Optimal. Leaf size=425 \[ \frac{2 (d+e x)^{7/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^8}+\frac{6 c^2 (d+e x)^{11/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^8}-\frac{10 c (d+e x)^{9/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac{14 c^3 (d+e x)^{13/2} (2 c d-b e)}{13 e^8}+\frac{4 c^4 (d+e x)^{15/2}}{15 e^8} \]
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Rubi [A] time = 0.233837, antiderivative size = 425, 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} \[ \frac{2 (d+e x)^{7/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{7 e^8}+\frac{6 c^2 (d+e x)^{11/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{11 e^8}-\frac{10 c (d+e x)^{9/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{9 e^8}-\frac{6 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^8}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac{14 c^3 (d+e x)^{13/2} (2 c d-b e)}{13 e^8}+\frac{4 c^4 (d+e x)^{15/2}}{15 e^8} \]
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}{\sqrt{d+e x}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 \sqrt{d+e x}}+\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 ) \sqrt{d+e x}}{e^7}+\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 ) (d+e x)^{3/2}}{e^7}+\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 ) (d+e x)^{5/2}}{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)^{7/2}}{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)^{9/2}}{e^7}-\frac{7 c^3 (2 c d-b e) (d+e x)^{11/2}}{e^7}+\frac{2 c^4 (d+e x)^{13/2}}{e^7}\right ) \, dx\\ &=-\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 \sqrt{d+e x}}{e^8}+\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 ) (d+e x)^{3/2}}{3 e^8}-\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 ) (d+e x)^{5/2}}{5 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)^{7/2}}{7 e^8}-\frac{10 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)^{9/2}}{9 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)^{11/2}}{11 e^8}-\frac{14 c^3 (2 c d-b e) (d+e x)^{13/2}}{13 e^8}+\frac{4 c^4 (d+e x)^{15/2}}{15 e^8}\\ \end{align*}
Mathematica [A] time = 0.635182, size = 599, normalized size = 1.41 \[ \frac{2 \sqrt{d+e x} \left (-39 c^2 e^2 \left (198 a^2 e^2 \left (-8 d^2 e x+16 d^3+6 d e^2 x^2-5 e^3 x^3\right )-55 a b e \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )+15 b^2 \left (96 d^3 e^2 x^2-80 d^2 e^3 x^3-128 d^4 e x+256 d^5+70 d e^4 x^4-63 e^5 x^5\right )\right )+143 c e^3 \left (189 a^2 b e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+210 a^3 e^3 (e x-2 d)+108 a b^2 e \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )+5 b^3 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )+1287 b e^4 \left (35 a^2 b e^2 (e x-2 d)+35 a^3 e^3+7 a b^2 e \left (8 d^2-4 d e x+3 e^2 x^2\right )+b^3 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )\right )+15 c^3 e \left (26 a e \left (-96 d^3 e^2 x^2+80 d^2 e^3 x^3+128 d^4 e x-256 d^5-70 d e^4 x^4+63 e^5 x^5\right )+7 b \left (384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-512 d^5 e x+1024 d^6-252 d e^5 x^5+231 e^6 x^6\right )\right )-14 c^4 \left (768 d^5 e^2 x^2-640 d^4 e^3 x^3+560 d^3 e^4 x^4-504 d^2 e^5 x^5-1024 d^6 e x+2048 d^7+462 d e^6 x^6-429 e^7 x^7\right )\right )}{45045 e^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 795, normalized size = 1.9 \begin{align*}{\frac{12012\,{c}^{4}{x}^{7}{e}^{7}+48510\,b{c}^{3}{e}^{7}{x}^{6}-12936\,{c}^{4}d{e}^{6}{x}^{6}+49140\,a{c}^{3}{e}^{7}{x}^{5}+73710\,{b}^{2}{c}^{2}{e}^{7}{x}^{5}-52920\,b{c}^{3}d{e}^{6}{x}^{5}+14112\,{c}^{4}{d}^{2}{e}^{5}{x}^{5}+150150\,ab{c}^{2}{e}^{7}{x}^{4}-54600\,a{c}^{3}d{e}^{6}{x}^{4}+50050\,{b}^{3}c{e}^{7}{x}^{4}-81900\,{b}^{2}{c}^{2}d{e}^{6}{x}^{4}+58800\,b{c}^{3}{d}^{2}{e}^{5}{x}^{4}-15680\,{c}^{4}{d}^{3}{e}^{4}{x}^{4}+77220\,{a}^{2}{c}^{2}{e}^{7}{x}^{3}+154440\,a{b}^{2}c{e}^{7}{x}^{3}-171600\,ab{c}^{2}d{e}^{6}{x}^{3}+62400\,a{c}^{3}{d}^{2}{e}^{5}{x}^{3}+12870\,{b}^{4}{e}^{7}{x}^{3}-57200\,{b}^{3}cd{e}^{6}{x}^{3}+93600\,{b}^{2}{c}^{2}{d}^{2}{e}^{5}{x}^{3}-67200\,b{c}^{3}{d}^{3}{e}^{4}{x}^{3}+17920\,{c}^{4}{d}^{4}{e}^{3}{x}^{3}+162162\,{a}^{2}bc{e}^{7}{x}^{2}-92664\,{a}^{2}{c}^{2}d{e}^{6}{x}^{2}+54054\,a{b}^{3}{e}^{7}{x}^{2}-185328\,a{b}^{2}cd{e}^{6}{x}^{2}+205920\,ab{c}^{2}{d}^{2}{e}^{5}{x}^{2}-74880\,a{c}^{3}{d}^{3}{e}^{4}{x}^{2}-15444\,{b}^{4}d{e}^{6}{x}^{2}+68640\,{b}^{3}c{d}^{2}{e}^{5}{x}^{2}-112320\,{b}^{2}{c}^{2}{d}^{3}{e}^{4}{x}^{2}+80640\,b{c}^{3}{d}^{4}{e}^{3}{x}^{2}-21504\,{c}^{4}{d}^{5}{e}^{2}{x}^{2}+60060\,{a}^{3}c{e}^{7}x+90090\,{a}^{2}{b}^{2}{e}^{7}x-216216\,{a}^{2}bcd{e}^{6}x+123552\,{a}^{2}{c}^{2}{d}^{2}{e}^{5}x-72072\,a{b}^{3}d{e}^{6}x+247104\,a{b}^{2}c{d}^{2}{e}^{5}x-274560\,ab{c}^{2}{d}^{3}{e}^{4}x+99840\,a{c}^{3}{d}^{4}{e}^{3}x+20592\,{b}^{4}{d}^{2}{e}^{5}x-91520\,{b}^{3}c{d}^{3}{e}^{4}x+149760\,{b}^{2}{c}^{2}{d}^{4}{e}^{3}x-107520\,b{c}^{3}{d}^{5}{e}^{2}x+28672\,{c}^{4}{d}^{6}ex+90090\,b{a}^{3}{e}^{7}-120120\,{a}^{3}cd{e}^{6}-180180\,{a}^{2}{b}^{2}d{e}^{6}+432432\,{a}^{2}bc{d}^{2}{e}^{5}-247104\,{a}^{2}{c}^{2}{d}^{3}{e}^{4}+144144\,a{b}^{3}{d}^{2}{e}^{5}-494208\,a{b}^{2}c{d}^{3}{e}^{4}+549120\,ab{c}^{2}{d}^{4}{e}^{3}-199680\,a{c}^{3}{d}^{5}{e}^{2}-41184\,{b}^{4}{d}^{3}{e}^{4}+183040\,{b}^{3}c{d}^{4}{e}^{3}-299520\,{b}^{2}{c}^{2}{d}^{5}{e}^{2}+215040\,b{c}^{3}{d}^{6}e-57344\,{c}^{4}{d}^{7}}{45045\,{e}^{8}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02841, size = 871, normalized size = 2.05 \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 = 1.27552, size = 1501, normalized size = 3.53 \begin{align*} \frac{2 \,{\left (6006 \, c^{4} e^{7} x^{7} - 28672 \, c^{4} d^{7} + 107520 \, b c^{3} d^{6} e + 45045 \, a^{3} b e^{7} - 49920 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 91520 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 20592 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 72072 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 30030 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 1617 \,{\left (4 \, c^{4} d e^{6} - 15 \, b c^{3} e^{7}\right )} x^{6} + 63 \,{\left (112 \, c^{4} d^{2} e^{5} - 420 \, b c^{3} d e^{6} + 195 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 35 \,{\left (224 \, c^{4} d^{3} e^{4} - 840 \, b c^{3} d^{2} e^{5} + 390 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 715 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + 5 \,{\left (1792 \, c^{4} d^{4} e^{3} - 6720 \, b c^{3} d^{3} e^{4} + 3120 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 5720 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 1287 \,{\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} - 13440 \, b c^{3} d^{4} e^{3} + 6240 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 11440 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 2574 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 9009 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} +{\left (14336 \, c^{4} d^{6} e - 53760 \, b c^{3} d^{5} e^{2} + 24960 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 45760 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 10296 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 36036 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 15015 \,{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{8}} \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 = 1.23978, size = 1135, normalized size = 2.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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