Optimal. Leaf size=75 \[ \frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3} \]
[Out]
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Rubi [A] time = 0.0844432, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}{3 e^3}-\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(a + b*x + c*x^2),x]
[Out]
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Rubi in Sympy [A] time = 14.444, size = 70, normalized size = 0.93 \[ \frac{2 c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.059073, size = 55, normalized size = 0.73 \[ \frac{2 (d+e x)^{3/2} \left (7 e (5 a e-2 b d+3 b e x)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(a + b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.006, size = 53, normalized size = 0.7 \[{\frac{30\,c{e}^{2}{x}^{2}+42\,b{e}^{2}x-24\,cdex+70\,a{e}^{2}-28\,bde+16\,c{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.677768, size = 80, normalized size = 1.07 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c - 21 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c d^{2} - b d e + a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205688, size = 113, normalized size = 1.51 \[ \frac{2 \,{\left (15 \, c e^{3} x^{3} + 8 \, c d^{3} - 14 \, b d^{2} e + 35 \, a d e^{2} + 3 \,{\left (c d e^{2} + 7 \, b e^{3}\right )} x^{2} -{\left (4 \, c d^{2} e - 7 \, b d e^{2} - 35 \, a e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.37174, size = 71, normalized size = 0.95 \[ \frac{2 \left (\frac{c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )}{3 e^{2}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)*(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.204736, size = 119, normalized size = 1.59 \[ \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b e^{\left (-1\right )} +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c e^{\left (-14\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*sqrt(e*x + d),x, algorithm="giac")
[Out]