Optimal. Leaf size=111 \[ -\frac {\sqrt {a} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x (A b-a C)}{b^2}-\frac {a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}+\frac {x^2 (b B-a D)}{2 b^2}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b} \]
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Rubi [A] time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1802, 635, 205, 260} \[ \frac {x (A b-a C)}{b^2}-\frac {\sqrt {a} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}+\frac {x^2 (b B-a D)}{2 b^2}-\frac {a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 635
Rule 1802
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x+C x^2+D x^3\right )}{a+b x^2} \, dx &=\int \left (\frac {A b-a C}{b^2}+\frac {(b B-a D) x}{b^2}+\frac {C x^2}{b}+\frac {D x^3}{b}-\frac {a (A b-a C)+a (b B-a D) x}{b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {(A b-a C) x}{b^2}+\frac {(b B-a D) x^2}{2 b^2}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b}-\frac {\int \frac {a (A b-a C)+a (b B-a D) x}{a+b x^2} \, dx}{b^2}\\ &=\frac {(A b-a C) x}{b^2}+\frac {(b B-a D) x^2}{2 b^2}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b}-\frac {(a (A b-a C)) \int \frac {1}{a+b x^2} \, dx}{b^2}-\frac {(a (b B-a D)) \int \frac {x}{a+b x^2} \, dx}{b^2}\\ &=\frac {(A b-a C) x}{b^2}+\frac {(b B-a D) x^2}{2 b^2}+\frac {C x^3}{3 b}+\frac {D x^4}{4 b}-\frac {\sqrt {a} (A b-a C) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2}}-\frac {a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 95, normalized size = 0.86 \[ \frac {b x \left (-6 a (2 C+D x)+12 A b+b x \left (6 B+4 C x+3 D x^2\right )\right )+12 \sqrt {a} \sqrt {b} (a C-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+6 a (a D-b B) \log \left (a+b x^2\right )}{12 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 238, normalized size = 2.14 \[ \left [\frac {3 \, D b^{2} x^{4} + 4 \, C b^{2} x^{3} - 6 \, {\left (D a b - B b^{2}\right )} x^{2} - 6 \, {\left (C a b - A b^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 12 \, {\left (C a b - A b^{2}\right )} x + 6 \, {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{3}}, \frac {3 \, D b^{2} x^{4} + 4 \, C b^{2} x^{3} - 6 \, {\left (D a b - B b^{2}\right )} x^{2} + 12 \, {\left (C a b - A b^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 12 \, {\left (C a b - A b^{2}\right )} x + 6 \, {\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 112, normalized size = 1.01 \[ \frac {{\left (C a^{2} - A a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {{\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac {3 \, D b^{3} x^{4} + 4 \, C b^{3} x^{3} - 6 \, D a b^{2} x^{2} + 6 \, B b^{3} x^{2} - 12 \, C a b^{2} x + 12 \, A b^{3} x}{12 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 128, normalized size = 1.15 \[ \frac {D x^{4}}{4 b}+\frac {C \,x^{3}}{3 b}-\frac {A a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {B \,x^{2}}{2 b}+\frac {C \,a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {D a \,x^{2}}{2 b^{2}}+\frac {A x}{b}-\frac {B a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}-\frac {C a x}{b^{2}}+\frac {D a^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 98, normalized size = 0.88 \[ \frac {{\left (C a^{2} - A a b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {3 \, D b x^{4} + 4 \, C b x^{3} - 6 \, {\left (D a - B b\right )} x^{2} - 12 \, {\left (C a - A b\right )} x}{12 \, b^{2}} + \frac {{\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]
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 \[ \int \frac {x^2\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.65, size = 245, normalized size = 2.21 \[ \frac {C x^{3}}{3 b} + \frac {D x^{4}}{4 b} + x^{2} \left (\frac {B}{2 b} - \frac {D a}{2 b^{2}}\right ) + x \left (\frac {A}{b} - \frac {C a}{b^{2}}\right ) + \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} - \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right ) \log {\left (x + \frac {B a b - D a^{2} + 2 b^{3} \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} - \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right )}{- A b^{2} + C a b} \right )} + \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} + \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right ) \log {\left (x + \frac {B a b - D a^{2} + 2 b^{3} \left (\frac {a \left (- B b + D a\right )}{2 b^{3}} + \frac {\sqrt {- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right )}{- A b^{2} + C a b} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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