x^6}-\frac {a (-2 b c+a d)}{c^2 x^4}+\frac {(b c-a d)^2}{c^3 x^2}-\frac {d (b c-a d)^2}{c^3 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {a^2}{5 c x^5}-\frac {a (2 b c-a d)}{3 c^2 x^3}-\frac {(b c-a d)^2}{c^3 x}-\frac {\left (d (b c-a d)^2\right ) \int \frac {1}{c+d x^2} \, dx}{c^3}\\ &=-\frac {a^2}{5 c x^5}-\frac {a (2 b c-a d)}{3 c^2 x^3}-\frac {(b c-a d)^2}{c^3 x}-\frac {\sqrt {d} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 86, normalized size = 0.99 \begin {gather*} -\frac {a^2}{5 c x^5}+\frac {a (-2 b c+a d)}{3 c^2 x^3}-\frac {(b c-a d)^2}{c^3 x}-\frac {\sqrt {d} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 100, normalized size = 1.15
method | result | size |
default | \(-\frac {d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{c^{3} \sqrt {c d}}-\frac {a^{2}}{5 c \,x^{5}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{c^{3} x}+\frac {a \left (a d -2 b c \right )}{3 c^{2} x^{3}}\) | \(100\) |
risch | \(\frac {-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) x^{4}}{c^{3}}+\frac {a \left (a d -2 b c \right ) x^{2}}{3 c^{2}}-\frac {a^{2}}{5 c}}{x^{5}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c^{7} \textit {\_Z}^{2}+a^{4} d^{5}-4 a^{3} b c \,d^{4}+6 a^{2} b^{2} c^{2} d^{3}-4 a \,b^{3} c^{3} d^{2}+b^{4} c^{4} d \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} c^{7}+2 a^{4} d^{5}-8 a^{3} b c \,d^{4}+12 a^{2} b^{2} c^{2} d^{3}-8 a \,b^{3} c^{3} d^{2}+2 b^{4} c^{4} d \right ) x +\left (a^{2} c^{4} d^{2}-2 a b \,c^{5} d +b^{2} c^{6}\right ) \textit {\_R} \right )\right )}{2}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 107, normalized size = 1.23 \begin {gather*} -\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{3}} - \frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 5 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{15 \, c^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.33, size = 236, normalized size = 2.71 \begin {gather*} \left [\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) - 30 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} - 6 \, a^{2} c^{2} - 10 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{30 \, c^{3} x^{5}}, -\frac {15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{5} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} + 5 \, {\left (2 \, a b c^{2} - a^{2} c d\right )} x^{2}}{15 \, c^{3} x^{5}}\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. 207 vs.
\(2 (75) = 150\).
time = 0.49, size = 207, normalized size = 2.38 \begin {gather*} \frac {\sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2} \log {\left (- \frac {c^{4} \sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} - \frac {\sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2} \log {\left (\frac {c^{4} \sqrt {- \frac {d}{c^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{3} - 2 a b c d^{2} + b^{2} c^{2} d} + x \right )}}{2} + \frac {- 3 a^{2} c^{2} + x^{4} \left (- 15 a^{2} d^{2} + 30 a b c d - 15 b^{2} c^{2}\right ) + x^{2} \cdot \left (5 a^{2} c d - 10 a b c^{2}\right )}{15 c^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 112, normalized size = 1.29 \begin {gather*} -\frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} c^{3}} - \frac {15 \, b^{2} c^{2} x^{4} - 30 \, a b c d x^{4} + 15 \, a^{2} d^{2} x^{4} + 10 \, a b c^{2} x^{2} - 5 \, a^{2} c d x^{2} + 3 \, a^{2} c^{2}}{15 \, c^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 129, normalized size = 1.48 \begin {gather*} \frac {a^2\,d}{3\,c^2\,x^3}-\frac {b^2}{c\,x}-\frac {a^2}{5\,c\,x^5}-\frac {a^2\,d^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{7/2}}-\frac {b^2\,\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{3/2}}-\frac {a^2\,d^2}{c^3\,x}-\frac {2\,a\,b}{3\,c\,x^3}+\frac {2\,a\,b\,d}{c^2\,x}+\frac {2\,a\,b\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {c}}\right )}{c^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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