Optimal. Leaf size=432 \[ -\frac {\left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{15/2}}+\frac {\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{131072 c^7}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{49152 c^6}+\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{15360 c^5}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{40320 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c} \]
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Rubi [A] time = 0.49, antiderivative size = 432, normalized size of antiderivative = 1.00, number of steps used = 8, 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) \left (a+b x+c x^2\right )^{5/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{15360 c^5}-\frac {\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{49152 c^6}+\frac {\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{131072 c^7}-\frac {\left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{15/2}}-\frac {\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{40320 c^4}-\frac {x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 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) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}+\frac {\int x^2 \left (-3 a B-\frac {1}{2} (13 b B-20 A c) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{10 c}\\ &=-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}+\frac {\int x \left (a (13 b B-20 A c)+\frac {1}{4} \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{90 c^2}\\ &=-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{1280 c^4}\\ &=\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{6144 c^5}\\ &=-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}+\frac {\left (\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right )\right ) \int \sqrt {a+b x+c x^2} \, dx}{32768 c^6}\\ &=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{262144 c^7}\\ &=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 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 )}{131072 c^7}\\ &=\frac {\left (b^2-4 a c\right )^2 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^7}-\frac {\left (b^2-4 a c\right ) \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^6}+\frac {\left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 a^2 B c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^5}-\frac {(13 b B-20 A c) x^2 \left (a+b x+c x^2\right )^{7/2}}{180 c^2}+\frac {B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c}-\frac {\left (1287 b^3 B-1980 A b^2 c-1804 a b B c+1280 a A c^2-14 c \left (143 b^2 B-220 A b c-108 a B c\right ) x\right ) \left (a+b x+c x^2\right )^{7/2}}{40320 c^4}-\frac {\left (b^2-4 a c\right )^3 \left (143 b^4 B-220 A b^3 c-264 a b^2 B c+240 a A b c^2+48 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 )}{262144 c^{15/2}}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 315, normalized size = 0.73 \begin {gather*} \frac {\frac {\left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\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 )\right )\right )}{393216 c^{13/2}}+\frac {(a+x (b+c x))^{7/2} \left (44 b c (41 a B-70 A c x)-8 a c^2 (160 A+189 B x)+22 b^2 c (90 A+91 B x)-1287 b^3 B\right )}{4032 c^3}+\frac {x^2 (a+x (b+c x))^{7/2} (20 A c-13 b B)}{18 c}+B x^3 (a+x (b+c x))^{7/2}}{10 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 5.17, size = 804, normalized size = 1.86 \begin {gather*} \frac {\sqrt {c x^2+b x+a} \left (45045 B b^9-69300 A c b^8-30030 B c x b^8+24024 B c^2 x^2 b^7-563640 a B c b^7+46200 A c^2 x b^7-20592 B c^3 x^3 b^6+814800 a A c^2 b^6-36960 A c^3 x^2 b^6+343728 a B c^2 x b^6+18304 B c^4 x^4 b^5+31680 A c^4 x^3 b^5+2487744 a^2 B c^2 b^5-250272 a B c^3 x^2 b^5-493920 a A c^3 x b^5-16640 B c^5 x^5 b^4-28160 A c^5 x^4 b^4-3245760 a^2 A c^3 b^4+193600 a B c^4 x^3 