Optimal. Leaf size=185 \[ \frac {3 (A b-5 a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{7/2}}-\frac {3 x (A b-5 a C)}{8 a b^3}+\frac {x^3 (4 x (b B-2 a D)-5 a C+A b)}{8 a b^2 \left (a+b x^2\right )}-\frac {x^4 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac {(b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}-\frac {x^2 (b B-3 a D)}{2 a b^3} \]
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Rubi [A] time = 0.34, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1804, 801, 635, 205, 260} \[ \frac {x^3 (4 x (b B-2 a D)-5 a C+A b)}{8 a b^2 \left (a+b x^2\right )}-\frac {3 x (A b-5 a C)}{8 a b^3}+\frac {3 (A b-5 a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{7/2}}-\frac {x^4 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}-\frac {x^2 (b B-3 a D)}{2 a b^3}+\frac {(b B-3 a D) \log \left (a+b x^2\right )}{2 b^4} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 801
Rule 1804
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^4 \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 {x^3 \left (-4 a \left (B-\frac {a D}{b}\right )+(A b-5 a C) x-4 a D x^2\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=-\frac {x^4 \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 {x^3 (A b-5 a C+4 (b B-2 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac {\int \frac {x^2 (-3 a (A b-5 a C)-8 a (b B-3 a D) x)}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=-\frac {x^4 \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 {x^3 (A b-5 a C+4 (b B-2 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac {\int \left (-\frac {3 a (A b-5 a C)}{b}-\frac {8 a (b B-3 a D) x}{b}+\frac {3 a^2 (A b-5 a C)+8 a^2 (b B-3 a D) x}{b \left (a+b x^2\right )}\right ) \, dx}{8 a^2 b^2}\\ &=-\frac {3 (A b-5 a C) x}{8 a b^3}-\frac {(b B-3 a D) x^2}{2 a b^3}-\frac {x^4 \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 {x^3 (A b-5 a C+4 (b B-2 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac {\int \frac {3 a^2 (A b-5 a C)+8 a^2 (b B-3 a D) x}{a+b x^2} \, dx}{8 a^2 b^3}\\ &=-\frac {3 (A b-5 a C) x}{8 a b^3}-\frac {(b B-3 a D) x^2}{2 a b^3}-\frac {x^4 \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 {x^3 (A b-5 a C+4 (b B-2 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac {(3 (A b-5 a C)) \int \frac {1}{a+b x^2} \, dx}{8 b^3}+\frac {(b B-3 a D) \int \frac {x}{a+b x^2} \, dx}{b^3}\\ &=-\frac {3 (A b-5 a C) x}{8 a b^3}-\frac {(b B-3 a D) x^2}{2 a b^3}-\frac {x^4 \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 {x^3 (A b-5 a C+4 (b B-2 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac {3 (A b-5 a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{7/2}}+\frac {(b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 139, normalized size = 0.75 \[ \frac {\frac {2 a \left (a^2 D-a b (B+C x)+A b^2 x\right )}{\left (a+b x^2\right )^2}+\frac {-12 a^2 D+8 a b B+9 a b C x-5 A b^2 x}{a+b x^2}+\frac {3 \sqrt {b} (A b-5 a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+4 (b B-3 a D) \log \left (a+b x^2\right )+8 b C x+4 b D x^2}{8 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 574, normalized size = 3.10 \[ \left [\frac {8 \, D a b^{3} x^{6} + 16 \, C a b^{3} x^{5} + 16 \, D a^{2} b^{2} x^{4} - 20 \, D a^{4} + 12 \, B a^{3} b + 10 \, {\left (5 \, C a^{2} b^{2} - A a b^{3}\right )} x^{3} - 16 \, {\left (D a^{3} b - B a^{2} b^{2}\right )} x^{2} + 3 \, {\left ({\left (5 \, C a b^{2} - A b^{3}\right )} x^{4} + 5 \, C a^{3} - A a^{2} b + 2 \, {\left (5 \, C a^{2} b - A a b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (5 \, C a^{3} b - A a^{2} b^{2}\right )} x - 8 \, {\left (3 \, D a^{4} - B a^{3} b + {\left (3 \, D a^{2} b^{2} - B a b^{3}\right )} x^{4} + 2 \, {\left (3 \, D a^{3} b - B a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{16 \, {\left (a b^{6} x^{4} + 2 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, \frac {4 \, D a b^{3} x^{6} + 8 \, C a b^{3} x^{5} + 8 \, D a^{2} b^{2} x^{4} - 10 \, D a^{4} + 6 \, B a^{3} b + 5 \, {\left (5 \, C a^{2} b^{2} - A a b^{3}\right )} x^{3} - 8 \, {\left (D a^{3} b - B a^{2} b^{2}\right )} x^{2} - 3 \, {\left ({\left (5 \, C a b^{2} - A