Optimal. Leaf size=253 \[ \frac{4}{5} \sqrt{2 x+1}+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
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Rubi [A] time = 0.357923, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {703, 826, 1169, 634, 618, 204, 628} \[ \frac{4}{5} \sqrt{2 x+1}+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{5} \sqrt{\frac{2}{155} \left (35 \sqrt{35}-178\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 703
Rule 826
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx &=\frac{4}{5} \sqrt{1+2 x}+\frac{1}{5} \int \frac{-3+8 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac{4}{5} \sqrt{1+2 x}+\frac{2}{5} \operatorname{Subst}\left (\int \frac{-14+8 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{4}{5} \sqrt{1+2 x}+\frac{\operatorname{Subst}\left (\int \frac{-14 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (-14-8 \sqrt{\frac{7}{5}}\right ) x}{\sqrt{\frac{7}{5}}-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )}{5 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{-14 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (-14-8 \sqrt{\frac{7}{5}}\right ) x}{\sqrt{\frac{7}{5}}+\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )}{5 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=\frac{4}{5} \sqrt{1+2 x}+\frac{1}{25} \left (4-\sqrt{35}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{7}{5}}-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )+\frac{1}{25} \left (4-\sqrt{35}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{7}{5}}+\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \operatorname{Subst}\left (\int \frac{-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 x}{\sqrt{\frac{7}{5}}-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \operatorname{Subst}\left (\int \frac{\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 x}{\sqrt{\frac{7}{5}}+\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )} x+x^2} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{4}{5} \sqrt{1+2 x}+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (\sqrt{35}-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )-\frac{1}{25} \left (2 \left (4-\sqrt{35}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{2}{5} \left (2-\sqrt{35}\right )-x^2} \, dx,x,-\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 \sqrt{1+2 x}\right )-\frac{1}{25} \left (2 \left (4-\sqrt{35}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{2}{5} \left (2-\sqrt{35}\right )-x^2} \, dx,x,\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 \sqrt{1+2 x}\right )\\ &=\frac{4}{5} \sqrt{1+2 x}+\frac{1}{5} \sqrt{\frac{2}{155} \left (-178+35 \sqrt{35}\right )} \tan ^{-1}\left (\sqrt{\frac{5}{2 \left (-2+\sqrt{35}\right )}} \left (\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-2 \sqrt{1+2 x}\right )\right )-\frac{1}{5} \sqrt{\frac{2}{155} \left (-178+35 \sqrt{35}\right )} \tan ^{-1}\left (\sqrt{\frac{5}{2 \left (-2+\sqrt{35}\right )}} \left (\sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+2 \sqrt{1+2 x}\right )\right )+\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (\sqrt{35}-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )-\frac{1}{5} \sqrt{\frac{1}{310} \left (178+35 \sqrt{35}\right )} \log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )\\ \end{align*}
Mathematica [C] time = 0.265968, size = 128, normalized size = 0.51 \[ \frac{2}{775} \left (310 \sqrt{2 x+1}-i \sqrt{10-5 i \sqrt{31}} \left (2 \sqrt{31}-31 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )+i \sqrt{10+5 i \sqrt{31}} \left (2 \sqrt{31}+31 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 616, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x + 1\right )}^{\frac{3}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55124, size = 2006, normalized size = 7.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.6129, size = 85, normalized size = 0.34 \begin{align*} \frac{4 \sqrt{2 x + 1}}{5} + \frac{16 \operatorname{RootSum}{\left (1230080 t^{4} + 1984 t^{2} + 7, \left ( t \mapsto t \log{\left (9920 t^{3} + 8 t + \sqrt{2 x + 1} \right )} \right )\right )}}{5} - \frac{28 \operatorname{RootSum}{\left (1722112 t^{4} + 1984 t^{2} + 5, \left ( t \mapsto t \log{\left (- \frac{27776 t^{3}}{5} + \frac{108 t}{5} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, x + 1\right )}^{\frac{3}{2}}}{5 \, x^{2} + 3 \, x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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