Optimal. Leaf size=166 \[ \frac{2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac{4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5} \]
[Out]
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Rubi [A] time = 0.261563, antiderivative size = 166, 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 \[ \frac{2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac{4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac{4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 42.1725, size = 162, normalized size = 0.98 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{13}{2}} \left (b e - 2 c d\right )}{13 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{11}{2}} \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{11 e^{5}} + \frac{4 \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{9 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e^{2} - b d e + c d^{2}\right )^{2}}{7 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.245621, size = 173, normalized size = 1.04 \[ \frac{2 (d+e x)^{7/2} \left (65 e^2 \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-10 c e \left (3 b \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )-13 a e \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.01, size = 194, normalized size = 1.2 \[{\frac{6006\,{x}^{4}{c}^{2}{e}^{4}+13860\,bc{e}^{4}{x}^{3}-3696\,{x}^{3}{c}^{2}d{e}^{3}+16380\,{x}^{2}ac{e}^{4}+8190\,{b}^{2}{e}^{4}{x}^{2}-7560\,bcd{e}^{3}{x}^{2}+2016\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+20020\,ab{e}^{4}x-7280\,xacd{e}^{3}-3640\,{b}^{2}d{e}^{3}x+3360\,bc{d}^{2}{e}^{2}x-896\,x{c}^{2}{d}^{3}e+12870\,{a}^{2}{e}^{4}-5720\,abd{e}^{3}+2080\,ac{d}^{2}{e}^{2}+1040\,{b}^{2}{d}^{2}{e}^{2}-960\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x)
[Out]
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Maxima [A] time = 0.689552, size = 238, normalized size = 1.43 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 6930 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 4095 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 10010 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212622, size = 493, normalized size = 2.97 \[ \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e - 2860 \, a b d^{4} e^{3} + 6435 \, a^{2} d^{3} e^{4} + 520 \,{\left (b^{2} + 2 \, a c\right )} d^{5} e^{2} + 231 \,{\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \,{\left (b^{2} + 2 \, a c\right )} e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 286 \, a b e^{7} + 299 \,{\left (b^{2} + 2 \, a c\right )} d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 5434 \, a b d e^{6} - 1287 \, a^{2} e^{7} - 1469 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 7150 \, a b d^{2} e^{5} + 6435 \, a^{2} d e^{6} + 65 \,{\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{2} -{\left (64 \, c^{2} d^{6} e - 240 \, b c d^{5} e^{2} - 1430 \, a b d^{3} e^{4} - 19305 \, a^{2} d^{2} e^{5} + 260 \,{\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.4625, size = 1129, normalized size = 6.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.225322, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(e*x + d)^(5/2),x, algorithm="giac")
[Out]