b^{3}\right )} x^{4} + 5 \, C a^{3} - A a^{2} b + 2 \, {\left (5 \, C a^{2} b - A a b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (5 \, C a^{3} b - A a^{2} b^{2}\right )} x - 4 \, {\left (3 \, D a^{4} - B a^{3} b + {\left (3 \, D a^{2} b^{2} - B a b^{3}\right )} x^{4} + 2 \, {\left (3 \, D a^{3} b - B a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{8 \, {\left (a b^{6} x^{4} + 2 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 157, normalized size = 0.85 \[ -\frac {3 \, {\left (5 \, C a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} - \frac {{\left (3 \, D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac {D b^{3} x^{2} + 2 \, C b^{3} x}{2 \, b^{6}} - \frac {10 \, D a^{3} - 6 \, B a^{2} b - {\left (9 \, C a b^{2} - 5 \, A b^{3}\right )} x^{3} + 4 \, {\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} x^{2} - {\left (7 \, C a^{2} b - 3 \, A a b^{2}\right )} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 235, normalized size = 1.27 \[ -\frac {5 A \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b}+\frac {9 C a \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {B a \,x^{2}}{\left (b \,x^{2}+a \right )^{2} b^{2}}-\frac {3 D a^{2} x^{2}}{2 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {3 A a x}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}+\frac {7 C \,a^{2} x}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {3 A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{2}}+\frac {3 B \,a^{2}}{4 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {15 C a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{3}}-\frac {5 D a^{3}}{4 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {D x^{2}}{2 b^{3}}+\frac {B \ln \left (b \,x^{2}+a \right )}{2 b^{3}}+\frac {C x}{b^{3}}-\frac {3 D a \ln \left (b \,x^{2}+a \right )}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.07, size = 165, normalized size = 0.89 \[ -\frac {10 \, D a^{3} - 6 \, B a^{2} b - {\left (9 \, C a b^{2} - 5 \, A b^{3}\right )} x^{3} + 4 \, {\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} x^{2} - {\left (7 \, C a^{2} b - 3 \, A a b^{2}\right )} x}{8 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} - \frac {3 \, {\left (5 \, C a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} + \frac {D x^{2} + 2 \, C x}{2 \, b^{3}} - \frac {{\left (3 \, D a - B b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.56, size = 232, normalized size = 1.25 \[ \frac {\frac {7\,C\,a^2\,x}{8}+\frac {9\,C\,b\,a\,x^3}{8}}{a^2\,b^3+2\,a\,b^4\,x^2+b^5\,x^4}-\frac {\frac {5\,A\,x^3}{8\,b}+\frac {3\,A\,a\,x}{8\,b^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\frac {3\,B\,a^2}{4\,b^3}+\frac {B\,a\,x^2}{b^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {D\,\left (3\,a\,\ln \left (b\,x^2+a\right )-b\,x^2+\frac {3\,a^2}{b\,x^2+a}-\frac {a^3}{2\,{\left (b\,x^2+a\right )}^2}\right )}{2\,b^4}+\frac {B\,\ln \left (b\,x^2+a\right )}{2\,b^3}+\frac {C\,x}{b^3}+\frac {3\,A\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,b^{5/2}}-\frac {15\,C\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 29.59, size = 357, normalized size = 1.93 \[ \frac {C x}{b^{3}} + \frac {D x^{2}}{2 b^{3}} + \left (- \frac {- B b + 3 D a}{2 b^{4}} - \frac {3 \sqrt {- a b^{9}} \left (- A b + 5 C a\right )}{16 a b^{8}}\right ) \log {\left (x + \frac {8 B a b - 24 D a^{2} - 16 a b^{4} \left (- \frac {- B b + 3 D a}{2 b^{4}} - \frac {3 \sqrt {- a b^{9}} \left (- A b + 5 C a\right )}{16 a b^{8}}\right )}{- 3 A b^{2} + 15 C a b} \right )} + \left (- \frac {- B b + 3 D a}{2 b^{4}} + \frac {3 \sqrt {- a b^{9}} \left (- A b + 5 C a\right )}{16 a b^{8}}\right ) \log {\left (x + \frac {8 B a b - 24 D a^{2} - 16 a b^{4} \left (- \frac {- B b + 3 D a}{2 b^{4}} + \frac {3 \sqrt {- a b^{9}} \left (- A b + 5 C a\right )}{16 a b^{8}}\right )}{- 3 A b^{2} + 15 C a b} \right )} + \frac {6 B a^{2} b - 10 D a^{3} + x^{3} \left (- 5 A b^{3} + 9 C a b^{2}\right ) + x^{2} \left (8 B a b^{2} - 12 D a^{2} b\right ) + x \left (- 3 A a b^{2} + 7 C a^{2} b\right )}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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