Optimal. Leaf size=119 \[ \frac {(3 a D+b B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{5/2}}-\frac {2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left (a+b x^2\right )}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1804, 1814, 12, 205} \[ \frac {(3 a D+b B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{5/2}}-\frac {2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left (a+b x^2\right )}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 1804
Rule 1814
Rubi steps
\begin {align*} \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac {\int \frac {-a \left (B-\frac {a D}{b}\right )-2 (A b+a C) x-4 a D x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac {2 (A b+a C)-(b B-5 a D) x}{8 a b^2 \left (a+b x^2\right )}+\frac {\int \frac {a \left (B+\frac {3 a D}{b}\right )}{a+b x^2} \, dx}{8 a^2 b}\\ &=-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac {2 (A b+a C)-(b B-5 a D) x}{8 a b^2 \left (a+b x^2\right )}+\frac {(b B+3 a D) \int \frac {1}{a+b x^2} \, dx}{8 a b^2}\\ &=-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac {2 (A b+a C)-(b B-5 a D) x}{8 a b^2 \left (a+b x^2\right )}+\frac {(b B+3 a D) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{3/2} b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 99, normalized size = 0.83 \[ \frac {\frac {(3 a D+b B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {\sqrt {b} \left (-a^2 (2 C+3 D x)-a b \left (2 A+x \left (B+4 C x+5 D x^2\right )\right )+b^2 B x^3\right )}{a \left (a+b x^2\right )^2}}{8 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 357, normalized size = 3.00 \[ \left [-\frac {8 \, C a^{2} b^{2} x^{2} + 4 \, C a^{3} b + 4 \, A a^{2} b^{2} + 2 \, {\left (5 \, D a^{2} b^{2} - B a b^{3}\right )} x^{3} + {\left ({\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + 3 \, D a^{3} + B a^{2} b + 2 \, {\left (3 \, D a^{2} b + B a b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (3 \, D a^{3} b + B a^{2} b^{2}\right )} x}{16 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac {4 \, C a^{2} b^{2} x^{2} + 2 \, C a^{3} b + 2 \, A a^{2} b^{2} + {\left (5 \, D a^{2} b^{2} - B a b^{3}\right )} x^{3} - {\left ({\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + 3 \, D a^{3} + B a^{2} b + 2 \, {\left (3 \, D a^{2} b + B a b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, D a^{3} b + B a^{2} b^{2}\right )} x}{8 \, {\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 97, normalized size = 0.82 \[ \frac {{\left (3 \, D a + B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{2}} - \frac {5 \, D a b x^{3} - B b^{2} x^{3} + 4 \, C a b x^{2} + 3 \, D a^{2} x + B a b x + 2 \, C a^{2} + 2 \, A a b}{8 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 110, normalized size = 0.92 \[ \frac {B \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {3 D \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{2}}+\frac {-\frac {C \,x^{2}}{2 b}+\frac {\left (b B -5 a D\right ) x^{3}}{8 a b}-\frac {\left (b B +3 a D\right ) x}{8 b^{2}}-\frac {A b +a C}{4 b^{2}}}{\left (b \,x^{2}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 111, normalized size = 0.93 \[ -\frac {4 \, C a b x^{2} + {\left (5 \, D a b - B b^{2}\right )} x^{3} + 2 \, C a^{2} + 2 \, A a b + {\left (3 \, D a^{2} + B a b\right )} x}{8 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {{\left (3 \, D a + B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.37, size = 178, normalized size = 1.50 \[ - \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (B b + 3 D a\right ) \log {\left (- a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{3} b^{5}}} \left (B b + 3 D a\right ) \log {\left (a^{2} b^{2} \sqrt {- \frac {1}{a^{3} b^{5}}} + x \right )}}{16} + \frac {- 2 A a b - 2 C a^{2} - 4 C a b x^{2} + x^{3} \left (B b^{2} - 5 D a b\right ) + x \left (- B a b - 3 D a^{2}\right )}{8 a^{3} b^{2} + 16 a^{2} b^{3} x^{2} + 8 a b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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