b^4+357120 a A c^4 x^2 b^4-1324800 a^2 B c^3 x b^4+15360 B c^6 x^6 b^3+25600 A c^6 x^5 b^3-153600 a B c^5 x^4 b^3-4406400 a^3 B c^3 b^3-273920 a A c^5 x^3 b^3+827520 a^2 B c^4 x^2 b^3+1687680 a^2 A c^4 x b^3+5490688 B c^7 x^7 b^2+6328320 A c^7 x^6 b^2+122880 a B c^6 x^5 b^2+4688640 a^3 A c^4 b^2+215040 a A c^6 x^4 b^2-533760 a^2 B c^5 x^3 b^2-1021440 a^2 A c^5 x^2 b^2+1834240 a^3 B c^4 x b^2+9404416 B c^8 x^8 b+10608640 A c^8 x^7 b+13029376 a B c^7 x^6 b+15421440 a A c^7 x^5 b+2379520 a^4 B c^4 b+337920 a^2 B c^6 x^4 b+629760 a^2 A c^6 x^3 b-826880 a^3 B c^5 x^2 b-1763840 a^3 A c^5 x b+4128768 B c^9 x^9+4587520 A c^9 x^8+10838016 a B c^8 x^7+12451840 a A c^8 x^6-1310720 a^4 A c^5+7999488 a^2 B c^7 x^5+9830400 a^2 A c^7 x^4+322560 a^3 B c^6 x^3+655360 a^3 A c^6 x^2-483840 a^4 B c^5 x\right )}{41287680 c^7}+\frac {\left (143 B b^{10}-220 A c b^9-1980 a B c b^8+2880 a A c^2 b^7+10080 a^2 B c^2 b^6-13440 a^2 A c^3 b^5-22400 a^3 B c^3 b^4+25600 a^3 A c^4 b^3+19200 a^4 B c^4 b^2-15360 a^4 A c^5 b-3072 a^5 B c^5\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {c x^2+b x+a}\right )}{262144 c^{15/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 1511, normalized size = 3.50
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 769, normalized size = 1.78 \begin {gather*} \frac {1}{41287680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, {\left (18 \, B c^{2} x + \frac {41 \, B b c^{10} + 20 \, A c^{11}}{c^{9}}\right )} x + \frac {383 \, B b^{2} c^{9} + 756 \, B a c^{10} + 740 \, A b c^{10}}{c^{9}}\right )} x + \frac {15 \, B b^{3} c^{8} + 12724 \, B a b c^{9} + 6180 \, A b^{2} c^{9} + 12160 \, A a c^{10}}{c^{9}}\right )} x - \frac {65 \, B b^{4} c^{7} - 480 \, B a b^{2} c^{8} - 100 \, A b^{3} c^{8} - 31248 \, B a^{2} c^{9} - 60240 \, A a b c^{9}}{c^{9}}\right )} x + \frac {143 \, B b^{5} c^{6} - 1200 \, B a b^{3} c^{7} - 220 \, A b^{4} c^{7} + 2640 \, B a^{2} b c^{8} + 1680 \, A a b^{2} c^{8} + 76800 \, A a^{2} c^{9}}{c^{9}}\right )} x - \frac {1287 \, B b^{6} c^{5} - 12100 \, B a b^{4} c^{6} - 1980 \, A b^{5} c^{6} + 33360 \, B a^{2} b^{2} c^{7} + 17120 \, A a b^{3} c^{7} - 20160 \, B a^{3} c^{8} - 39360 \, A a^{2} b c^{8}}{c^{9}}\right )} x + \frac {3003 \, B b^{7} c^{4} - 31284 \, B a b^{5} c^{5} - 4620 \, A b^{6} c^{5} + 103440 \, B a^{2} b^{3} c^{6} + 44640 \, A a b^{4} c^{6} - 103360 \, B a^{3} b c^{7} - 127680 \, A a^{2} b^{2} c^{7} + 81920 \, A a^{3} c^{8}}{c^{9}}\right )} x - \frac {15015 \, B b^{8} c^{3} - 171864 \, B a b^{6} c^{4} - 23100 \, A b^{7} c^{4} + 662400 \, B a^{2} b^{4} c^{5} + 246960 \, A a b^{5} c^{5} - 917120 \, B a^{3} b^{2} c^{6} - 843840 \, A a^{2} b^{3} c^{6} + 241920 \, B a^{4} c^{7} + 881920 \, A a^{3} b c^{7}}{c^{9}}\right )} x + \frac {45045 \, B b^{9} c^{2} - 563640 \, B a b^{7} c^{3} - 69300 \, A b^{8} c^{3} + 2487744 \, B a^{2} b^{5} c^{4} + 814800 \, A a b^{6} c^{4} - 4406400 \, B a^{3} b^{3} c^{5} - 3245760 \, A a^{2} b^{4} c^{5} + 2379520 \, B a^{4} b c^{6} + 4688640 \, A a^{3} b^{2} c^{6} - 1310720 \, A a^{4} c^{7}}{c^{9}}\right )} + \frac {{\left (143 \, B b^{10} - 1980 \, B a b^{8} c - 220 \, A b^{9} c + 10080 \, B a^{2} b^{6} c^{2} + 2880 \, A a b^{7} c^{2} - 22400 \, B a^{3} b^{4} c^{3} - 13440 \, A a^{2} b^{5} c^{3} + 19200 \, B a^{4} b^{2} c^{4} + 25600 \, A a^{3} b^{3} c^{4} - 3072 \, B a^{5} c^{5} - 15360 \, A a^{4} b c^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{262144 \, c^{\frac {15}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 1549, normalized size = 3.59
result too large to display
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^3\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,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^{3} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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