Optimal. Leaf size=126 \[ -\frac {a (6 b c-5 a d)}{3 c^3 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}+\frac {(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}} \]
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Rubi [A]
time = 0.09, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {473, 467, 464,
211} \begin {gather*} \frac {x \left (5 a^2 d^2-6 a b c d+3 b^2 c^2\right )}{6 c^3 \left (c+d x^2\right )}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {(b c-5 a d) (b c-a d) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}}-\frac {a (6 b c-5 a d)}{3 c^3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 464
Rule 467
Rule 473
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )^2} \, dx &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\int \frac {a (6 b c-5 a d)+3 b^2 c x^2}{x^2 \left (c+d x^2\right )^2} \, dx}{3 c}\\ &=-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}-\frac {\int \frac {-\frac {2 a (6 b c-5 a d)}{c}-\left (3 b^2-\frac {6 a b d}{c}+\frac {5 a^2 d^2}{c^2}\right ) x^2}{x^2 \left (c+d x^2\right )} \, dx}{6 c}\\ &=-\frac {a (6 b c-5 a d)}{3 c^3 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}+\frac {((b c-5 a d) (b c-a d)) \int \frac {1}{c+d x^2} \, dx}{2 c^3}\\ &=-\frac {a (6 b c-5 a d)}{3 c^3 x}-\frac {a^2}{3 c x^3 \left (c+d x^2\right )}+\frac {\left (3 b^2 c^2-6 a b c d+5 a^2 d^2\right ) x}{6 c^3 \left (c+d x^2\right )}+\frac {(b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 107, normalized size = 0.85 \begin {gather*} -\frac {a^2}{3 c^2 x^3}+\frac {2 a (-b c+a d)}{c^3 x}+\frac {(b c-a d)^2 x}{2 c^3 \left (c+d x^2\right )}+\frac {\left (b^2 c^2-6 a b c d+5 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{7/2} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 107, normalized size = 0.85
method | result | size |
default | \(\frac {\frac {\left (\frac {1}{2} a^{2} d^{2}-a b c d +\frac {1}{2} b^{2} c^{2}\right ) x}{d \,x^{2}+c}+\frac {\left (5 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \sqrt {c d}}}{c^{3}}-\frac {a^{2}}{3 c^{2} x^{3}}+\frac {2 a \left (a d -b c \right )}{c^{3} x}\) | \(107\) |
risch | \(\frac {\frac {\left (5 a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) x^{4}}{2 c^{3}}+\frac {a \left (5 a d -6 b c \right ) x^{2}}{3 c^{2}}-\frac {a^{2}}{3 c}}{x^{3} \left (d \,x^{2}+c \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c^{7} d \,\textit {\_Z}^{2}+25 a^{4} d^{4}-60 a^{3} b c \,d^{3}+46 a^{2} b^{2} c^{2} d^{2}-12 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} c^{7} d +50 a^{4} d^{4}-120 a^{3} b c \,d^{3}+92 a^{2} b^{2} c^{2} d^{2}-24 a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x +\left (-5 a^{2} c^{4} d^{2}+6 a b \,c^{5} d -b^{2} c^{6}\right ) \textit {\_R} \right )\right )}{4}\) | \(232\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 118, normalized size = 0.94 \begin {gather*} \frac {3 \, {\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} - 2 \, a^{2} c^{2} - 2 \, {\left (6 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}}{6 \, {\left (c^{3} d x^{5} + c^{4} x^{3}\right )}} + \frac {{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.32, size = 356, normalized size = 2.83 \begin {gather*} \left [-\frac {4 \, a^{2} c^{3} d - 6 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{4} + 4 \, {\left (6 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2} + 3 \, {\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right )}{12 \, {\left (c^{4} d^{2} x^{5} + c^{5} d x^{3}\right )}}, -\frac {2 \, a^{2} c^{3} d - 3 \, {\left (b^{2} c^{3} d - 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{4} + 2 \, {\left (6 \, a b c^{3} d - 5 \, a^{2} c^{2} d^{2}\right )} x^{2} - 3 \, {\left ({\left (b^{2} c^{2} d - 6 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{5} + {\left (b^{2} c^{3} - 6 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} x^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right )}{6 \, {\left (c^{4} d^{2} x^{5} + c^{5} d x^{3}\right )}}\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. 248 vs.
\(2 (114) = 228\).
time = 0.54, size = 248, normalized size = 1.97 \begin {gather*} - \frac {\sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log {\left (- \frac {c^{4} \sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right ) \log {\left (\frac {c^{4} \sqrt {- \frac {1}{c^{7} d}} \left (a d - b c\right ) \left (5 a d - b c\right )}{5 a^{2} d^{2} - 6 a b c d + b^{2} c^{2}} + x \right )}}{4} + \frac {- 2 a^{2} c^{2} + x^{4} \cdot \left (15 a^{2} d^{2} - 18 a b c d + 3 b^{2} c^{2}\right ) + x^{2} \cdot \left (10 a^{2} c d - 12 a b c^{2}\right )}{6 c^{4} x^{3} + 6 c^{3} d x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 111, normalized size = 0.88 \begin {gather*} \frac {{\left (b^{2} c^{2} - 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, \sqrt {c d} c^{3}} + \frac {b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \, {\left (d x^{2} + c\right )} c^{3}} - \frac {6 \, a b c x^{2} - 6 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 147, normalized size = 1.17 \begin {gather*} \frac {\frac {x^4\,\left (5\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{2\,c^3}-\frac {a^2}{3\,c}+\frac {a\,x^2\,\left (5\,a\,d-6\,b\,c\right )}{3\,c^2}}{d\,x^5+c\,x^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-b\,c\right )}{\sqrt {c}\,\left (5\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (5\,a\,d-b\,c\right )}{2\,c^{7/2}\,\sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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