Optimal. Leaf size=185 \[ -\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7} \]
[Out]
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Rubi [A] time = 0.182906, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{12 b^5 (d+e x)^{17/2} (b d-a e)}{17 e^7}+\frac{2 b^4 (d+e x)^{15/2} (b d-a e)^2}{e^7}-\frac{40 b^3 (d+e x)^{13/2} (b d-a e)^3}{13 e^7}+\frac{30 b^2 (d+e x)^{11/2} (b d-a e)^4}{11 e^7}-\frac{4 b (d+e x)^{9/2} (b d-a e)^5}{3 e^7}+\frac{2 (d+e x)^{7/2} (b d-a e)^6}{7 e^7}+\frac{2 b^6 (d+e x)^{19/2}}{19 e^7} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 84.5476, size = 172, normalized size = 0.93 \[ \frac{2 b^{6} \left (d + e x\right )^{\frac{19}{2}}}{19 e^{7}} + \frac{12 b^{5} \left (d + e x\right )^{\frac{17}{2}} \left (a e - b d\right )}{17 e^{7}} + \frac{2 b^{4} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )^{2}}{e^{7}} + \frac{40 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{3}}{13 e^{7}} + \frac{30 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{4}}{11 e^{7}} + \frac{4 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{5}}{3 e^{7}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{6}}{7 e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.399254, size = 291, normalized size = 1.57 \[ \frac{2 (d+e x)^{7/2} \left (138567 a^6 e^6+92378 a^5 b e^5 (7 e x-2 d)+20995 a^4 b^2 e^4 \left (8 d^2-28 d e x+63 e^2 x^2\right )+6460 a^3 b^3 e^3 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+323 a^2 b^4 e^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+38 a b^5 e \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )+b^6 \left (1024 d^6-3584 d^5 e x+8064 d^4 e^2 x^2-14784 d^3 e^3 x^3+24024 d^2 e^4 x^4-36036 d e^5 x^5+51051 e^6 x^6\right )\right )}{969969 e^7} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [B] time = 0.013, size = 377, normalized size = 2. \[{\frac{102102\,{x}^{6}{b}^{6}{e}^{6}+684684\,{x}^{5}a{b}^{5}{e}^{6}-72072\,{x}^{5}{b}^{6}d{e}^{5}+1939938\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-456456\,{x}^{4}a{b}^{5}d{e}^{5}+48048\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+2984520\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-1193808\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+280896\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-29568\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+2645370\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-1627920\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+651168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-153216\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+16128\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+1293292\,x{a}^{5}b{e}^{6}-1175720\,x{a}^{4}{b}^{2}d{e}^{5}+723520\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-289408\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+68096\,xa{b}^{5}{d}^{4}{e}^{2}-7168\,x{b}^{6}{d}^{5}e+277134\,{a}^{6}{e}^{6}-369512\,{a}^{5}bd{e}^{5}+335920\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-206720\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+82688\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-19456\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{969969\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.73431, size = 473, normalized size = 2.56 \[ \frac{2 \,{\left (51051 \,{\left (e x + d\right )}^{\frac{19}{2}} b^{6} - 342342 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 1492260 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 646646 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 138567 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{969969 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208747, size = 857, normalized size = 4.63 \[ \frac{2 \,{\left (51051 \, b^{6} e^{9} x^{9} + 1024 \, b^{6} d^{9} - 9728 \, a b^{5} d^{8} e + 41344 \, a^{2} b^{4} d^{7} e^{2} - 103360 \, a^{3} b^{3} d^{6} e^{3} + 167960 \, a^{4} b^{2} d^{5} e^{4} - 184756 \, a^{5} b d^{4} e^{5} + 138567 \, a^{6} d^{3} e^{6} + 9009 \,{\left (13 \, b^{6} d e^{8} + 38 \, a b^{5} e^{9}\right )} x^{8} + 3003 \,{\left (23 \, b^{6} d^{2} e^{7} + 266 \, a b^{5} d e^{8} + 323 \, a^{2} b^{4} e^{9}\right )} x^{7} + 231 \,{\left (b^{6} d^{3} e^{6} + 2090 \, a b^{5} d^{2} e^{7} + 10013 \, a^{2} b^{4} d e^{8} + 6460 \, a^{3} b^{3} e^{9}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{4} e^{5} - 38 \, a b^{5} d^{3} e^{6} - 22933 \, a^{2} b^{4} d^{2} e^{7} - 58140 \, a^{3} b^{3} d e^{8} - 20995 \, a^{4} b^{2} e^{9}\right )} x^{5} + 7 \,{\left (40 \, b^{6} d^{5} e^{4} - 380 \, a b^{5} d^{4} e^{5} + 1615 \, a^{2} b^{4} d^{3} e^{6} + 342380 \, a^{3} b^{3} d^{2} e^{7} + 482885 \, a^{4} b^{2} d e^{8} + 92378 \, a^{5} b e^{9}\right )} x^{4} -{\left (320 \, b^{6} d^{6} e^{3} - 3040 \, a b^{5} d^{5} e^{4} + 12920 \, a^{2} b^{4} d^{4} e^{5} - 32300 \, a^{3} b^{3} d^{3} e^{6} - 2372435 \, a^{4} b^{2} d^{2} e^{7} - 1755182 \, a^{5} b d e^{8} - 138567 \, a^{6} e^{9}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{7} e^{2} - 1216 \, a b^{5} d^{6} e^{3} + 5168 \, a^{2} b^{4} d^{5} e^{4} - 12920 \, a^{3} b^{3} d^{4} e^{5} + 20995 \, a^{4} b^{2} d^{3} e^{6} + 461890 \, a^{5} b d^{2} e^{7} + 138567 \, a^{6} d e^{8}\right )} x^{2} -{\left (512 \, b^{6} d^{8} e - 4864 \, a b^{5} d^{7} e^{2} + 20672 \, a^{2} b^{4} d^{6} e^{3} - 51680 \, a^{3} b^{3} d^{5} e^{4} + 83980 \, a^{4} b^{2} d^{4} e^{5} - 92378 \, a^{5} b d^{3} e^{6} - 415701 \, a^{6} d^{2} e^{7}\right )} x\right )} \sqrt{e x + d}}{969969 \, e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.1829, size = 1671, normalized size = 9.03 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.242442, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(e*x + d)^(5/2),x, algorithm="giac")
[Out]