Optimal. Leaf size=161 \[ \frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 161, 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 x (3 b c-7 a d)}{8 a^2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (15 a^2 d^2-10 a b c d+3 b^2 c^2\right )}{8 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}+\frac {b x}{4 a \left (a+b x^2\right )^2 (b c-a d)} \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 )^3 \left (c+d x^2\right )} \, dx &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}-\frac {\int \frac {-3 b c+4 a d-3 b d x^2}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx}{4 a (b c-a d)}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\int \frac {3 b^2 c^2-7 a b c d+8 a^2 d^2+b d (3 b c-7 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{8 a^2 (b c-a d)^2}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {d^3 \int \frac {1}{c+d x^2} \, dx}{(b c-a d)^3}+\frac {\left (b \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 (b c-a d)^3}\\ &=\frac {b x}{4 a (b c-a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{8 a^2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} (b c-a d)^3}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 158, normalized size = 0.98 \begin {gather*} \frac {1}{8} \left (-\frac {2 b x}{a (-b c+a d) \left (a+b x^2\right )^2}+\frac {b (3 b c-7 a d) x}{a^2 (b c-a d)^2 \left (a+b x^2\right )}-\frac {\sqrt {b} \left (3 b^2 c^2-10 a b c d+15 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2} (-b c+a d)^3}-\frac {8 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} (b c-a d)^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 158, normalized size = 0.98
method | result | size |
default | \(-\frac {b \left (\frac {\frac {b \left (7 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) x^{3}}{8 a^{2}}+\frac {\left (9 a^{2} d^{2}-14 a b c d +5 b^{2} c^{2}\right ) x}{8 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (15 a^{2} d^{2}-10 a b c d +3 b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}\right )}{\left (a d -b c \right )^{3}}+\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{3} \sqrt {c d}}\) | \(158\) |
risch | \(\text {Expression too large to display}\) | \(2285\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 278, normalized size = 1.73 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{3} c^{2} - 10 \, a b^{2} c d + 15 \, a^{2} b d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} + \frac {{\left (3 \, b^{3} c - 7 \, a b^{2} d\right )} x^{3} + {\left (5 \, a b^{2} c - 9 \, a^{2} b d\right )} x}{8 \, {\left (a^{4} b^{2} c^{2} - 2 \, a^{5} b c d + a^{6} d^{2} + {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\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. 372 vs.
\(2 (139) = 278\).
time = 1.04, size = 1587, normalized size = 9.86 \begin {gather*} \left [\frac {2 \, {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{3} - {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 15 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 15 \, a^{3} b 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 ) - 8 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{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 (5 \, a b^{3} c^{2} - 14 \, a^{2} b^{2} c d + 9 \, a^{3} b d^{2}\right )} x}{16 \, {\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} + {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} x^{2}\right )}}, \frac {2 \, {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{3} - 16 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) - {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 15 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 15 \, a^{3} b 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 (5 \, a b^{3} c^{2} - 14 \, a^{2} b^{2} c d + 9 \, a^{3} b d^{2}\right )} x}{16 \, {\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} + {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} x^{2}\right )}}, \frac {{\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{3} + {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 15 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 15 \, a^{3} b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - 4 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{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 ) + {\left (5 \, a b^{3} c^{2} - 14 \, a^{2} b^{2} c d + 9 \, a^{3} b d^{2}\right )} x}{8 \, {\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} + {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} x^{2}\right )}}, \frac {{\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{3} + {\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 15 \, a^{4} d^{2} + {\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 15 \, a^{2} b^{2} d^{2}\right )} x^{4} + 2 \, {\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 15 \, a^{3} b d^{2}\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - 8 \, {\left (a^{2} b^{2} d^{2} x^{4} + 2 \, a^{3} b d^{2} x^{2} + a^{4} d^{2}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + {\left (5 \, a b^{3} c^{2} - 14 \, a^{2} b^{2} c d + 9 \, a^{3} b d^{2}\right )} x}{8 \, {\left (a^{4} b^{3} c^{3} - 3 \, a^{5} b^{2} c^{2} d + 3 \, a^{6} b c d^{2} - a^{7} d^{3} + {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} x^{4} + 2 \, {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\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 = 1.45, size = 218, normalized size = 1.35 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d}} + \frac {{\left (3 \, b^{3} c^{2} - 10 \, a b^{2} c d + 15 \, a^{2} b d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \sqrt {a b}} + \frac {3 \, b^{3} c x^{3} - 7 \, a b^{2} d x^{3} + 5 \, a b^{2} c x - 9 \, a^{2} b d x}{8 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} {\left (b x^{2} + a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.89, size = 2500, normalized size = 15.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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