Optimal. Leaf size=55 \[ \frac {1}{7} \left (7 x^2-5\right )^{3/2}+x \sqrt {7 x^2-5}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {7 x^2-5}}\right )}{\sqrt {7}} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {641, 195, 217, 206} \begin {gather*} \frac {1}{7} \left (7 x^2-5\right )^{3/2}+x \sqrt {7 x^2-5}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {7 x^2-5}}\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 641
Rubi steps
\begin {align*} \int (2+3 x) \sqrt {-5+7 x^2} \, dx &=\frac {1}{7} \left (-5+7 x^2\right )^{3/2}+2 \int \sqrt {-5+7 x^2} \, dx\\ &=x \sqrt {-5+7 x^2}+\frac {1}{7} \left (-5+7 x^2\right )^{3/2}-5 \int \frac {1}{\sqrt {-5+7 x^2}} \, dx\\ &=x \sqrt {-5+7 x^2}+\frac {1}{7} \left (-5+7 x^2\right )^{3/2}-5 \operatorname {Subst}\left (\int \frac {1}{1-7 x^2} \, dx,x,\frac {x}{\sqrt {-5+7 x^2}}\right )\\ &=x \sqrt {-5+7 x^2}+\frac {1}{7} \left (-5+7 x^2\right )^{3/2}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {-5+7 x^2}}\right )}{\sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 50, normalized size = 0.91 \begin {gather*} \left (x^2+x-\frac {5}{7}\right ) \sqrt {7 x^2-5}-\frac {5 \log \left (\sqrt {7} \sqrt {7 x^2-5}+7 x\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 61, normalized size = 1.11 \begin {gather*} \frac {1}{7} \sqrt {7 x^2-5} \left (7 x^2+7 x-5\right )-\frac {10 \tanh ^{-1}\left (\frac {\sqrt {7 x^2-5}}{\sqrt {7} x+\sqrt {5}}\right )}{\sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 50, normalized size = 0.91 \begin {gather*} \frac {1}{7} \, {\left (7 \, x^{2} + 7 \, x - 5\right )} \sqrt {7 \, x^{2} - 5} + \frac {5}{14} \, \sqrt {7} \log \left (-2 \, \sqrt {7} \sqrt {7 \, x^{2} - 5} x + 14 \, x^{2} - 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 43, normalized size = 0.78 \begin {gather*} \frac {1}{7} \, {\left (7 \, {\left (x + 1\right )} x - 5\right )} \sqrt {7 \, x^{2} - 5} + \frac {5}{7} \, \sqrt {7} \log \left ({\left | -\sqrt {7} x + \sqrt {7 \, x^{2} - 5} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 0.82 \begin {gather*} \sqrt {7 x^{2}-5}\, x -\frac {5 \sqrt {7}\, \ln \left (\sqrt {7}\, x +\sqrt {7 x^{2}-5}\right )}{7}+\frac {\left (7 x^{2}-5\right )^{\frac {3}{2}}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.85, size = 47, normalized size = 0.85 \begin {gather*} \frac {1}{7} \, {\left (7 \, x^{2} - 5\right )}^{\frac {3}{2}} + \sqrt {7 \, x^{2} - 5} x - \frac {5}{7} \, \sqrt {7} \log \left (2 \, \sqrt {7} \sqrt {7 \, x^{2} - 5} + 14 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 44, normalized size = 0.80 \begin {gather*} x\,\sqrt {7\,x^2-5}-\frac {5\,\sqrt {7}\,\ln \left (\sqrt {7}\,x+\sqrt {7\,x^2-5}\right )}{7}+\frac {{\left (7\,x^2-5\right )}^{3/2}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 56, normalized size = 1.02 \begin {gather*} x^{2} \sqrt {7 x^{2} - 5} + x \sqrt {7 x^{2} - 5} - \frac {5 \sqrt {7 x^{2} - 5}}{7} - \frac {5 \sqrt {7} \operatorname {acosh}{\left (\frac {\sqrt {35} x}{5} \right )}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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