Optimal. Leaf size=110 \[ -\frac {a (2 A b-3 a B) x}{b^4}+\frac {(A b-2 a B) x^3}{3 b^3}+\frac {B x^5}{5 b^2}-\frac {a^2 (A b-a B) x}{2 b^4 \left (a+b x^2\right )}+\frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {466, 1824, 211}
\begin {gather*} \frac {a^{3/2} (5 A b-7 a B) \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}-\frac {a^2 x (A b-a B)}{2 b^4 \left (a+b x^2\right )}-\frac {a x (2 A b-3 a B)}{b^4}+\frac {x^3 (A b-2 a B)}{3 b^3}+\frac {B x^5}{5 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 466
Rule 1824
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac {a^2 (A b-a B) x}{2 b^4 \left (a+b x^2\right )}-\frac {\int \frac {-a^2 (A b-a B)+2 a b (A b-a B) x^2-2 b^2 (A b-a B) x^4-2 b^3 B x^6}{a+b x^2} \, dx}{2 b^4}\\ &=-\frac {a^2 (A b-a B) x}{2 b^4 \left (a+b x^2\right )}-\frac {\int \left (2 a (2 A b-3 a B)-2 b (A b-2 a B) x^2-2 b^2 B x^4+\frac {-5 a^2 A b+7 a^3 B}{a+b x^2}\right ) \, dx}{2 b^4}\\ &=-\frac {a (2 A b-3 a B) x}{b^4}+\frac {(A b-2 a B) x^3}{3 b^3}+\frac {B x^5}{5 b^2}-\frac {a^2 (A b-a B) x}{2 b^4 \left (a+b x^2\right )}+\frac {\left (a^2 (5 A b-7 a B)\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^4}\\ &=-\frac {a (2 A b-3 a B) x}{b^4}+\frac {(A b-2 a B) x^3}{3 b^3}+\frac {B x^5}{5 b^2}-\frac {a^2 (A b-a B) x}{2 b^4 \left (a+b x^2\right )}+\frac {a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 111, normalized size = 1.01 \begin {gather*} \frac {a (-2 A b+3 a B) x}{b^4}+\frac {(A b-2 a B) x^3}{3 b^3}+\frac {B x^5}{5 b^2}-\frac {\left (a^2 A b-a^3 B\right ) x}{2 b^4 \left (a+b x^2\right )}-\frac {a^{3/2} (-5 A b+7 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 100, normalized size = 0.91
method | result | size |
default | \(-\frac {-\frac {1}{5} b^{2} B \,x^{5}-\frac {1}{3} A \,b^{2} x^{3}+\frac {2}{3} B a b \,x^{3}+2 a b A x -3 a^{2} B x}{b^{4}}+\frac {a^{2} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) x}{b \,x^{2}+a}+\frac {\left (5 A b -7 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{4}}\) | \(100\) |
risch | \(\frac {B \,x^{5}}{5 b^{2}}+\frac {A \,x^{3}}{3 b^{2}}-\frac {2 B a \,x^{3}}{3 b^{3}}-\frac {2 a A x}{b^{3}}+\frac {3 a^{2} B x}{b^{4}}+\frac {\left (-\frac {1}{2} A \,a^{2} b +\frac {1}{2} B \,a^{3}\right ) x}{b^{4} \left (b \,x^{2}+a \right )}+\frac {5 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right ) A}{4 b^{4}}-\frac {7 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) B}{4 b^{5}}-\frac {5 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right ) A}{4 b^{4}}+\frac {7 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x +a \right ) B}{4 b^{5}}\) | \(178\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 112, normalized size = 1.02 \begin {gather*} \frac {{\left (B a^{3} - A a^{2} b\right )} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} - \frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, B b^{2} x^{5} - 5 \, {\left (2 \, B a b - A b^{2}\right )} x^{3} + 15 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} x}{15 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.72, size = 298, normalized size = 2.71 \begin {gather*} \left [\frac {12 \, B b^{3} x^{7} - 4 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 20 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{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 ) + 30 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x}{60 \, {\left (b^{5} x^{2} + a b^{4}\right )}}, \frac {6 \, B b^{3} x^{7} - 2 \, {\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{5} + 10 \, {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3} - 15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b + {\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 15 \, {\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} x}{30 \, {\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs.
\(2 (104) = 208\).
time = 0.48, size = 211, normalized size = 1.92 \begin {gather*} \frac {B x^{5}}{5 b^{2}} + x^{3} \left (\frac {A}{3 b^{2}} - \frac {2 B a}{3 b^{3}}\right ) + x \left (- \frac {2 A a}{b^{3}} + \frac {3 B a^{2}}{b^{4}}\right ) + \frac {x \left (- A a^{2} b + B a^{3}\right )}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right ) \log {\left (- \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right )}{- 5 A a b + 7 B a^{2}} + x \right )}}{4} - \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right ) \log {\left (\frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (- 5 A b + 7 B a\right )}{- 5 A a b + 7 B a^{2}} + x \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 115, normalized size = 1.05 \begin {gather*} -\frac {{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {B a^{3} x - A a^{2} b x}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, B b^{8} x^{5} - 10 \, B a b^{7} x^{3} + 5 \, A b^{8} x^{3} + 45 \, B a^{2} b^{6} x - 30 \, A a b^{7} x}{15 \, b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 141, normalized size = 1.28 \begin {gather*} x^3\,\left (\frac {A}{3\,b^2}-\frac {2\,B\,a}{3\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )+\frac {B\,x^5}{5\,b^2}+\frac {x\,\left (\frac {B\,a^3}{2}-\frac {A\,a^2\,b}{2}\right )}{b^5\,x^2+a\,b^4}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (5\,A\,b-7\,B\,a\right )}{7\,B\,a^3-5\,A\,a^2\,b}\right )\,\left (5\,A\,b-7\,B\,a\right )}{2\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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