Optimal. Leaf size=189 \[ -\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^2}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2} \]
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Rubi [A]
time = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {483, 597, 536,
211} \begin {gather*} \frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (5 b c-7 a d)}{2 a^{7/2} (b c-a d)^2}-\frac {5 b c-2 a d}{6 a^2 c x^3 (b c-a d)}+\frac {-2 a^2 d^2-2 a b c d+5 b^2 c^2}{2 a^3 c^2 x (b c-a d)}+\frac {d^{7/2} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2}+\frac {b}{2 a x^3 \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 483
Rule 536
Rule 597
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int \frac {-5 b c+2 a d-5 b d x^2}{x^4 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {\int \frac {-3 \left (5 b^2 c^2-2 a b c d-2 a^2 d^2\right )-3 b d (5 b c-2 a d) x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{6 a^2 c (b c-a d)}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}-\frac {\int \frac {-3 \left (5 b^3 c^3-2 a b^2 c^2 d-2 a^2 b c d^2-2 a^3 d^3\right )-3 b d \left (5 b^2 c^2-2 a b c d-2 a^2 d^2\right ) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{6 a^3 c^2 (b c-a d)}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {d^4 \int \frac {1}{c+d x^2} \, dx}{c^2 (b c-a d)^2}+\frac {\left (b^3 (5 b c-7 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^3 (b c-a d)^2}\\ &=-\frac {5 b c-2 a d}{6 a^2 c (b c-a d) x^3}+\frac {5 b^2 c^2-2 a b c d-2 a^2 d^2}{2 a^3 c^2 (b c-a d) x}+\frac {b}{2 a (b c-a d) x^3 \left (a+b x^2\right )}+\frac {b^{5/2} (5 b c-7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (b c-a d)^2}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 142, normalized size = 0.75 \begin {gather*} -\frac {1}{3 a^2 c x^3}+\frac {2 b c+a d}{a^3 c^2 x}-\frac {b^3 x}{2 a^3 (-b c+a d) \left (a+b x^2\right )}-\frac {b^{5/2} (-5 b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} (-b c+a d)^2}+\frac {d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{5/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 128, normalized size = 0.68
method | result | size |
default | \(-\frac {b^{3} \left (\frac {\left (\frac {a d}{2}-\frac {b c}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (7 a d -5 b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3} \left (a d -b c \right )^{2}}+\frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{c^{2} \left (a d -b c \right )^{2} \sqrt {c d}}-\frac {1}{3 a^{2} c \,x^{3}}-\frac {-a d -2 b c}{a^{3} c^{2} x}\) | \(128\) |
risch | \(\text {Expression too large to display}\) | \(1579\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 236, normalized size = 1.25 \begin {gather*} \frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt {c d}} + \frac {{\left (5 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {a b}} - \frac {2 \, a^{2} b c^{2} - 2 \, a^{3} c d - 3 \, {\left (5 \, b^{3} c^{2} - 2 \, a b^{2} c d - 2 \, a^{2} b d^{2}\right )} x^{4} - 2 \, {\left (5 \, a b^{2} c^{2} - 2 \, a^{2} b c d - 3 \, a^{3} d^{2}\right )} x^{2}}{6 \, {\left ({\left (a^{3} b^{2} c^{3} - a^{4} b c^{2} d\right )} x^{5} + {\left (a^{4} b c^{3} - a^{5} c^{2} d\right )} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.37, size = 1281, normalized size = 6.78 \begin {gather*} \left [-\frac {4 \, a^{2} b^{2} c^{3} - 8 \, a^{3} b c^{2} d + 4 \, a^{4} c d^{2} - 6 \, {\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d + 2 \, a^{3} b d^{3}\right )} x^{4} - 4 \, {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + 3 \, a^{4} d^{3}\right )} x^{2} + 3 \, {\left ({\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 6 \, {\left (a^{3} b d^{3} x^{5} + a^{4} d^{3} x^{3}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{12 \, {\left ({\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{3} d + a^{5} b c^{2} d^{2}\right )} x^{5} + {\left (a^{4} b^{2} c^{4} - 2 \, a^{5} b c^{3} d + a^{6} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {4 \, a^{2} b^{2} c^{3} - 8 \, a^{3} b c^{2} d + 4 \, a^{4} c d^{2} - 6 \, {\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d + 2 \, a^{3} b d^{3}\right )} x^{4} - 4 \, {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + 3 \, a^{4} d^{3}\right )} x^{2} - 12 \, {\left (a^{3} b d^{3} x^{5} + a^{4} d^{3} x^{3}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + 3 \, {\left ({\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{12 \, {\left ({\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{3} d + a^{5} b c^{2} d^{2}\right )} x^{5} + {\left (a^{4} b^{2} c^{4} - 2 \, a^{5} b c^{3} d + a^{6} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, a^{2} b^{2} c^{3} - 4 \, a^{3} b c^{2} d + 2 \, a^{4} c d^{2} - 3 \, {\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d + 2 \, a^{3} b d^{3}\right )} x^{4} - 2 \, {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + 3 \, a^{4} d^{3}\right )} x^{2} - 3 \, {\left ({\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - 3 \, {\left (a^{3} b d^{3} x^{5} + a^{4} d^{3} x^{3}\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} + 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{6 \, {\left ({\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{3} d + a^{5} b c^{2} d^{2}\right )} x^{5} + {\left (a^{4} b^{2} c^{4} - 2 \, a^{5} b c^{3} d + a^{6} c^{2} d^{2}\right )} x^{3}\right )}}, -\frac {2 \, a^{2} b^{2} c^{3} - 4 \, a^{3} b c^{2} d + 2 \, a^{4} c d^{2} - 3 \, {\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d + 2 \, a^{3} b d^{3}\right )} x^{4} - 2 \, {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + 3 \, a^{4} d^{3}\right )} x^{2} - 3 \, {\left ({\left (5 \, b^{4} c^{3} - 7 \, a b^{3} c^{2} d\right )} x^{5} + {\left (5 \, a b^{3} c^{3} - 7 \, a^{2} b^{2} c^{2} d\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - 6 \, {\left (a^{3} b d^{3} x^{5} + a^{4} d^{3} x^{3}\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right )}{6 \, {\left ({\left (a^{3} b^{3} c^{4} - 2 \, a^{4} b^{2} c^{3} d + a^{5} b c^{2} d^{2}\right )} x^{5} + {\left (a^{4} b^{2} c^{4} - 2 \, a^{5} b c^{3} d + a^{6} c^{2} d^{2}\right )} x^{3}\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 = 1.53, size = 165, normalized size = 0.87 \begin {gather*} \frac {d^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \sqrt {c d}} + \frac {b^{3} x}{2 \, {\left (a^{3} b c - a^{4} d\right )} {\left (b x^{2} + a\right )}} + \frac {{\left (5 \, b^{4} c - 7 \, a b^{3} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{3} b^{2} c^{2} - 2 \, a^{4} b c d + a^{5} d^{2}\right )} \sqrt {a b}} + \frac {6 \, b c x^{2} + 3 \, a d x^{2} - a c}{3 \, a^{3} c^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.64, size = 2500, normalized size = 13.23 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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