Optimal. Leaf size=278 \[ \frac{2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7} \]
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Rubi [A] time = 0.134166, antiderivative size = 278, 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} \[ \frac{2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7} \]
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)^{7/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^{7/2}}+\frac{3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^{5/2}}+\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 )}{e^6 (d+e x)^{3/2}}+\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 )}{e^6 \sqrt{d+e x}}+\frac{3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \sqrt{d+e x}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{3/2}}{e^6}+\frac{c^3 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^{5/2}}+\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{6 \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 )}{e^7 \sqrt{d+e x}}-\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 ) \sqrt{d+e x}}{e^7}+\frac{2 c \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{6 c^2 (2 c d-b e) (d+e x)^{5/2}}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7}\\ \end{align*}
Mathematica [A] time = 0.364776, size = 391, normalized size = 1.41 \[ -\frac{2 \left (7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )+7 e^3 \left (a^2 b e^2 (2 d+5 e x)+a^3 e^3+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )+b^3 \left (-\left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )-7 c^2 e \left (a e \left (-240 d^2 e^2 x^2-320 d^3 e x-128 d^4-40 d e^3 x^3+5 e^4 x^4\right )+b \left (480 d^3 e^2 x^2+80 d^2 e^3 x^3+640 d^4 e x+256 d^5-10 d e^4 x^4+3 e^5 x^5\right )\right )+c^3 \left (1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+2560 d^5 e x+1024 d^6+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 495, normalized size = 1.8 \begin{align*} -{\frac{-10\,{c}^{3}{x}^{6}{e}^{6}-42\,b{c}^{2}{e}^{6}{x}^{5}+24\,{c}^{3}d{e}^{5}{x}^{5}-70\,a{c}^{2}{e}^{6}{x}^{4}-70\,{b}^{2}c{e}^{6}{x}^{4}+140\,b{c}^{2}d{e}^{5}{x}^{4}-80\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-420\,abc{e}^{6}{x}^{3}+560\,a{c}^{2}d{e}^{5}{x}^{3}-70\,{b}^{3}{e}^{6}{x}^{3}+560\,{b}^{2}cd{e}^{5}{x}^{3}-1120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+210\,{a}^{2}c{e}^{6}{x}^{2}+210\,a{b}^{2}{e}^{6}{x}^{2}-2520\,abcd{e}^{5}{x}^{2}+3360\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-420\,{b}^{3}d{e}^{5}{x}^{2}+3360\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-6720\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+3840\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+70\,{a}^{2}b{e}^{6}x+280\,{a}^{2}cd{e}^{5}x+280\,a{b}^{2}d{e}^{5}x-3360\,abc{d}^{2}{e}^{4}x+4480\,a{c}^{2}{d}^{3}{e}^{3}x-560\,{b}^{3}{d}^{2}{e}^{4}x+4480\,{b}^{2}c{d}^{3}{e}^{3}x-8960\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+14\,{a}^{3}{e}^{6}+28\,{a}^{2}bd{e}^{5}+112\,{a}^{2}c{d}^{2}{e}^{4}+112\,a{b}^{2}{d}^{2}{e}^{4}-1344\,abc{d}^{3}{e}^{3}+1792\,a{c}^{2}{d}^{4}{e}^{2}-224\,{b}^{3}{d}^{3}{e}^{3}+1792\,{b}^{2}c{d}^{4}{e}^{2}-3584\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03108, size = 558, normalized size = 2.01 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 21 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\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{3}{2}} - 35 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\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} + 15 \,{\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 )}^{2} - 5 \,{\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 )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{6}}\right )}}{35 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99361, size = 950, normalized size = 3.42 \begin{align*} \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 14 \, a^{2} b d e^{5} - 7 \, a^{3} e^{6} - 896 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 112 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 56 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 7 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 15 \,{\left (128 \, c^{3} d^{4} e^{2} - 224 \, b c^{2} d^{3} e^{3} + 112 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 14 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 7 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \,{\left (512 \, c^{3} d^{5} e - 896 \, b c^{2} d^{4} e^{2} + 7 \, a^{2} b e^{6} + 448 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 56 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 28 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\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 = 1.13184, size = 822, normalized size = 2.96 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} e^{43} - 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt{x e + d} b c^{2} d^{2} e^{43} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c e^{44} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{44} - 420 \, \sqrt{x e + d} b^{2} c d e^{44} - 420 \, \sqrt{x e + d} a c^{2} d e^{44} + 35 \, \sqrt{x e + d} b^{3} e^{45} + 210 \, \sqrt{x e + d} a b c e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \,{\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \,{\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \,{\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} + 90 \,{\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} - 20 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - 15 \,{\left (x e + d\right )}^{2} b^{3} d e^{3} - 90 \,{\left (x e + d\right )}^{2} a b c d e^{3} + 5 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} + 30 \,{\left (x e + d\right )} a b c d^{2} e^{3} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 15 \,{\left (x e + d\right )}^{2} a b^{2} e^{4} + 15 \,{\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \,{\left (x e + d\right )} a b^{2} d e^{4} - 10 \,{\left (x e + d\right )} a^{2} c d e^{4} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} + 5 \,{\left (x e + d\right )} a^{2} b e^{5} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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