Optimal. Leaf size=66 \[ \frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {819, 641, 217, 206} \begin {gather*} \frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 641
Rule 819
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a+b x^2\right )^{3/2}} \, dx &=-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {\int \frac {a A+2 a B x}{\sqrt {a+b x^2}} \, dx}{a b}\\ &=-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b}\\ &=-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b}\\ &=-\frac {x (A+B x)}{b \sqrt {a+b x^2}}+\frac {2 B \sqrt {a+b x^2}}{b^2}+\frac {A \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 67, normalized size = 1.02 \begin {gather*} \frac {A \sqrt {b} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )+2 a B+b x (B x-A)}{b^2 \sqrt {a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 61, normalized size = 0.92 \begin {gather*} \frac {2 a B-A b x+b B x^2}{b^2 \sqrt {a+b x^2}}-\frac {A \log \left (\sqrt {a+b x^2}-\sqrt {b} x\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 164, normalized size = 2.48 \begin {gather*} \left [\frac {{\left (A b x^{2} + A a\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt {b x^{2} + a}}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, -\frac {{\left (A b x^{2} + A a\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (B b x^{2} - A b x + 2 \, B a\right )} \sqrt {b x^{2} + a}}{b^{3} x^{2} + a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 58, normalized size = 0.88 \begin {gather*} \frac {{\left (\frac {B x}{b} - \frac {A}{b}\right )} x + \frac {2 \, B a}{b^{2}}}{\sqrt {b x^{2} + a}} - \frac {A \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 1.09 \begin {gather*} \frac {B \,x^{2}}{\sqrt {b \,x^{2}+a}\, b}-\frac {A x}{\sqrt {b \,x^{2}+a}\, b}+\frac {A \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}+\frac {2 B a}{\sqrt {b \,x^{2}+a}\, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 64, normalized size = 0.97 \begin {gather*} \frac {B x^{2}}{\sqrt {b x^{2} + a} b} - \frac {A x}{\sqrt {b x^{2} + a} b} + \frac {A \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, B a}{\sqrt {b x^{2} + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 61, normalized size = 0.92 \begin {gather*} \frac {A\,\ln \left (\sqrt {b}\,x+\sqrt {b\,x^2+a}\right )}{b^{3/2}}-\frac {A\,x}{b\,\sqrt {b\,x^2+a}}+\frac {B\,\left (b\,x^2+2\,a\right )}{b^2\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 16.61, size = 83, normalized size = 1.26 \begin {gather*} A \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{b^{\frac {3}{2}}} - \frac {x}{\sqrt {a} b \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + B \left (\begin {cases} \frac {2 a}{b^{2} \sqrt {a + b x^{2}}} + \frac {x^{2}}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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