Optimal. Leaf size=144 \[ -\frac {b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac {3 b c-2 a d}{2 a^2 c x (b c-a d)}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2}+\frac {b}{2 a x \left (a+b x^2\right ) (b c-a d)} \]
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Rubi [A] time = 0.21, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {472, 583, 522, 205} \begin {gather*} -\frac {b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac {3 b c-2 a d}{2 a^2 c x (b c-a d)}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2}+\frac {b}{2 a x \left (a+b x^2\right ) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 472
Rule 522
Rule 583
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx &=\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {\int \frac {-3 b c+2 a d-3 b d x^2}{x^2 \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a (b c-a d)}\\ &=-\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}+\frac {\int \frac {-3 b^2 c^2+2 a b c d+2 a^2 d^2-b d (3 b c-2 a d) x^2}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{2 a^2 c (b c-a d)}\\ &=-\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {d^3 \int \frac {1}{c+d x^2} \, dx}{c (b c-a d)^2}-\frac {\left (b^2 (3 b c-5 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^2 (b c-a d)^2}\\ &=-\frac {3 b c-2 a d}{2 a^2 c (b c-a d) x}+\frac {b}{2 a (b c-a d) x \left (a+b x^2\right )}-\frac {b^{3/2} (3 b c-5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (b c-a d)^2}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 123, normalized size = 0.85 \begin {gather*} \frac {b^{3/2} (5 a d-3 b c) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2} (a d-b c)^2}+\frac {b^2 x}{2 a^2 \left (a+b x^2\right ) (a d-b c)}-\frac {1}{a^2 c x}-\frac {d^{5/2} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{c^{3/2} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.63, size = 1003, normalized size = 6.97 \begin {gather*} \left [-\frac {4 \, a b^{2} c^{2} - 8 \, a^{2} b c d + 4 \, a^{3} d^{2} + 2 \, {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\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 (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {-\frac {d}{c}} \log \left (\frac {d x^{2} - 2 \, c x \sqrt {-\frac {d}{c}} - c}{d x^{2} + c}\right )}{4 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}, -\frac {4 \, a b^{2} c^{2} - 8 \, a^{2} b c d + 4 \, a^{3} d^{2} + 2 \, {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + 4 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right ) + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{4 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}, -\frac {2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\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 ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}, -\frac {2 \, a b^{2} c^{2} - 4 \, a^{2} b c d + 2 \, a^{3} d^{2} + {\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} x^{2} + {\left ({\left (3 \, b^{3} c^{2} - 5 \, a b^{2} c d\right )} x^{3} + {\left (3 \, a b^{2} c^{2} - 5 \, a^{2} b c d\right )} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt {\frac {d}{c}} \arctan \left (x \sqrt {\frac {d}{c}}\right )}{2 \, {\left ({\left (a^{2} b^{3} c^{3} - 2 \, a^{3} b^{2} c^{2} d + a^{4} b c d^{2}\right )} x^{3} + {\left (a^{3} b^{2} c^{3} - 2 \, a^{4} b c^{2} d + a^{5} c d^{2}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 164, normalized size = 1.14 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {c d}} - \frac {{\left (3 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b}} - \frac {3 \, b^{2} c x^{2} - 2 \, a b d x^{2} + 2 \, a b c - 2 \, a^{2} d}{2 \, {\left (a^{2} b c^{2} - a^{3} c d\right )} {\left (b x^{3} + a x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 169, normalized size = 1.17 \begin {gather*} \frac {b^{2} d x}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a}+\frac {5 b^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {a b}\, a}-\frac {b^{3} c x}{2 \left (a d -b c \right )^{2} \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b^{3} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \left (a d -b c \right )^{2} \sqrt {a b}\, a^{2}}-\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right )^{2} \sqrt {c d}\, c}-\frac {1}{a^{2} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.26, size = 178, normalized size = 1.24 \begin {gather*} -\frac {d^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {c d}} - \frac {{\left (3 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \sqrt {a b}} - \frac {2 \, a b c - 2 \, a^{2} d + {\left (3 \, b^{2} c - 2 \, a b d\right )} x^{2}}{2 \, {\left ({\left (a^{2} b^{2} c^{2} - a^{3} b c d\right )} x^{3} + {\left (a^{3} b c^{2} - a^{4} c d\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 2400, normalized size = 16.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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