Optimal. Leaf size=269 \[ \frac {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac {\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]
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Rubi [A] time = 0.25, antiderivative size = 269, 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 {\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac {\left (b^2-4 a c\right ) (b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac {\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 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^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\int x \left (-2 a B-\frac {1}{2} (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{256 c^4}\\ &=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^5}\\ &=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A 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 )}{1024 c^5}\\ &=\frac {\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}-\frac {\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac {\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 206, normalized size = 0.77 \begin {gather*} \frac {\frac {(a+x (b+c x))^{5/2} \left (4 c (35 A c x-12 a B)-2 b c (49 A+45 B x)+63 b^2 B\right )}{120 c^2}-\frac {7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\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 )}{6144 c^{9/2}}+B x^2 (a+x (b+c x))^{5/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.80, size = 423, normalized size = 1.57 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-6144 a^3 B c^3-18144 a^2 A b c^3+6720 a^2 A c^4 x+16464 a^2 b^2 B c^2-7008 a^2 b B c^3 x+3072 a^2 B c^4 x^2+10640 a A b^3 c^2-6048 a A b^2 c^3 x+4032 a A b c^4 x^2+31360 a A c^5 x^3-7560 a b^4 B c+4368 a b^3 B c^2 x-2976 a b^2 B c^3 x^2+2112 a b B c^4 x^3+24576 a B c^5 x^4-1470 A b^5 c+980 A b^4 c^2 x-784 A b^3 c^3 x^2+672 A b^2 c^4 x^3+23296 A b c^5 x^4+17920 A c^6 x^5+945 b^6 B-630 b^5 B c x+504 b^4 B c^2 x^2-432 b^3 B c^3 x^3+384 b^2 B c^4 x^4+19200 b B c^5 x^5+15360 B c^6 x^6\right )}{107520 c^5}+\frac {\left (128 a^3 A c^4-192 a^3 b B c^3-288 a^2 A b^2 c^3+240 a^2 b^3 B c^2+120 a A b^4 c^2-84 a b^5 B c-14 A b^6 c+9 b^7 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2048 c^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 845, normalized size = 3.14 \begin {gather*} \left [\frac {105 \, {\left (9 \, B b^{7} + 128 \, A a^{3} c^{4} - 96 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 120 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 14 \, {\left (6 \, B a b^{5} + A b^{6}\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 (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} - 96 \, {\left (64 \, B a^{3} + 189 \, A a^{2} b\right )} c^{4} + 128 \, {\left (3 \, B b^{2} c^{5} + 2 \, {\left (96 \, B a + 91 \, A b\right )} c^{6}\right )} x^{4} + 112 \, {\left (147 \, B a^{2} b^{2} + 95 \, A a b^{3}\right )} c^{3} - 16 \, {\left (27 \, B b^{3} c^{4} - 1960 \, A a c^{6} - 6 \, {\left (22 \, B a b + 7 \, A b^{2}\right )} c^{5}\right )} x^{3} - 210 \, {\left (36 \, B a b^{4} + 7 \, A b^{5}\right )} c^{2} + 8 \, {\left (63 \, B b^{4} c^{3} + 24 \, {\left (16 \, B a^{2} + 21 \, A a b\right )} c^{5} - 2 \, {\left (186 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 48 \, {\left (73 \, B a^{2} b + 63 \, A a b^{2}\right )} c^{4} - 14 \, {\left (156 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{430080 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{7} + 128 \, A a^{3} c^{4} - 96 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} c^{3} + 120 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} c^{2} - 14 \, {\left (6 \, B a b^{5} + A b^{6}\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 (15360 \, B c^{7} x^{6} + 945 \, B b^{6} c + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{5} - 96 \, {\left (64 \, B a^{3} + 189 \, A a^{2} b\right )} c^{4} + 128 \, {\left (3 \, B b^{2} c^{5} + 2 \, {\left (96 \, B a + 91 \, A b\right )} c^{6}\right )} x^{4} + 112 \, {\left (147 \, B a^{2} b^{2} + 95 \, A a b^{3}\right )} c^{3} - 16 \, {\left (27 \, B b^{3} c^{4} - 1960 \, A a c^{6} - 6 \, {\left (22 \, B a b + 7 \, A b^{2}\right )} c^{5}\right )} x^{3} - 210 \, {\left (36 \, B a b^{4} + 7 \, A b^{5}\right )} c^{2} + 8 \, {\left (63 \, B b^{4} c^{3} + 24 \, {\left (16 \, B a^{2} + 21 \, A a b\right )} c^{5} - 2 \, {\left (186 \, B a b^{2} + 49 \, A b^{3}\right )} c^{4}\right )} x^{2} - 2 \, {\left (315 \, B b^{5} c^{2} - 3360 \, A a^{2} c^{5} + 48 \, {\left (73 \, B a^{2} b + 63 \, A a b^{2}\right )} c^{4} - 14 \, {\left (156 \, B a b^{3} + 35 \, A b^{4}\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{215040 \, c^{6}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 422, normalized size = 1.57 \begin {gather*} \frac {1}{107520} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x + \frac {15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac {3 \, B b^{2} c^{5} + 192 \, B a c^{6} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac {27 \, B b^{3} c^{4} - 132 \, B a b c^{5} - 42 \, A b^{2} c^{5} - 1960 \, A a c^{6}}{c^{6}}\right )} x + \frac {63 \, B b^{4} c^{3} - 372 \, B a b^{2} c^{4} - 98 \, A b^{3} c^{4} + 384 \, B a^{2} c^{5} + 504 \, A a b c^{5}}{c^{6}}\right )} x - \frac {315 \, B b^{5} c^{2} - 2184 \, B a b^{3} c^{3} - 490 \, A b^{4} c^{3} + 3504 \, B a^{2} b c^{4} + 3024 \, A a b^{2} c^{4} - 3360 \, A a^{2} c^{5}}{c^{6}}\right )} x + \frac {945 \, B b^{6} c - 7560 \, B a b^{4} c^{2} - 1470 \, A b^{5} c^{2} + 16464 \, B a^{2} b^{2} c^{3} + 10640 \, A a b^{3} c^{3} - 6144 \, B a^{3} c^{4} - 18144 \, A a^{2} b c^{4}}{c^{6}}\right )} + \frac {{\left (9 \, B b^{7} - 84 \, B a b^{5} c - 14 \, A b^{6} c + 240 \, B a^{2} b^{3} c^{2} + 120 \, A a b^{4} c^{2} - 192 \, B a^{3} b c^{3} - 288 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 838, normalized size = 3.12 \begin {gather*} -\frac {A \,a^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {3}{2}}}+\frac {9 A \,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 {5}{2}}}-\frac {15 A a \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {7}{2}}}+\frac {7 A \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}+\frac {3 B \,a^{3} b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32 c^{\frac {5}{2}}}-\frac {15 B \,a^{2} b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {7}{2}}}+\frac {21 B a \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{512 c^{\frac {9}{2}}}-\frac {9 B \,b^{7} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {11}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} x}{16 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A a \,b^{2} x}{8 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{4} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b x}{32 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{3} x}{32 c^{3}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, B \,b^{5} x}{512 c^{4}}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,a^{2} b}{32 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, A a \,b^{3}}{16 c^{3}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a x}{24 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, A \,b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} x}{96 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,a^{2} b^{2}}{64 c^{3}}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,b^{4}}{64 c^{4}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a b x}{16 c^{2}}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, B \,b^{6}}{1024 c^{5}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,x^{2}}{7 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A a b}{48 c^{2}}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A x}{6 c}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B a \,b^{2}}{32 c^{3}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{4}}{128 c^{4}}-\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b x}{28 c^{2}}-\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{60 c^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B a}{35 c^{2}}+\frac {3 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B \,b^{2}}{40 c^{3}} \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 [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\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 x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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