Optimal. Leaf size=270 \[ -\frac{\sqrt{2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
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Rubi [A] time = 0.363569, antiderivative size = 270, 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 = {738, 826, 1169, 634, 618, 204, 628} \[ -\frac{\sqrt{2 x+1} (5-4 x)}{31 \left (5 x^2+3 x+2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{31} \sqrt{\frac{1}{310} \left (47 \sqrt{35}-218\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )+\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \sqrt{35}\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 738
Rule 826
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(1+2 x)^{3/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \frac{9+4 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{2}{31} \operatorname{Subst}\left (\int \frac{14+4 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{14 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (14-4 \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 )}{31 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{14 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (14-4 \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 )}{31 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=-\frac{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{155} \left (2+\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}{155} \left (2+\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}{31} \sqrt{\frac{1}{310} \left (-218+47 \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}{31} \sqrt{\frac{1}{310} \left (-218+47 \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{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac{1}{31} \sqrt{\frac{1}{310} \left (-218+47 \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}{31} \sqrt{\frac{1}{310} \left (-218+47 \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}{155} \left (2 \left (2+\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}{155} \left (2 \left (2+\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{(5-4 x) \sqrt{1+2 x}}{31 \left (2+3 x+5 x^2\right )}-\frac{1}{31} \sqrt{\frac{2}{155} \left (218+47 \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}{31} \sqrt{\frac{2}{155} \left (218+47 \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}{31} \sqrt{\frac{1}{310} \left (-218+47 \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}{31} \sqrt{\frac{1}{310} \left (-218+47 \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 = 0.718012, size = 141, normalized size = 0.52 \[ \frac{\frac{155 \sqrt{2 x+1} (4 x-5)}{5 x^2+3 x+2}+2 \left (31-4 i \sqrt{31}\right ) \sqrt{10-5 i \sqrt{31}} \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )+2 \left (31+4 i \sqrt{31}\right ) \sqrt{10+5 i \sqrt{31}} \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )}{4805} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.082, size = 642, 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}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.19425, size = 2195, normalized size = 8.13 \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 = 58.9013, size = 211, normalized size = 0.78 \begin{align*} \frac{64 \left (2 x + 1\right )^{\frac{3}{2}}}{- 992 x + 620 \left (2 x + 1\right )^{2} + 372} - \frac{224 \left (2 x + 1\right )^{\frac{3}{2}}}{- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604} - \frac{128 \sqrt{2 x + 1}}{5 \left (- 992 x + 620 \left (2 x + 1\right )^{2} + 372\right )} - \frac{3024 \sqrt{2 x + 1}}{5 \left (- 6944 x + 4340 \left (2 x + 1\right )^{2} + 2604\right )} + \frac{64 \operatorname{RootSum}{\left (407144088666112 t^{4} + 3325152256 t^{2} + 11045, \left ( t \mapsto t \log{\left (\frac{33312534528 t^{3}}{235} + \frac{166784 t}{235} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} - \frac{112 \operatorname{RootSum}{\left (19950060344639488 t^{4} + 498437272576 t^{2} + 10878125, \left ( t \mapsto t \log{\left (- \frac{11049511452672 t^{3}}{2205125} + \frac{307918256 t}{2205125} + \sqrt{2 x + 1} \right )} \right )\right )}}{5} + \frac{16 \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}}}{{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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