Optimal. Leaf size=147 \[ \frac{2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac{4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5} \]
[Out]
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Rubi [A] time = 0.195676, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 (d+e x)^{7/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{7 e^5}+\frac{2 d^2 (d+e x)^{3/2} (c d-b e)^2}{3 e^5}-\frac{4 c (d+e x)^{9/2} (2 c d-b e)}{9 e^5}-\frac{4 d (d+e x)^{5/2} (c d-b e) (2 c d-b e)}{5 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 33.1211, size = 141, normalized size = 0.96 \[ \frac{2 c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{5}} + \frac{4 c \left (d + e x\right )^{\frac{9}{2}} \left (b e - 2 c d\right )}{9 e^{5}} + \frac{2 d^{2} \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )^{2}}{3 e^{5}} - \frac{4 d \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right ) \left (b e - c d\right )}{5 e^{5}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**2*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.109628, size = 124, normalized size = 0.84 \[ \frac{2 (d+e x)^{3/2} \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.009, size = 141, normalized size = 1. \[{\frac{630\,{c}^{2}{x}^{4}{e}^{4}+1540\,bc{e}^{4}{x}^{3}-560\,{c}^{2}d{e}^{3}{x}^{3}+990\,{b}^{2}{e}^{4}{x}^{2}-1320\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-792\,{b}^{2}d{e}^{3}x+1056\,bc{d}^{2}{e}^{2}x-384\,{c}^{2}{d}^{3}ex+528\,{b}^{2}{d}^{2}{e}^{2}-704\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^2*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.695546, size = 188, normalized size = 1.28 \[ \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} - 770 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 1386 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208373, size = 236, normalized size = 1.61 \[ \frac{2 \,{\left (315 \, c^{2} e^{5} x^{5} + 128 \, c^{2} d^{5} - 352 \, b c d^{4} e + 264 \, b^{2} d^{3} e^{2} + 35 \,{\left (c^{2} d e^{4} + 22 \, b c e^{5}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{2} e^{3} - 22 \, b c d e^{4} - 99 \, b^{2} e^{5}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{3} e^{2} - 44 \, b c d^{2} e^{3} + 33 \, b^{2} d e^{4}\right )} x^{2} - 4 \,{\left (16 \, c^{2} d^{4} e - 44 \, b c d^{3} e^{2} + 33 \, b^{2} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.87415, size = 173, normalized size = 1.18 \[ \frac{2 \left (\frac{c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (2 b c e - 4 c^{2} d\right )}{9 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**2*(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208891, size = 259, normalized size = 1.76 \[ \frac{2}{3465} \,{\left (33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} b^{2} e^{\left (-14\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} b c e^{\left (-27\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} c^{2} e^{\left (-44\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*sqrt(e*x + d),x, algorithm="giac")
[Out]