Optimal. Leaf size=333 \[ \frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}+\frac {5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}}-\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
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Rubi [A] time = 0.31, antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}-\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac {5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac {5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac {5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
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 )^{5/2} \, dx &=\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\int x \left (-2 a B-\frac {1}{2} (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=-\frac {\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=\frac {5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac {\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac {\left (5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}{8192 c^5}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}}{32768 c^6}+\frac {5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac {\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 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}{65536 c^6}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}}{32768 c^6}+\frac {5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac {\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {\left (5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 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 )}{32768 c^6}\\ &=-\frac {5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}}{32768 c^6}+\frac {5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac {\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac {B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac {\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac {5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 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 )}{65536 c^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 254, normalized size = 0.76 \[ \frac {\frac {(a+x (b+c x))^{7/2} \left (4 c (63 A c x-16 a B)-2 b c (81 A+77 B x)+99 b^2 B\right )}{224 c^2}+\frac {3 \left (-8 a A c^2+12 a b B c+18 A b^2 c-11 b^3 B\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )}{65536 c^{11/2}}+B x^2 (a+x (b+c x))^{7/2}}{9 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.77, size = 1263, normalized size = 3.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 643, normalized size = 1.93 \[ \frac {1}{2064384} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (14 \, {\left (16 \, B c^{2} x + \frac {37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac {309 \, B b^{2} c^{8} + 608 \, B a c^{9} + 594 \, A b c^{9}}{c^{8}}\right )} x + \frac {5 \, B b^{3} c^{7} + 3012 \, B a b c^{8} + 1458 \, A b^{2} c^{8} + 2856 \, A a c^{9}}{c^{8}}\right )} x - \frac {11 \, B b^{4} c^{6} - 84 \, B a b^{2} c^{7} - 18 \, A b^{3} c^{7} - 3840 \, B a^{2} c^{8} - 7368 \, A a b c^{8}}{c^{8}}\right )} x + \frac {99 \, B b^{5} c^{5} - 856 \, B a b^{3} c^{6} - 162 \, A b^{4} c^{6} + 1968 \, B a^{2} b c^{7} + 1296 \, A a b^{2} c^{7} + 39648 \, A a^{2} c^{8}}{c^{8}}\right )} x - \frac {231 \, B b^{6} c^{4} - 2232 \, B a b^{4} c^{5} - 378 \, A b^{5} c^{5} + 6384 \, B a^{2} b^{2} c^{6} + 3408 \, A a b^{3} c^{6} - 4096 \, B a^{3} c^{7} - 8352 \, A a^{2} b c^{7}}{c^{8}}\right )} x + \frac {1155 \, B b^{7} c^{3} - 12348 \, B a b^{5} c^{4} - 1890 \, A b^{6} c^{4} + 42192 \, B a^{2} b^{3} c^{5} + 18984 \, A a b^{4} c^{5} - 44096 \, B a^{3} b c^{6} - 57312 \, A a^{2} b^{2} c^{6} + 40320 \, A a^{3} c^{7}}{c^{8}}\right )} x - \frac {3465 \, B b^{8} c^{2} - 40740 \, B a b^{6} c^{3} - 5670 \, A b^{7} c^{3} + 162288 \, B a^{2} b^{4} c^{4} + 63000 \, A a b^{5} c^{4} - 234432 \, B a^{3} b^{2} c^{5} - 226464 \, A a^{2} b^{3} c^{5} + 65536 \, B a^{4} c^{6} + 254592 \, A a^{3} b c^{6}}{c^{8}}\right )} - \frac {5 \, {\left (11 \, B b^{9} - 144 \, B a b^{7} c - 18 \, A b^{8} c + 672 \, B a^{2} b^{5} c^{2} + 224 \, A a b^{6} c^{2} - 1280 \, B a^{3} b^{3} c^{3} - 960 \, A a^{2} b^{4} c^{3} + 768 \, B a^{4} b c^{4} + 1536 \, A a^{3} b^{2} c^{4} - 512 \, A a^{4} c^{5}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{65536 \, c^{\frac {13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 1277, normalized size = 3.83 \[ \text {result too large to display} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int x^2\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \]
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 \[ \int x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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