Optimal. Leaf size=77 \[ -\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^5}{5 b}+\frac {a^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, 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 = {470, 308, 211}
\begin {gather*} \frac {a^{3/2} (A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}-\frac {a x (A b-a B)}{b^3}+\frac {x^3 (A b-a B)}{3 b^2}+\frac {B x^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 308
Rule 470
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{a+b x^2} \, dx &=\frac {B x^5}{5 b}-\frac {(-5 A b+5 a B) \int \frac {x^4}{a+b x^2} \, dx}{5 b}\\ &=\frac {B x^5}{5 b}-\frac {(-5 A b+5 a B) \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{5 b}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^5}{5 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {1}{a+b x^2} \, dx}{b^3}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^5}{5 b}+\frac {a^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 77, normalized size = 1.00 \begin {gather*} \frac {a (-A b+a B) x}{b^3}+\frac {(A b-a B) x^3}{3 b^2}+\frac {B x^5}{5 b}-\frac {a^{3/2} (-A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 75, normalized size = 0.97
method | result | size |
default | \(-\frac {-\frac {1}{5} b^{2} B \,x^{5}-\frac {1}{3} A \,b^{2} x^{3}+\frac {1}{3} B a b \,x^{3}+a b A x -a^{2} B x}{b^{3}}+\frac {a^{2} \left (A b -B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(75\) |
risch | \(\frac {B \,x^{5}}{5 b}+\frac {A \,x^{3}}{3 b}-\frac {B a \,x^{3}}{3 b^{2}}-\frac {a A x}{b^{2}}+\frac {a^{2} B x}{b^{3}}+\frac {\sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right ) A}{2 b^{3}}-\frac {\sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) B}{2 b^{4}}-\frac {\sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right ) A}{2 b^{3}}+\frac {\sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x +a \right ) B}{2 b^{4}}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.59, size = 78, normalized size = 1.01 \begin {gather*} -\frac {{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, B b^{2} x^{5} - 5 \, {\left (B a b - A b^{2}\right )} x^{3} + 15 \, {\left (B a^{2} - A a b\right )} x}{15 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.17, size = 178, normalized size = 2.31 \begin {gather*} \left [\frac {6 \, B b^{2} x^{5} - 10 \, {\left (B a b - A b^{2}\right )} x^{3} - 15 \, {\left (B a^{2} - A a b\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 (B a^{2} - A a b\right )} x}{30 \, b^{3}}, \frac {3 \, B b^{2} x^{5} - 5 \, {\left (B a b - A b^{2}\right )} x^{3} - 15 \, {\left (B a^{2} - A a b\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 15 \, {\left (B a^{2} - A a b\right )} x}{15 \, b^{3}}\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. 153 vs.
\(2 (68) = 136\).
time = 0.24, size = 153, normalized size = 1.99 \begin {gather*} \frac {B x^{5}}{5 b} + x^{3} \left (\frac {A}{3 b} - \frac {B a}{3 b^{2}}\right ) + x \left (- \frac {A a}{b^{2}} + \frac {B a^{2}}{b^{3}}\right ) + \frac {\sqrt {- \frac {a^{3}}{b^{7}}} \left (- A b + B a\right ) \log {\left (- \frac {b^{3} \sqrt {- \frac {a^{3}}{b^{7}}} \left (- A b + B a\right )}{- A a b + B a^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {a^{3}}{b^{7}}} \left (- A b + B a\right ) \log {\left (\frac {b^{3} \sqrt {- \frac {a^{3}}{b^{7}}} \left (- A b + B a\right )}{- A a b + B a^{2}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.83, size = 85, normalized size = 1.10 \begin {gather*} -\frac {{\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, B b^{4} x^{5} - 5 \, B a b^{3} x^{3} + 5 \, A b^{4} x^{3} + 15 \, B a^{2} b^{2} x - 15 \, A a b^{3} x}{15 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 96, normalized size = 1.25 \begin {gather*} x^3\,\left (\frac {A}{3\,b}-\frac {B\,a}{3\,b^2}\right )+\frac {B\,x^5}{5\,b}-\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,\left (A\,b-B\,a\right )}{B\,a^3-A\,a^2\,b}\right )\,\left (A\,b-B\,a\right )}{b^{7/2}}-\frac {a\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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