Optimal. Leaf size=364 \[ -\frac {b \left (1168 a^2 c^2-728 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (-2048 a^3 c^3+5488 a^2 b^2 c^2-2520 a b^4 c+315 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}-\frac {3 b x \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {x \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b x^2 \left (9 b^2-44 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (24 a c+b^2+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2} \]
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Rubi [A] time = 1.04, antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1906, 1945, 1949, 12, 1914, 621, 206} \begin {gather*} -\frac {b \left (1168 a^2 c^2-728 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (5488 a^2 b^2 c^2-2048 a^3 c^3-2520 a b^4 c+315 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}-\frac {b x^2 \left (9 b^2-44 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {3 b x \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {x^3 \left (24 a c+b^2+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1906
Rule 1914
Rule 1945
Rule 1949
Rubi steps
\begin {align*} \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx &=\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {3}{14} \int x^2 (2 a+b x) \sqrt {a x^2+b x^3+c x^4} \, dx\\ &=\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {\int \frac {x^4 \left (-4 a \left (b^2-6 a c\right )-\frac {1}{2} b \left (9 b^2-44 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{280 c}\\ &=-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\int \frac {x^3 \left (-\frac {3}{2} a b \left (9 b^2-44 a c\right )-\frac {3}{4} \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{1120 c^2}\\ &=\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {\int \frac {x^2 \left (-\frac {3}{2} a \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right )-\frac {3}{8} b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{3360 c^3}\\ &=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\int \frac {x \left (-\frac {3}{8} a b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right )-\frac {3}{16} \left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{6720 c^4}\\ &=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {\int -\frac {315 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x}{32 \sqrt {a x^2+b x^3+c x^4}} \, dx}{6720 c^5}\\ &=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2048 c^5}\\ &=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^5 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{1024 c^5 \sqrt {a x^2+b x^3+c x^4}}\\ &=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt {a x^2+b x^3+c x^4}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 197, normalized size = 0.54 \begin {gather*} \frac {\left (x^2 (a+x (b+c x))\right )^{3/2} \left (\frac {\left (-16 a c+21 b^2-30 b c x\right ) (a+x (b+c x))}{40 c^2}+\frac {7 \left (4 a b c-3 b^3\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{2048 c^{9/2} (a+x (b+c x))^{3/2}}+x^2 (a+x (b+c x))\right )}{7 c x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.08, size = 266, normalized size = 0.73 \begin {gather*} \frac {3 \left (-64 a^3 b c^3+80 a^2 b^3 c^2-28 a b^5 c+3 b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a} x-\sqrt {a x^2+b x^3+c x^4}}\right )}{1024 c^{11/2}}+\frac {\sqrt {a x^2+b x^3+c x^4} \left (-2048 a^3 c^3+5488 a^2 b^2 c^2-2336 a^2 b c^3 x+1024 a^2 c^4 x^2-2520 a b^4 c+1456 a b^3 c^2 x-992 a b^2 c^3 x^2+704 a b c^4 x^3+8192 a c^5 x^4+315 b^6-210 b^5 c x+168 b^4 c^2 x^2-144 b^3 c^3 x^3+128 b^2 c^4 x^4+6400 b c^5 x^5+5120 c^6 x^6\right )}{35840 c^5 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 558, normalized size = 1.53 \begin {gather*} \left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{143360 \, c^{6} x}, \frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{71680 \, c^{6} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 429, normalized size = 1.18 \begin {gather*} \frac {1}{35840} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c x \mathrm {sgn}\relax (x) + 5 \, b \mathrm {sgn}\relax (x)\right )} x + \frac {b^{2} c^{5} \mathrm {sgn}\relax (x) + 64 \, a c^{6} \mathrm {sgn}\relax (x)}{c^{6}}\right )} x - \frac {9 \, b^{3} c^{4} \mathrm {sgn}\relax (x) - 44 \, a b c^{5} \mathrm {sgn}\relax (x)}{c^{6}}\right )} x + \frac {21 \, b^{4} c^{3} \mathrm {sgn}\relax (x) - 124 \, a b^{2} c^{4} \mathrm {sgn}\relax (x) + 128 \, a^{2} c^{5} \mathrm {sgn}\relax (x)}{c^{6}}\right )} x - \frac {105 \, b^{5} c^{2} \mathrm {sgn}\relax (x) - 728 \, a b^{3} c^{3} \mathrm {sgn}\relax (x) + 1168 \, a^{2} b c^{4} \mathrm {sgn}\relax (x)}{c^{6}}\right )} x + \frac {315 \, b^{6} c \mathrm {sgn}\relax (x) - 2520 \, a b^{4} c^{2} \mathrm {sgn}\relax (x) + 5488 \, a^{2} b^{2} c^{3} \mathrm {sgn}\relax (x) - 2048 \, a^{3} c^{4} \mathrm {sgn}\relax (x)}{c^{6}}\right )} + \frac {3 \, {\left (3 \, b^{7} \mathrm {sgn}\relax (x) - 28 \, a b^{5} c \mathrm {sgn}\relax (x) + 80 \, a^{2} b^{3} c^{2} \mathrm {sgn}\relax (x) - 64 \, a^{3} b c^{3} \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {{\left (315 \, b^{7} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 2940 \, a b^{5} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 8400 \, a^{2} b^{3} c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 6720 \, a^{3} b c^{3} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 630 \, \sqrt {a} b^{6} \sqrt {c} - 5040 \, a^{\frac {3}{2}} b^{4} c^{\frac {3}{2}} + 10976 \, a^{\frac {5}{2}} b^{2} c^{\frac {5}{2}} - 4096 \, a^{\frac {7}{2}} c^{\frac {7}{2}}\right )} \mathrm {sgn}\relax (x)}{71680 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 479, normalized size = 1.32 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (6720 a^{3} b \,c^{4} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-8400 a^{2} b^{3} c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+2940 a \,b^{5} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-315 b^{7} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+6720 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{\frac {9}{2}} x -6720 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{\frac {7}{2}} x +1260 \sqrt {c \,x^{2}+b x +a}\, b^{5} c^{\frac {5}{2}} x +3360 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c^{\frac {7}{2}}-3360 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} c^{\frac {5}{2}}+4480 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,c^{\frac {9}{2}} x +630 \sqrt {c \,x^{2}+b x +a}\, b^{6} c^{\frac {3}{2}}-3360 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} c^{\frac {7}{2}} x +10240 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {11}{2}} x^{2}+2240 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} c^{\frac {7}{2}}-1680 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} c^{\frac {5}{2}}-7680 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,c^{\frac {9}{2}} x -4096 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,c^{\frac {9}{2}}+5376 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} c^{\frac {7}{2}}\right )}{71680 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {13}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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