Optimal. Leaf size=167 \[ \frac {d (b c+a d) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac {d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {425, 541, 536,
211} \begin {gather*} \frac {b^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-5 a d)}{2 a^{3/2} (b c-a d)^3}+\frac {d^{3/2} (5 b c-a d) \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^3}+\frac {b x}{2 a \left (a+b x^2\right ) \left (c+d x^2\right ) (b c-a d)}+\frac {d x (a d+b c)}{2 a c \left (c+d x^2\right ) (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 425
Rule 536
Rule 541
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx &=\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {-b c+2 a d-3 b d x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{2 a (b c-a d)}\\ &=\frac {d (b c+a d) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}-\frac {\int \frac {-2 \left (b^2 c^2-4 a b c d+a^2 d^2\right )-2 b d (b c+a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{4 a c (b c-a d)^2}\\ &=\frac {d (b c+a d) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {\left (b^2 (b c-5 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a (b c-a d)^3}+\frac {\left (d^2 (5 b c-a d)\right ) \int \frac {1}{c+d x^2} \, dx}{2 c (b c-a d)^3}\\ &=\frac {d (b c+a d) x}{2 a c (b c-a d)^2 \left (c+d x^2\right )}+\frac {b x}{2 a (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b^{3/2} (b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} (b c-a d)^3}+\frac {d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{2 c^{3/2} (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 136, normalized size = 0.81 \begin {gather*} \frac {1}{2} \left (\frac {b^{3/2} (-b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} (-b c+a d)^3}+\frac {(b c-a d) x \left (\frac {b^2}{a^2+a b x^2}+\frac {d^2}{c^2+c d x^2}\right )+\frac {d^{3/2} (5 b c-a d) \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2}}}{(b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 133, normalized size = 0.80
method | result | size |
default | \(\frac {b^{2} \left (\frac {\left (a d -b c \right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (5 a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}\right )}{\left (a d -b c \right )^{3}}+\frac {d^{2} \left (\frac {\left (a d -b c \right ) x}{2 c \left (d \,x^{2}+c \right )}+\frac {\left (a d -5 b c \right ) \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 c \sqrt {c d}}\right )}{\left (a d -b c \right )^{3}}\) | \(133\) |
risch | \(\text {Expression too large to display}\) | \(2124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs.
\(2 (143) = 286\).
time = 0.50, size = 294, normalized size = 1.76 \begin {gather*} \frac {{\left (b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {a b}} + \frac {{\left (5 \, b c d^{2} - a d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt {c d}} + \frac {{\left (b^{2} c d + a b d^{2}\right )} x^{3} + {\left (b^{2} c^{2} + a^{2} d^{2}\right )} x}{2 \, {\left (a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + a^{4} c^{2} d^{2} + {\left (a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3}\right )} x^{4} + {\left (a b^{3} c^{4} - a^{2} b^{2} c^{3} d - a^{3} b c^{2} d^{2} + a^{4} c d^{3}\right )} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs.
\(2 (143) = 286\).
time = 1.19, size = 1681, normalized size = 10.07 \begin {gather*} \left [\frac {2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{3} + {\left (a b^{2} c^{3} - 5 \, a^{2} b c^{2} d + {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + {\left (5 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (5 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + 2 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{4 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{3} + 2 \, {\left (5 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (5 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + {\left (a b^{2} c^{3} - 5 \, a^{2} b c^{2} d + {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 2 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{4 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )}}, \frac {2 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{3} + 2 \, {\left (a b^{2} c^{3} - 5 \, a^{2} b c^{2} d + {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + {\left (5 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (5 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right ) + 2 \, {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{4 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )}}, \frac {{\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{3} + {\left (a b^{2} c^{3} - 5 \, a^{2} b c^{2} d + {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2}\right )} x^{4} + {\left (b^{3} c^{3} - 4 \, a b^{2} c^{2} d - 5 \, a^{2} b c d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + {\left (5 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (5 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{4} + {\left (5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + {\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{2 \, {\left (a^{2} b^{3} c^{5} - 3 \, a^{3} b^{2} c^{4} d + 3 \, a^{4} b c^{3} d^{2} - a^{5} c^{2} d^{3} + {\left (a b^{4} c^{4} d - 3 \, a^{2} b^{3} c^{3} d^{2} + 3 \, a^{3} b^{2} c^{2} d^{3} - a^{4} b c d^{4}\right )} x^{4} + {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + 2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.67, size = 232, normalized size = 1.39 \begin {gather*} \frac {{\left (b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt {a b}} + \frac {{\left (5 \, b c d^{2} - a d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{2 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \sqrt {c d}} + \frac {b^{2} c d x^{3} + a b d^{2} x^{3} + b^{2} c^{2} x + a^{2} d^{2} x}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} {\left (b d x^{4} + b c x^{2} + a d x^{2} + a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.87, size = 2500, normalized size = 14.97 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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