Optimal. Leaf size=296 \[ -\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
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Rubi [A] time = 0.445667, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {738, 824, 826, 1169, 634, 618, 204, 628} \[ -\frac{(5-4 x) (2 x+1)^{5/2}}{31 \left (5 x^2+3 x+2\right )}-\frac{8}{155} (2 x+1)^{3/2}+\frac{604}{775} \sqrt{2 x+1}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)-\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )-\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (5 (2 x+1)+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{2 x+1}+\sqrt{35}\right )+\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{\sqrt{10 \left (2+\sqrt{35}\right )}-10 \sqrt{2 x+1}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right )-\frac{1}{775} \sqrt{\frac{2}{155} \left (968975 \sqrt{35}-5682718\right )} \tan ^{-1}\left (\frac{10 \sqrt{2 x+1}+\sqrt{10 \left (2+\sqrt{35}\right )}}{\sqrt{10 \left (\sqrt{35}-2\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 738
Rule 824
Rule 826
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{(1+2 x)^{7/2}}{\left (2+3 x+5 x^2\right )^2} \, dx &=-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \frac{(29-12 x) (1+2 x)^{3/2}}{2+3 x+5 x^2} \, dx\\ &=-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{155} \int \frac{\sqrt{1+2 x} (193+302 x)}{2+3 x+5 x^2} \, dx\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{775} \int \frac{-243+1628 x}{\sqrt{1+2 x} \left (2+3 x+5 x^2\right )} \, dx\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{2}{775} \operatorname{Subst}\left (\int \frac{-2114+1628 x^2}{7-4 x^2+5 x^4} \, dx,x,\sqrt{1+2 x}\right )\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-2114 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}-\left (-2114-1628 \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 )}{775 \sqrt{14 \left (2+\sqrt{35}\right )}}+\frac{\operatorname{Subst}\left (\int \frac{-2114 \sqrt{\frac{2}{5} \left (2+\sqrt{35}\right )}+\left (-2114-1628 \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 )}{775 \sqrt{14 \left (2+\sqrt{35}\right )}}\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}-\frac{\sqrt{1460631-245828 \sqrt{35}} \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 )}{3875}-\frac{\sqrt{1460631-245828 \sqrt{35}} \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 )}{3875}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \sqrt{35}\right )} \log \left (\sqrt{35}+\sqrt{10 \left (2+\sqrt{35}\right )} \sqrt{1+2 x}+5 (1+2 x)\right )+\frac{\left (2 \sqrt{1460631-245828 \sqrt{35}}\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 )}{3875}+\frac{\left (2 \sqrt{1460631-245828 \sqrt{35}}\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 )}{3875}\\ &=\frac{604}{775} \sqrt{1+2 x}-\frac{8}{155} (1+2 x)^{3/2}-\frac{(5-4 x) (1+2 x)^{5/2}}{31 \left (2+3 x+5 x^2\right )}+\frac{1}{775} \sqrt{\frac{2}{155} \left (-5682718+968975 \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}{775} \sqrt{\frac{2}{155} \left (-5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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}{775} \sqrt{\frac{1}{310} \left (5682718+968975 \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 = 0.433634, size = 199, normalized size = 0.67 \[ \frac{1}{217} \left (\frac{(20 x+37) (2 x+1)^{9/2}}{5 x^2+3 x+2}-8 (2 x+1)^{7/2}-28 (2 x+1)^{5/2}-\frac{56}{5} (2 x+1)^{3/2}+\frac{4228}{25} \sqrt{2 x+1}-\frac{14 i \left (\sqrt{2-i \sqrt{31}} \left (512 \sqrt{31}-4681 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2-i \sqrt{31}}}\right )-\sqrt{2+i \sqrt{31}} \left (512 \sqrt{31}+4681 i\right ) \tanh ^{-1}\left (\frac{\sqrt{10 x+5}}{\sqrt{2+i \sqrt{31}}}\right )\right )}{775 \sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.078, size = 651, normalized size = 2.2 \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{7}{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 = 3.00175, size = 2880, normalized size = 9.73 \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{7}{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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