Optimal. Leaf size=280 \[ -\frac {\left (b^2-4 a c\right ) \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}}+\frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right )}{512 c^5}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{960 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \]
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Rubi [A] time = 0.32, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \begin {gather*} \frac {(b+2 c x) \sqrt {a+b x+c x^2} \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right )}{512 c^5}-\frac {\left (b^2-4 a c\right ) \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{960 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int x^3 (A+B x) \sqrt {a+b x+c x^2} \, dx &=\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int x^2 \left (-3 a B-\frac {3}{2} (3 b B-4 A c) x\right ) \sqrt {a+b x+c x^2} \, dx}{6 c}\\ &=-\frac {(3 b B-4 A c) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}+\frac {\int x \left (3 a (3 b B-4 A c)+\frac {3}{4} \left (21 b^2 B-28 A b c-20 a B c\right ) x\right ) \sqrt {a+b x+c x^2} \, dx}{30 c^2}\\ &=-\frac {(3 b B-4 A c) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (105 b^3 B-140 A b^2 c-196 a b B c+128 a A c^2-6 c \left (21 b^2 B-28 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}+\frac {\left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) \int \sqrt {a+b x+c x^2} \, dx}{128 c^4}\\ &=\frac {\left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}-\frac {(3 b B-4 A c) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (105 b^3 B-140 A b^2 c-196 a b B c+128 a A c^2-6 c \left (21 b^2 B-28 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{1024 c^5}\\ &=\frac {\left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}-\frac {(3 b B-4 A c) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (105 b^3 B-140 A b^2 c-196 a b B c+128 a A c^2-6 c \left (21 b^2 B-28 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right )\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 )}{512 c^5}\\ &=\frac {\left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}-\frac {(3 b B-4 A c) x^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (105 b^3 B-140 A b^2 c-196 a b B c+128 a A c^2-6 c \left (21 b^2 B-28 A b c-20 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (b^2-4 a c\right ) \left (21 b^4 B-28 A b^3 c-56 a b^2 B c+48 a A b c^2+16 a^2 B c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 241, normalized size = 0.86 \begin {gather*} \frac {\frac {3 \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{512 c^{9/2}}+\frac {(a+x (b+c x))^{3/2} \left (28 b c (7 a B-6 A c x)-8 a c^2 (16 A+15 B x)+14 b^2 c (10 A+9 B x)-105 b^3 B\right )}{160 c^3}+\frac {3 x^2 (a+x (b+c x))^{3/2} (4 A c-3 b B)}{10 c}+B x^3 (a+x (b+c x))^{3/2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.29, size = 326, normalized size = 1.16 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-1024 a^2 A c^3+1808 a^2 b B c^2-480 a^2 B c^3 x+1840 a A b^2 c^2-928 a A b c^3 x+512 a A c^4 x^2-1680 a b^3 B c+896 a b^2 B c^2 x-544 a b B c^3 x^2+320 a B c^4 x^3-420 A b^4 c+280 A b^3 c^2 x-224 A b^2 c^3 x^2+192 A b c^4 x^3+1536 A c^5 x^4+315 b^5 B-210 b^4 B c x+168 b^3 B c^2 x^2-144 b^2 B c^3 x^3+128 b B c^4 x^4+1280 B c^5 x^5\right )}{7680 c^5}+\frac {\left (-64 a^3 B c^3-192 a^2 A b c^3+240 a^2 b^2 B c^2+160 a A b^3 c^2-140 a b^4 B c-28 A b^5 c+21 b^6 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{1024 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 667, normalized size = 2.38 \begin {gather*} \left [-\frac {15 \, {\left (21 \, B b^{6} - 64 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c^{3} + 80 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} c^{2} - 28 \, {\left (5 \, B a b^{4} + A b^{5}\right )} c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (1280 \, B c^{6} x^{5} + 315 \, B b^{5} c - 1024 \, A a^{2} c^{4} + 128 \, {\left (B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 16 \, {\left (113 \, B a^{2} b + 115 \, A a b^{2}\right )} c^{3} - 16 \, {\left (9 \, B b^{2} c^{4} - 4 \, {\left (5 \, B a + 3 \, A b\right )} c^{5}\right )} x^{3} - 420 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c^{2} + 8 \, {\left (21 \, B b^{3} c^{3} + 64 \, A a c^{5} - 4 \, {\left (17 \, B a b + 7 \, A b^{2}\right )} c^{4}\right )} x^{2} - 2 \, {\left (105 \, B b^{4} c^{2} + 16 \, {\left (15 \, B a^{2} + 29 \, A a b\right )} c^{4} - 28 \, {\left (16 \, B a b^{2} + 5 \, A b^{3}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{30720 \, c^{6}}, \frac {15 \, {\left (21 \, B