Optimal. Leaf size=87 \[ \frac {a (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2}}+\frac {x \sqrt {a+b x^2} (4 A b-a B)}{8 b}+\frac {B x \left (a+b x^2\right )^{3/2}}{4 b} \]
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Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {388, 195, 217, 206} \[ \frac {a (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2}}+\frac {x \sqrt {a+b x^2} (4 A b-a B)}{8 b}+\frac {B x \left (a+b x^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 388
Rubi steps
\begin {align*} \int \sqrt {a+b x^2} \left (A+B x^2\right ) \, dx &=\frac {B x \left (a+b x^2\right )^{3/2}}{4 b}-\frac {(-4 A b+a B) \int \sqrt {a+b x^2} \, dx}{4 b}\\ &=\frac {(4 A b-a B) x \sqrt {a+b x^2}}{8 b}+\frac {B x \left (a+b x^2\right )^{3/2}}{4 b}+\frac {(a (4 A b-a B)) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b}\\ &=\frac {(4 A b-a B) x \sqrt {a+b x^2}}{8 b}+\frac {B x \left (a+b x^2\right )^{3/2}}{4 b}+\frac {(a (4 A b-a B)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b}\\ &=\frac {(4 A b-a B) x \sqrt {a+b x^2}}{8 b}+\frac {B x \left (a+b x^2\right )^{3/2}}{4 b}+\frac {a (4 A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 85, normalized size = 0.98 \[ \frac {\sqrt {a+b x^2} \left (\sqrt {b} x \left (B \left (a+2 b x^2\right )+4 A b\right )-\frac {\sqrt {a} (a B-4 A b) \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{8 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 155, normalized size = 1.78 \[ \left [-\frac {{\left (B a^{2} - 4 \, A a b\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (2 \, B b^{2} x^{3} + {\left (B a b + 4 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, b^{2}}, \frac {{\left (B a^{2} - 4 \, A a b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (2 \, B b^{2} x^{3} + {\left (B a b + 4 \, A b^{2}\right )} x\right )} \sqrt {b x^{2} + a}}{8 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 69, normalized size = 0.79 \[ \frac {1}{8} \, {\left (2 \, B x^{2} + \frac {B a b + 4 \, A b^{2}}{b^{2}}\right )} \sqrt {b x^{2} + a} x + \frac {{\left (B a^{2} - 4 \, A a b\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 96, normalized size = 1.10 \[ \frac {A a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}-\frac {B \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {3}{2}}}+\frac {\sqrt {b \,x^{2}+a}\, A x}{2}-\frac {\sqrt {b \,x^{2}+a}\, B a x}{8 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B x}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 81, normalized size = 0.93 \[ \frac {1}{2} \, \sqrt {b x^{2} + a} A x + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B x}{4 \, b} - \frac {\sqrt {b x^{2} + a} B a x}{8 \, b} - \frac {B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {A a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b}} \]
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 \[ \int \left (B\,x^2+A\right )\,\sqrt {b\,x^2+a} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.30, size = 144, normalized size = 1.66 \[ \frac {A \sqrt {a} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {A a \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{2 \sqrt {b}} + \frac {B a^{\frac {3}{2}} x}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B \sqrt {a} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + \frac {B b x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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