Optimal. Leaf size=208 \[ -\frac {83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac {804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac {27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac {359471503 (1-4 x) \sqrt {2 x^2-x+3}}{67108864}+\frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac {14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac {233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac {4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac {8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac {8267844569 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{134217728 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac {125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac {14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac {233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac {4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac {8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac {83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac {804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac {27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac {359471503 (1-4 x) \sqrt {2 x^2-x+3}}{67108864}-\frac {8267844569 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{134217728 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 612
Rule 619
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx &=\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {1}{20} \int \sqrt {3-x+2 x^2} \left (320+1920 x+7520 x^2+18720 x^3+35220 x^4+46800 x^5+33875 x^6+\frac {70625 x^7}{2}\right ) \, dx\\ &=\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {1}{360} \int \sqrt {3-x+2 x^2} \left (5760+34560 x+135360 x^2+336960 x^3+633960 x^4+206775 x^5+\frac {3498375 x^6}{4}\right ) \, dx\\ &=\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \sqrt {3-x+2 x^2} \left (92160+552960 x+2165760 x^2+5391360 x^3-\frac {11902185 x^4}{4}+\frac {71946075 x^5}{8}\right ) \, dx}{5760}\\ &=\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \sqrt {3-x+2 x^2} \left (1290240+7741440 x+30320640 x^2-\frac {64880145 x^3}{2}+\frac {124884465 x^4}{16}\right ) \, dx}{80640}\\ &=\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \sqrt {3-x+2 x^2} \left (15482880+92897280 x+\frac {4697602695 x^2}{16}-\frac {11333027655 x^3}{32}\right ) \, dx}{967680}\\ &=-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \sqrt {3-x+2 x^2} \left (154828800+\frac {48862647765 x}{16}+\frac {108572914215 x^2}{64}\right ) \, dx}{9676800}\\ &=\frac {804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \left (-\frac {246446397045}{64}+\frac {3670074028035 x}{128}\right ) \sqrt {3-x+2 x^2} \, dx}{77414400}\\ &=\frac {27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac {804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {359471503 \int \sqrt {3-x+2 x^2} \, dx}{8388608}\\ &=-\frac {359471503 (1-4 x) \sqrt {3-x+2 x^2}}{67108864}+\frac {27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac {804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {8267844569 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{134217728}\\ &=-\frac {359471503 (1-4 x) \sqrt {3-x+2 x^2}}{67108864}+\frac {27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac {804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac {\left (359471503 \sqrt {\frac {23}{2}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{134217728}\\ &=-\frac {359471503 (1-4 x) \sqrt {3-x+2 x^2}}{67108864}+\frac {27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac {804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac {83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac {8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac {4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac {233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac {14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}-\frac {8267844569 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{134217728 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 85, normalized size = 0.41 \[ \frac {4 \sqrt {2 x^2-x+3} \left (1321205760000 x^9+3486515200000 x^8+6327795712000 x^7+7725962035200 x^6+7612808028160 x^5+5354741991424 x^4+2211683657856 x^3-174418077792 x^2+537752185764 x+3801512106459\right )-2604371039235 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{84557168640} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.66, size = 98, normalized size = 0.47 \[ \frac {1}{21139292160} \, {\left (1321205760000 \, x^{9} + 3486515200000 \, x^{8} + 6327795712000 \, x^{7} + 7725962035200 \, x^{6} + 7612808028160 \, x^{5} + 5354741991424 \, x^{4} + 2211683657856 \, x^{3} - 174418077792 \, x^{2} + 537752185764 \, x + 3801512106459\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {8267844569}{536870912} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.25, size = 93, normalized size = 0.45 \[ \frac {1}{21139292160} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (40 \, {\left (140 \, {\left (160 \, {\left (36 \, x + 95\right )} x + 27587\right )} x + 4715553\right )} x + 185859571\right )} x + 2614620113\right )} x + 17278778577\right )} x - 5450564931\right )} x + 134438046441\right )} x + 3801512106459\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {8267844569}{268435456} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 166, normalized size = 0.80 \[ \frac {125 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{7}}{4}+\frac {14125 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{6}}{144}+\frac {233225 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{5}}{1536}+\frac {4796405 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{4}}{43008}+\frac {8325631 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{3}}{1032192}-\frac {83948353 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x^{2}}{2293760}+\frac {804243809 \left (2 x^{2}-x +3\right )^{\frac {3}{2}} x}{36700160}+\frac {8267844569 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{268435456}+\frac {27185733541 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{440401920}+\frac {359471503 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{67108864} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 177, normalized size = 0.85 \[ \frac {125}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{7} + \frac {14125}{144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{6} + \frac {233225}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} + \frac {4796405}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + \frac {8325631}{1032192} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {83948353}{2293760} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {804243809}{36700160} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {27185733541}{440401920} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {359471503}{16777216} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {8267844569}{268435456} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {359471503}{67108864} \, \sqrt {2 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.03, size = 221, normalized size = 1.06 \[ \frac {8325631\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{1032192}-\frac {83948353\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{2293760}+\frac {4796405\,x^4\,{\left (2\,x^2-x+3\right )}^{3/2}}{43008}+\frac {233225\,x^5\,{\left (2\,x^2-x+3\right )}^{3/2}}{1536}+\frac {14125\,x^6\,{\left (2\,x^2-x+3\right )}^{3/2}}{144}+\frac {125\,x^7\,{\left (2\,x^2-x+3\right )}^{3/2}}{4}-\frac {41987163941\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{1174405120}-\frac {1825528867\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{36700160}+\frac {27185733541\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{7046430720}+\frac {804243809\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{36700160}+\frac {625271871443\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{9395240960} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________