Optimal. Leaf size=147 \[ -\frac{7}{40} \left (5 x^2+2 x+3\right )^{5/2} x^3-\frac{1163 \left (5 x^2+2 x+3\right )^{5/2} x^2}{1400}+\frac{2809 \left (5 x^2+2 x+3\right )^{5/2} x}{5250}+\frac{149509 \left (5 x^2+2 x+3\right )^{5/2}}{262500}-\frac{18397 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{150000}-\frac{128779 (5 x+1) \sqrt{5 x^2+2 x+3}}{250000}-\frac{901453 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{125000 \sqrt{5}} \]
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Rubi [A] time = 0.129781, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {1661, 640, 612, 619, 215} \[ -\frac{7}{40} \left (5 x^2+2 x+3\right )^{5/2} x^3-\frac{1163 \left (5 x^2+2 x+3\right )^{5/2} x^2}{1400}+\frac{2809 \left (5 x^2+2 x+3\right )^{5/2} x}{5250}+\frac{149509 \left (5 x^2+2 x+3\right )^{5/2}}{262500}-\frac{18397 (5 x+1) \left (5 x^2+2 x+3\right )^{3/2}}{150000}-\frac{128779 (5 x+1) \sqrt{5 x^2+2 x+3}}{250000}-\frac{901453 \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{125000 \sqrt{5}} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \left (1+4 x-7 x^2\right ) \left (2+5 x+x^2\right ) \left (3+2 x+5 x^2\right )^{3/2} \, dx &=-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}+\frac{1}{40} \int \left (3+2 x+5 x^2\right )^{3/2} \left (80+520 x+343 x^2-1163 x^3\right ) \, dx\\ &=-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int \left (3+2 x+5 x^2\right )^{3/2} \left (2800+25178 x+22472 x^2\right ) \, dx}{1400}\\ &=\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}+\frac{\int (16584+598036 x) \left (3+2 x+5 x^2\right )^{3/2} \, dx}{42000}\\ &=\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{18397 \int \left (3+2 x+5 x^2\right )^{3/2} \, dx}{7500}\\ &=-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{128779 \int \sqrt{3+2 x+5 x^2} \, dx}{25000}\\ &=-\frac{128779 (1+5 x) \sqrt{3+2 x+5 x^2}}{250000}-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{901453 \int \frac{1}{\sqrt{3+2 x+5 x^2}} \, dx}{125000}\\ &=-\frac{128779 (1+5 x) \sqrt{3+2 x+5 x^2}}{250000}-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{\left (128779 \sqrt{\frac{7}{10}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{56}}} \, dx,x,2+10 x\right )}{250000}\\ &=-\frac{128779 (1+5 x) \sqrt{3+2 x+5 x^2}}{250000}-\frac{18397 (1+5 x) \left (3+2 x+5 x^2\right )^{3/2}}{150000}+\frac{149509 \left (3+2 x+5 x^2\right )^{5/2}}{262500}+\frac{2809 x \left (3+2 x+5 x^2\right )^{5/2}}{5250}-\frac{1163 x^2 \left (3+2 x+5 x^2\right )^{5/2}}{1400}-\frac{7}{40} x^3 \left (3+2 x+5 x^2\right )^{5/2}-\frac{901453 \sinh ^{-1}\left (\frac{1+5 x}{\sqrt{14}}\right )}{125000 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.148884, size = 75, normalized size = 0.51 \[ \frac{-5 \sqrt{5 x^2+2 x+3} \left (22968750 x^7+127406250 x^6+48237500 x^5+28373000 x^4-78608475 x^3-86464445 x^2-36695150 x-22275576\right )-37861026 \sqrt{5} \sinh ^{-1}\left (\frac{5 x+1}{\sqrt{14}}\right )}{26250000} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 117, normalized size = 0.8 \begin{align*} -{\frac{7\,{x}^{3}}{40} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{1163\,{x}^{2}}{1400} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{1287790\,x+257558}{500000}\sqrt{5\,{x}^{2}+2\,x+3}}-{\frac{901453\,\sqrt{5}}{625000}{\it Arcsinh} \left ({\frac{5\,\sqrt{14}}{14} \left ( x+{\frac{1}{5}} \right ) } \right ) }+{\frac{2809\,x}{5250} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{183970\,x+36794}{300000} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{149509}{262500} \left ( 5\,{x}^{2}+2\,x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47904, size = 186, normalized size = 1.27 \begin{align*} -\frac{7}{40} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{3} - \frac{1163}{1400} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{2809}{5250} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} x + \frac{149509}{262500} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{5}{2}} - \frac{18397}{30000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} x - \frac{18397}{150000} \,{\left (5 \, x^{2} + 2 \, x + 3\right )}^{\frac{3}{2}} - \frac{128779}{50000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} x - \frac{901453}{625000} \, \sqrt{5} \operatorname{arsinh}\left (\frac{1}{14} \, \sqrt{14}{\left (5 \, x + 1\right )}\right ) - \frac{128779}{250000} \, \sqrt{5 \, x^{2} + 2 \, x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26446, size = 323, normalized size = 2.2 \begin{align*} -\frac{1}{5250000} \,{\left (22968750 \, x^{7} + 127406250 \, x^{6} + 48237500 \, x^{5} + 28373000 \, x^{4} - 78608475 \, x^{3} - 86464445 \, x^{2} - 36695150 \, x - 22275576\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{901453}{1250000} \, \sqrt{5} \log \left (\sqrt{5} \sqrt{5 \, x^{2} + 2 \, x + 3}{\left (5 \, x + 1\right )} - 25 \, x^{2} - 10 \, x - 8\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - 43 x \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 57 x^{2} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 14 x^{3} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 48 x^{4} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 169 x^{5} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int 35 x^{6} \sqrt{5 x^{2} + 2 x + 3}\, dx - \int - 6 \sqrt{5 x^{2} + 2 x + 3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30397, size = 111, normalized size = 0.76 \begin{align*} -\frac{1}{5250000} \,{\left (5 \,{\left ({\left (5 \,{\left (10 \,{\left (25 \,{\left (15 \,{\left (245 \, x + 1359\right )} x + 7718\right )} x + 113492\right )} x - 3144339\right )} x - 17292889\right )} x - 7339030\right )} x - 22275576\right )} \sqrt{5 \, x^{2} + 2 \, x + 3} + \frac{901453}{625000} \, \sqrt{5} \log \left (-\sqrt{5}{\left (\sqrt{5} x - \sqrt{5 \, x^{2} + 2 \, x + 3}\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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