Optimal. Leaf size=80 \[ -\frac {a^2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2}}+\frac {\left (a+b x^2\right )^{3/2} (4 A+3 B x)}{12 b}-\frac {a B x \sqrt {a+b x^2}}{8 b} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {780, 195, 217, 206} \[ -\frac {a^2 B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 b^{3/2}}+\frac {\left (a+b x^2\right )^{3/2} (4 A+3 B x)}{12 b}-\frac {a B x \sqrt {a+b x^2}}{8 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 780
Rubi steps
\begin {align*} \int x (A+B x) \sqrt {a+b x^2} \, dx &=\frac {(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac {(a B) \int \sqrt {a+b x^2} \, dx}{4 b}\\ &=-\frac {a B x \sqrt {a+b x^2}}{8 b}+\frac {(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac {\left (a^2 B\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{8 b}\\ &=-\frac {a B x \sqrt {a+b x^2}}{8 b}+\frac {(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac {\left (a^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{8 b}\\ &=-\frac {a B x \sqrt {a+b x^2}}{8 b}+\frac {(4 A+3 B x) \left (a+b x^2\right )^{3/2}}{12 b}-\frac {a^2 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 = 0.15, size = 86, normalized size = 1.08 \[ \frac {\sqrt {a+b x^2} \left (\sqrt {b} \left (8 a A+3 a B x+8 A b x^2+6 b B x^3\right )-\frac {3 a^{3/2} B \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {\frac {b x^2}{a}+1}}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 157, normalized size = 1.96 \[ \left [\frac {3 \, B a^{2} \sqrt {b} \log \left (-2 \, b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (6 \, B b^{2} x^{3} + 8 \, A b^{2} x^{2} + 3 \, B a b x + 8 \, A a b\right )} \sqrt {b x^{2} + a}}{48 \, b^{2}}, \frac {3 \, B a^{2} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (6 \, B b^{2} x^{3} + 8 \, A b^{2} x^{2} + 3 \, B a b x + 8 \, A a b\right )} \sqrt {b x^{2} + a}}{24 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 68, normalized size = 0.85 \[ \frac {B a^{2} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{8 \, b^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left (3 \, B x + 4 \, A\right )} x + \frac {3 \, B a}{b}\right )} x + \frac {8 \, A a}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 75, normalized size = 0.94 \[ -\frac {B \,a^{2} \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{8 b^{\frac {3}{2}}}-\frac {\sqrt {b \,x^{2}+a}\, B a x}{8 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} B x}{4 b}+\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} A}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 67, normalized size = 0.84 \[ \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B x}{4 \, b} - \frac {\sqrt {b x^{2} + a} B a x}{8 \, b} - \frac {B a^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, b^{\frac {3}{2}}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} A}{3 \, b} \]
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 \[ \int x\,\sqrt {b\,x^2+a}\,\left (A+B\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.41, size = 124, normalized size = 1.55 \[ A \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{3 b} & \text {otherwise} \end {cases}\right ) + \frac {B a^{\frac {3}{2}} x}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 B \sqrt {a} x^{3}}{8 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {B a^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {3}{2}}} + \frac {B b x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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