Optimal. Leaf size=280 \[ \frac{6 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7} \]
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Rubi [A] time = 0.127591, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {698} \[ \frac{6 c (d+e x)^{7/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^7}-\frac{2 (d+e x)^{5/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7}+\frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{6 \sqrt{d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{e^7 \sqrt{d+e x}}-\frac{2 c^2 (d+e x)^{9/2} (2 c d-b e)}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{3/2}}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 \sqrt{d+e x}}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) \sqrt{d+e x}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{3/2}}{e^6}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{7/2}}{e^6}+\frac{c^3 (d+e x)^{9/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^3}{e^7 \sqrt{d+e x}}-\frac{6 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt{d+e x}}{e^7}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3/2}}{e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{5/2}}{5 e^7}+\frac{6 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{7 e^7}-\frac{2 c^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac{2 c^3 (d+e x)^{11/2}}{11 e^7}\\ \end{align*}
Mathematica [A] time = 0.411908, size = 394, normalized size = 1.41 \[ \frac{-66 c e^2 \left (35 a^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )-42 a b e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )+3 b^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )\right )+462 e^3 \left (15 a^2 b e^2 (2 d+e x)-5 a^3 e^3+5 a b^2 e \left (-8 d^2-4 d e x+e^2 x^2\right )+b^3 \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )\right )+22 c^2 e \left (9 a e \left (16 d^2 e^2 x^2-64 d^3 e x-128 d^4-8 d e^3 x^3+5 e^4 x^4\right )+5 b \left (-32 d^3 e^2 x^2+16 d^2 e^3 x^3+128 d^4 e x+256 d^5-10 d e^4 x^4+7 e^5 x^5\right )\right )-10 c^3 \left (-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+512 d^5 e x+1024 d^6+28 d e^5 x^5-21 e^6 x^6\right )}{1155 e^7 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 495, normalized size = 1.8 \begin{align*} -{\frac{-210\,{c}^{3}{x}^{6}{e}^{6}-770\,b{c}^{2}{e}^{6}{x}^{5}+280\,{c}^{3}d{e}^{5}{x}^{5}-990\,a{c}^{2}{e}^{6}{x}^{4}-990\,{b}^{2}c{e}^{6}{x}^{4}+1100\,b{c}^{2}d{e}^{5}{x}^{4}-400\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-2772\,abc{e}^{6}{x}^{3}+1584\,a{c}^{2}d{e}^{5}{x}^{3}-462\,{b}^{3}{e}^{6}{x}^{3}+1584\,{b}^{2}cd{e}^{5}{x}^{3}-1760\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}-2310\,{a}^{2}c{e}^{6}{x}^{2}-2310\,a{b}^{2}{e}^{6}{x}^{2}+5544\,abcd{e}^{5}{x}^{2}-3168\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+924\,{b}^{3}d{e}^{5}{x}^{2}-3168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+3520\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-1280\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-6930\,{a}^{2}b{e}^{6}x+9240\,{a}^{2}cd{e}^{5}x+9240\,a{b}^{2}d{e}^{5}x-22176\,abc{d}^{2}{e}^{4}x+12672\,a{c}^{2}{d}^{3}{e}^{3}x-3696\,{b}^{3}{d}^{2}{e}^{4}x+12672\,{b}^{2}c{d}^{3}{e}^{3}x-14080\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+2310\,{a}^{3}{e}^{6}-13860\,{a}^{2}bd{e}^{5}+18480\,{a}^{2}c{d}^{2}{e}^{4}+18480\,a{b}^{2}{d}^{2}{e}^{4}-44352\,abc{d}^{3}{e}^{3}+25344\,a{c}^{2}{d}^{4}{e}^{2}-7392\,{b}^{3}{d}^{3}{e}^{3}+25344\,{b}^{2}c{d}^{4}{e}^{2}-28160\,b{c}^{2}{d}^{5}e+10240\,{c}^{3}{d}^{6}}{1155\,{e}^{7}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04633, size = 560, normalized size = 2. \begin{align*} \frac{2 \,{\left (\frac{105 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{3} - 385 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 231 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 3465 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )} \sqrt{e x + d}}{e^{6}} - \frac{1155 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )}}{\sqrt{e x + d} e^{6}}\right )}}{1155 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01709, size = 950, normalized size = 3.39 \begin{align*} \frac{2 \,{\left (105 \, c^{3} e^{6} x^{6} - 5120 \, c^{3} d^{6} + 14080 \, b c^{2} d^{5} e + 6930 \, a^{2} b d e^{5} - 1155 \, a^{3} e^{6} - 12672 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3696 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 9240 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 35 \,{\left (4 \, c^{3} d e^{5} - 11 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (40 \, c^{3} d^{2} e^{4} - 110 \, b c^{2} d e^{5} + 99 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} -{\left (320 \, c^{3} d^{3} e^{3} - 880 \, b c^{2} d^{2} e^{4} + 792 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 231 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} +{\left (640 \, c^{3} d^{4} e^{2} - 1760 \, b c^{2} d^{3} e^{3} + 1584 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 462 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 1155 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} -{\left (2560 \, c^{3} d^{5} e - 7040 \, b c^{2} d^{4} e^{2} - 3465 \, a^{2} b e^{6} + 6336 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1848 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 4620 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{1155 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 99.2054, size = 428, normalized size = 1.53 \begin{align*} \frac{2 c^{3} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (6 b c^{2} e - 12 c^{3} d\right )}{9 e^{7}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (6 a c^{2} e^{2} + 6 b^{2} c e^{2} - 30 b c^{2} d e + 30 c^{3} d^{2}\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (12 a b c e^{3} - 24 a c^{2} d e^{2} + 2 b^{3} e^{3} - 24 b^{2} c d e^{2} + 60 b c^{2} d^{2} e - 40 c^{3} d^{3}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 a^{2} c e^{4} + 6 a b^{2} e^{4} - 36 a b c d e^{3} + 36 a c^{2} d^{2} e^{2} - 6 b^{3} d e^{3} + 36 b^{2} c d^{2} e^{2} - 60 b c^{2} d^{3} e + 30 c^{3} d^{4}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (6 a^{2} b e^{5} - 12 a^{2} c d e^{4} - 12 a b^{2} d e^{4} + 36 a b c d^{2} e^{3} - 24 a c^{2} d^{3} e^{2} + 6 b^{3} d^{2} e^{3} - 24 b^{2} c d^{3} e^{2} + 30 b c^{2} d^{4} e - 12 c^{3} d^{5}\right )}{e^{7}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{e^{7} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1384, size = 852, normalized size = 3.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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