Optimal. Leaf size=333 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac{5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]
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Rubi [A] time = 0.309255, antiderivative size = 333, normalized size of antiderivative = 1., 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 832
Rule 779
Rule 612
Rule 621
Rule 206
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*}
Mathematica [A] time = 0.488308, 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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Maple [B] time = 0.01, size = 1277, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.73838, size = 3071, normalized size = 9.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35669, size = 868, normalized size = 2.61 \begin{align*} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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