b^{6} - 64 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} c^{3} + 80 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} c^{2} - 28 \, {\left (5 \, B a b^{4} + A b^{5}\right )} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (1280 \, B c^{6} x^{5} + 315 \, B b^{5} c - 1024 \, A a^{2} c^{4} + 128 \, {\left (B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 16 \, {\left (113 \, B a^{2} b + 115 \, A a b^{2}\right )} c^{3} - 16 \, {\left (9 \, B b^{2} c^{4} - 4 \, {\left (5 \, B a + 3 \, A b\right )} c^{5}\right )} x^{3} - 420 \, {\left (4 \, B a b^{3} + A b^{4}\right )} c^{2} + 8 \, {\left (21 \, B b^{3} c^{3} + 64 \, A a c^{5} - 4 \, {\left (17 \, B a b + 7 \, A b^{2}\right )} c^{4}\right )} x^{2} - 2 \, {\left (105 \, B b^{4} c^{2} + 16 \, {\left (15 \, B a^{2} + 29 \, A a b\right )} c^{4} - 28 \, {\left (16 \, B a b^{2} + 5 \, A b^{3}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{15360 \, c^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 323, normalized size = 1.15 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B x + \frac {B b c^{4} + 12 \, A c^{5}}{c^{5}}\right )} x - \frac {9 \, B b^{2} c^{3} - 20 \, B a c^{4} - 12 \, A b c^{4}}{c^{5}}\right )} x + \frac {21 \, B b^{3} c^{2} - 68 \, B a b c^{3} - 28 \, A b^{2} c^{3} + 64 \, A a c^{4}}{c^{5}}\right )} x - \frac {105 \, B b^{4} c - 448 \, B a b^{2} c^{2} - 140 \, A b^{3} c^{2} + 240 \, B a^{2} c^{3} + 464 \, A a b c^{3}}{c^{5}}\right )} x + \frac {315 \, B b^{5} - 1680 \, B a b^{3} c - 420 \, A b^{4} c + 1808 \, B a^{2} b c^{2} + 1840 \, A a b^{2} c^{2} - 1024 \, A a^{2} c^{3}}{c^{5}}\right )} + \frac {{\left (21 \, B b^{6} - 140 \, B a b^{4} c - 28 \, A b^{5} c + 240 \, B a^{2} b^{2} c^{2} + 160 \, A a b^{3} c^{2} - 64 \, B a^{3} c^{3} - 192 \, A a^{2} b c^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 671, normalized size = 2.40 \begin {gather*} \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,x^{3}}{6 c}+\frac {3 A \,a^{2} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {5 A a \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {7}{2}}}+\frac {7 A \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}+\frac {B \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {5}{2}}}-\frac {15 B \,a^{2} b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {7}{2}}}+\frac {35 B a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}-\frac {21 B \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {11}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a b x}{16 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,x^{2}}{5 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,a^{2} x}{16 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{2} x}{32 c^{3}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, B \,b^{4} x}{256 c^{4}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b \,x^{2}}{20 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A a \,b^{2}}{32 c^{3}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4}}{128 c^{4}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b x}{40 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b}{32 c^{3}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3}}{64 c^{4}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a x}{8 c^{2}}+\frac {21 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5}}{512 c^{5}}+\frac {21 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2} x}{160 c^{3}}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a}{15 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{48 c^{3}}+\frac {49 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b}{240 c^{3}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3}}{64 c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.96, size = 781, normalized size = 2.79 \begin {gather*} \frac {A\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}+\frac {B\,x^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{6\,c}-\frac {2\,A\,a\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}+\frac {7\,A\,b\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}+\frac {B\,a\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{2\,c}-\frac {3\,B\,b\,\left (\frac {7\,b\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}-\frac {2\,a\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}+\frac {x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}\right )}{4\,c} \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 x^{3} \left (A + B x\right ) \sqrt {a + b x + c x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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