Optimal. Leaf size=421 \[ \frac{2 (d+e x)^{3/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac{6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac{2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac{14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac{4 c^4 (d+e x)^{11/2}}{11 e^8} \]
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Rubi [A] time = 0.228455, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {771} \[ \frac{2 (d+e x)^{3/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^8}+\frac{6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac{2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac{14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac{4 c^4 (d+e x)^{11/2}}{11 e^8} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^{5/2}}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^7 (d+e x)^{3/2}}+\frac{3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right )}{e^7 \sqrt{d+e x}}+\frac{\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \sqrt{d+e x}}{e^7}+\frac{5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{3/2}}{e^7}+\frac{3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{7/2}}{e^7}+\frac{2 c^4 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac{2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right )}{e^8 \sqrt{d+e x}}-\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \sqrt{d+e x}}{e^8}+\frac{2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{3/2}}{3 e^8}-\frac{2 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{5/2}}{e^8}+\frac{6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{7/2}}{7 e^8}-\frac{14 c^3 (2 c d-b e) (d+e x)^{9/2}}{9 e^8}+\frac{4 c^4 (d+e x)^{11/2}}{11 e^8}\\ \end{align*}
Mathematica [A] time = 0.608375, size = 598, normalized size = 1.42 \[ \frac{-198 c^2 e^2 \left (14 a^2 e^2 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )-7 a b e \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )+3 b^2 \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )\right )+462 c e^3 \left (9 a^2 b e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )-2 a^3 e^3 (2 d+3 e x)+12 a b^2 e \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )+b^3 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )-462 b e^4 \left (3 a^2 b e^2 (2 d+3 e x)+a^3 e^3-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )+22 c^3 e \left (7 b \left (384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4+1536 d^5 e x+1024 d^6-12 d e^5 x^5+7 e^6 x^6\right )-18 a e \left (96 d^3 e^2 x^2-16 d^2 e^3 x^3+384 d^4 e x+256 d^5+6 d e^4 x^4-3 e^5 x^5\right )\right )-28 c^4 \left (768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+3072 d^6 e x+2048 d^7+14 d e^6 x^6-9 e^7 x^7\right )}{693 e^8 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 795, normalized size = 1.9 \begin{align*} -{\frac{-252\,{c}^{4}{x}^{7}{e}^{7}-1078\,b{c}^{3}{e}^{7}{x}^{6}+392\,{c}^{4}d{e}^{6}{x}^{6}-1188\,a{c}^{3}{e}^{7}{x}^{5}-1782\,{b}^{2}{c}^{2}{e}^{7}{x}^{5}+1848\,b{c}^{3}d{e}^{6}{x}^{5}-672\,{c}^{4}{d}^{2}{e}^{5}{x}^{5}-4158\,ab{c}^{2}{e}^{7}{x}^{4}+2376\,a{c}^{3}d{e}^{6}{x}^{4}-1386\,{b}^{3}c{e}^{7}{x}^{4}+3564\,{b}^{2}{c}^{2}d{e}^{6}{x}^{4}-3696\,b{c}^{3}{d}^{2}{e}^{5}{x}^{4}+1344\,{c}^{4}{d}^{3}{e}^{4}{x}^{4}-2772\,{a}^{2}{c}^{2}{e}^{7}{x}^{3}-5544\,a{b}^{2}c{e}^{7}{x}^{3}+11088\,ab{c}^{2}d{e}^{6}{x}^{3}-6336\,a{c}^{3}{d}^{2}{e}^{5}{x}^{3}-462\,{b}^{4}{e}^{7}{x}^{3}+3696\,{b}^{3}cd{e}^{6}{x}^{3}-9504\,{b}