Optimal. Leaf size=89 \[ -\frac {a (4 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}+\frac {x \sqrt {a+b x^2} (4 A b-3 a B)}{8 b^2}+\frac {B x^3 \sqrt {a+b x^2}}{4 b} \]
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Rubi [A] time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {459, 321, 217, 206} \[ \frac {x \sqrt {a+b x^2} (4 A b-3 a B)}{8 b^2}-\frac {a (4 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}+\frac {B x^3 \sqrt {a+b x^2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 321
Rule 459
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {a+b x^2}} \, dx &=\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {(-4 A b+3 a B) \int \frac {x^2}{\sqrt {a+b x^2}} \, dx}{4 b}\\ &=\frac {(4 A b-3 a B) x \sqrt {a+b x^2}}{8 b^2}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {(a (4 A b-3 a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b^2}\\ &=\frac {(4 A b-3 a B) x \sqrt {a+b x^2}}{8 b^2}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {(a (4 A b-3 a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b^2}\\ &=\frac {(4 A b-3 a B) x \sqrt {a+b x^2}}{8 b^2}+\frac {B x^3 \sqrt {a+b x^2}}{4 b}-\frac {a (4 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 0.83 \[ \frac {\sqrt {b} x \sqrt {a+b x^2} \left (-3 a B+4 A b+2 b B x^2\right )+a (3 a B-4 A b) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 162, normalized size = 1.82 \[ \left [-\frac {{\left (3 \, B a^{2} - 4 \, A a b\right )} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2 \, B b^{2} x^{3} - {\left (3 \, B a b - 4 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, b^{3}}, -\frac {{\left (3 \, B a^{2} - 4 \, A a b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (2 \, B b^{2} x^{3} - {\left (3 \, B a b - 4 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 75, normalized size = 0.84 \[ \frac {1}{8} \, \sqrt {b x^{2} + a} {\left (\frac {2 \, B x^{2}}{b} - \frac {3 \, B a b - 4 \, A b^{2}}{b^{3}}\right )} x - \frac {{\left (3 \, B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 1.13 \[ \frac {\sqrt {b \,x^{2}+a}\, B \,x^{3}}{4 b}-\frac {A a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 b^{\frac {3}{2}}}+\frac {3 B \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {5}{2}}}+\frac {\sqrt {b \,x^{2}+a}\, A x}{2 b}-\frac {3 \sqrt {b \,x^{2}+a}\, B a x}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 86, normalized size = 0.97 \[ \frac {\sqrt {b x^{2} + a} B x^{3}}{4 \, b} - \frac {3 \, \sqrt {b x^{2} + a} B a x}{8 \, b^{2}} + \frac {\sqrt {b x^{2} + a} A x}{2 \, b} + \frac {3 \, B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {5}{2}}} - \frac {A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {3}{2}}} \]
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 \[ \int \frac {x^2\,\left (B\,x^2+A\right )}{\sqrt {b\,x^2+a}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.78, size = 150, normalized size = 1.69 \[ \frac {A \sqrt {a} x \sqrt {1 + \frac {b x^{2}}{a}}}{2 b} - \frac {A a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 b^{\frac {3}{2}}} - \frac {3 B a^{\frac {3}{2}} x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B \sqrt {a} x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {B x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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