Optimal. Leaf size=278 \[ \frac{2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7} \]
[Out]
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Rubi [A] time = 0.386852, 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 \[ \frac{2 c (d+e x)^{3/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac{2 \sqrt{d+e x} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}-\frac{6 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^{5/2}}-\frac{6 c^2 (d+e x)^{5/2} (2 c d-b e)}{5 e^7}+\frac{2 c^3 (d+e x)^{7/2}}{7 e^7} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 70.818, size = 275, normalized size = 0.99 \[ \frac{2 c^{3} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} + \frac{6 c^{2} \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{7}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}} \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7}} + \frac{2 \sqrt{d + e x} \left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{e^{7}} - \frac{6 \left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{7} \sqrt{d + e x}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**(7/2),x)
[Out]
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Mathematica [A] time = 0.512867, size = 391, normalized size = 1.41 \[ -\frac{2 \left (7 c e^2 \left (a^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 a b e \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+b^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )\right )+7 e^3 \left (a^3 e^3+a^2 b e^2 (2 d+5 e x)+a b^2 e \left (8 d^2+20 d e x+15 e^2 x^2\right )+b^3 \left (-\left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )\right )-7 c^2 e \left (a e \left (-128 d^4-320 d^3 e x-240 d^2 e^2 x^2-40 d e^3 x^3+5 e^4 x^4\right )+b \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )}{35 e^7 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3/(d + e*x)^(7/2),x]
[Out]
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Maple [A] time = 0.012, size = 495, normalized size = 1.8 \[ -{\frac{-10\,{c}^{3}{x}^{6}{e}^{6}-42\,b{c}^{2}{e}^{6}{x}^{5}+24\,{c}^{3}d{e}^{5}{x}^{5}-70\,{x}^{4}a{c}^{2}{e}^{6}-70\,{b}^{2}c{e}^{6}{x}^{4}+140\,b{c}^{2}d{e}^{5}{x}^{4}-80\,{x}^{4}{c}^{3}{d}^{2}{e}^{4}-420\,abc{e}^{6}{x}^{3}+560\,{x}^{3}a{c}^{2}d{e}^{5}-70\,{b}^{3}{e}^{6}{x}^{3}+560\,{b}^{2}cd{e}^{5}{x}^{3}-1120\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+640\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+210\,{x}^{2}{a}^{2}c{e}^{6}+210\,a{b}^{2}{e}^{6}{x}^{2}-2520\,abcd{e}^{5}{x}^{2}+3360\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-420\,{b}^{3}d{e}^{5}{x}^{2}+3360\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}-6720\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}+3840\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+70\,{a}^{2}b{e}^{6}x+280\,x{a}^{2}cd{e}^{5}+280\,a{b}^{2}d{e}^{5}x-3360\,abc{d}^{2}{e}^{4}x+4480\,xa{c}^{2}{d}^{3}{e}^{3}-560\,{b}^{3}{d}^{2}{e}^{4}x+4480\,{b}^{2}c{d}^{3}{e}^{3}x-8960\,b{c}^{2}{d}^{4}{e}^{2}x+5120\,{c}^{3}{d}^{5}ex+14\,{a}^{3}{e}^{6}+28\,{a}^{2}bd{e}^{5}+112\,{a}^{2}c{d}^{2}{e}^{4}+112\,a{b}^{2}{d}^{2}{e}^{4}-1344\,abc{d}^{3}{e}^{3}+1792\,{c}^{2}{d}^{4}a{e}^{2}-224\,{b}^{3}{d}^{3}{e}^{3}+1792\,{b}^{2}c{d}^{4}{e}^{2}-3584\,b{c}^{2}{d}^{5}e+2048\,{c}^{3}{d}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3/(e*x+d)^(7/2),x)
[Out]
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Maxima [A] time = 0.708722, size = 558, normalized size = 2.01 \[ \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} - 21 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 15 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} +{\left (a b^{2} + a^{2} c\right )} e^{4}\right )}{\left (e x + d\right )}^{2} - 5 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{6}}\right )}}{35 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211959, size = 579, normalized size = 2.08 \[ \frac{2 \,{\left (5 \, c^{3} e^{6} x^{6} - 1024 \, c^{3} d^{6} + 1792 \, b c^{2} d^{5} e - 14 \, a^{2} b d e^{5} - 7 \, a^{3} e^{6} - 896 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 112 \,{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 56 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \,{\left (4 \, c^{3} d e^{5} - 7 \, b c^{2} e^{6}\right )} x^{5} + 5 \,{\left (8 \, c^{3} d^{2} e^{4} - 14 \, b c^{2} d e^{5} + 7 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 5 \,{\left (64 \, c^{3} d^{3} e^{3} - 112 \, b c^{2} d^{2} e^{4} + 56 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} - 7 \,{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 15 \,{\left (128 \, c^{3} d^{4} e^{2} - 224 \, b c^{2} d^{3} e^{3} + 112 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 14 \,{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 7 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 5 \,{\left (512 \, c^{3} d^{5} e - 896 \, b c^{2} d^{4} e^{2} + 7 \, a^{2} b e^{6} + 448 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 56 \,{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 28 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )}}{35 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{3}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**3/(e*x+d)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.215122, size = 822, normalized size = 2.96 \[ \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{2} e^{42} - 700 \, \sqrt{x e + d} c^{3} d^{3} e^{42} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} b c^{2} e^{43} - 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b c^{2} d e^{43} + 1050 \, \sqrt{x e + d} b c^{2} d^{2} e^{43} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} c e^{44} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} e^{44} - 420 \, \sqrt{x e + d} b^{2} c d e^{44} - 420 \, \sqrt{x e + d} a c^{2} d e^{44} + 35 \, \sqrt{x e + d} b^{3} e^{45} + 210 \, \sqrt{x e + d} a b c e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} c^{3} d^{4} - 10 \,{\left (x e + d\right )} c^{3} d^{5} + c^{3} d^{6} - 150 \,{\left (x e + d\right )}^{2} b c^{2} d^{3} e + 25 \,{\left (x e + d\right )} b c^{2} d^{4} e - 3 \, b c^{2} d^{5} e + 90 \,{\left (x e + d\right )}^{2} b^{2} c d^{2} e^{2} + 90 \,{\left (x e + d\right )}^{2} a c^{2} d^{2} e^{2} - 20 \,{\left (x e + d\right )} b^{2} c d^{3} e^{2} - 20 \,{\left (x e + d\right )} a c^{2} d^{3} e^{2} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} - 15 \,{\left (x e + d\right )}^{2} b^{3} d e^{3} - 90 \,{\left (x e + d\right )}^{2} a b c d e^{3} + 5 \,{\left (x e + d\right )} b^{3} d^{2} e^{3} + 30 \,{\left (x e + d\right )} a b c d^{2} e^{3} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} + 15 \,{\left (x e + d\right )}^{2} a b^{2} e^{4} + 15 \,{\left (x e + d\right )}^{2} a^{2} c e^{4} - 10 \,{\left (x e + d\right )} a b^{2} d e^{4} - 10 \,{\left (x e + d\right )} a^{2} c d e^{4} + 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} c d^{2} e^{4} + 5 \,{\left (x e + d\right )} a^{2} b e^{5} - 3 \, a^{2} b d e^{5} + a^{3} e^{6}\right )} e^{\left (-7\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^(7/2),x, algorithm="giac")
[Out]