^{2}{c}^{2}{d}^{2}{e}^{5}{x}^{3}+9856\,b{c}^{3}{d}^{3}{e}^{4}{x}^{3}-3584\,{c}^{4}{d}^{4}{e}^{3}{x}^{3}-12474\,{a}^{2}bc{e}^{7}{x}^{2}+16632\,{a}^{2}{c}^{2}d{e}^{6}{x}^{2}-4158\,a{b}^{3}{e}^{7}{x}^{2}+33264\,a{b}^{2}cd{e}^{6}{x}^{2}-66528\,ab{c}^{2}{d}^{2}{e}^{5}{x}^{2}+38016\,a{c}^{3}{d}^{3}{e}^{4}{x}^{2}+2772\,{b}^{4}d{e}^{6}{x}^{2}-22176\,{b}^{3}c{d}^{2}{e}^{5}{x}^{2}+57024\,{b}^{2}{c}^{2}{d}^{3}{e}^{4}{x}^{2}-59136\,b{c}^{3}{d}^{4}{e}^{3}{x}^{2}+21504\,{c}^{4}{d}^{5}{e}^{2}{x}^{2}+2772\,{a}^{3}c{e}^{7}x+4158\,{a}^{2}{b}^{2}{e}^{7}x-49896\,{a}^{2}bcd{e}^{6}x+66528\,{a}^{2}{c}^{2}{d}^{2}{e}^{5}x-16632\,a{b}^{3}d{e}^{6}x+133056\,a{b}^{2}c{d}^{2}{e}^{5}x-266112\,ab{c}^{2}{d}^{3}{e}^{4}x+152064\,a{c}^{3}{d}^{4}{e}^{3}x+11088\,{b}^{4}{d}^{2}{e}^{5}x-88704\,{b}^{3}c{d}^{3}{e}^{4}x+228096\,{b}^{2}{c}^{2}{d}^{4}{e}^{3}x-236544\,b{c}^{3}{d}^{5}{e}^{2}x+86016\,{c}^{4}{d}^{6}ex+462\,b{a}^{3}{e}^{7}+1848\,{a}^{3}cd{e}^{6}+2772\,{a}^{2}{b}^{2}d{e}^{6}-33264\,{a}^{2}bc{d}^{2}{e}^{5}+44352\,{a}^{2}{c}^{2}{d}^{3}{e}^{4}-11088\,a{b}^{3}{d}^{2}{e}^{5}+88704\,a{b}^{2}c{d}^{3}{e}^{4}-177408\,ab{c}^{2}{d}^{4}{e}^{3}+101376\,a{c}^{3}{d}^{5}{e}^{2}+7392\,{b}^{4}{d}^{3}{e}^{4}-59136\,{b}^{3}c{d}^{4}{e}^{3}+152064\,{b}^{2}{c}^{2}{d}^{5}{e}^{2}-157696\,b{c}^{3}{d}^{6}e+57344\,{c}^{4}{d}^{7}}{693\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994385, size = 879, normalized size = 2.09 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43288, size = 1508, normalized size = 3.58 \begin{align*} \frac{2 \,{\left (126 \, c^{4} e^{7} x^{7} - 28672 \, c^{4} d^{7} + 78848 \, b c^{3} d^{6} e - 231 \, a^{3} b e^{7} - 25344 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 29568 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 3696 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 5544 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 462 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 49 \,{\left (4 \, c^{4} d e^{6} - 11 \, b c^{3} e^{7}\right )} x^{6} + 3 \,{\left (112 \, c^{4} d^{2} e^{5} - 308 \, b c^{3} d e^{6} + 99 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 3 \,{\left (224 \, c^{4} d^{3} e^{4} - 616 \, b c^{3} d^{2} e^{5} + 198 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 231 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} +{\left (1792 \, c^{4} d^{4} e^{3} - 4928 \, b c^{3} d^{3} e^{4} + 1584 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 1848 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 231 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 3 \,{\left (3584 \, c^{4} d^{5} e^{2} - 9856 \, b c^{3} d^{4} e^{3} + 3168 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 3696 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 462 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 693 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} - 3 \,{\left (14336 \, c^{4} d^{6} e - 39424 \, b c^{3} d^{5} e^{2} + 12672 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 14784 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 1848 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 2772 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 231 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )} \sqrt{e x + d}}{693 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28996, size = 1312, normalized size = 3.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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