3.1035 \(\int (2-5 x) x^{5/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=228 \[ -\frac{4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{6237}+\frac{8 (74313 x+57860) \sqrt{3 x^2+5 x+2} \sqrt{x}}{280665}-\frac{261784 (3 x+2) \sqrt{x}}{841995 \sqrt{3 x^2+5 x+2}}-\frac{13016 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{56133 \sqrt{3 x^2+5 x+2}}+\frac{261784 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{3 x^2+5 x+2}}-\frac{10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac{532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2} \]

[Out]

(-261784*Sqrt[x]*(2 + 3*x))/(841995*Sqrt[2 + 5*x + 3*x^2]) + (8*Sqrt[x]*(57860 +
74313*x)*Sqrt[2 + 5*x + 3*x^2])/280665 - (4420*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))
/6237 + (532*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/891 - (10*x^(5/2)*(2 + 5*x + 3*x^2
)^(3/2))/33 + (261784*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[S
qrt[x]], -1/2])/(841995*Sqrt[2 + 5*x + 3*x^2]) - (13016*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(56133*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.410852, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{4420 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{6237}+\frac{8 (74313 x+57860) \sqrt{3 x^2+5 x+2} \sqrt{x}}{280665}-\frac{261784 (3 x+2) \sqrt{x}}{841995 \sqrt{3 x^2+5 x+2}}-\frac{13016 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{56133 \sqrt{3 x^2+5 x+2}}+\frac{261784 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{841995 \sqrt{3 x^2+5 x+2}}-\frac{10}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}+\frac{532}{891} \left (3 x^2+5 x+2\right )^{3/2} x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(2 - 5*x)*x^(5/2)*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(-261784*Sqrt[x]*(2 + 3*x))/(841995*Sqrt[2 + 5*x + 3*x^2]) + (8*Sqrt[x]*(57860 +
74313*x)*Sqrt[2 + 5*x + 3*x^2])/280665 - (4420*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))
/6237 + (532*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/891 - (10*x^(5/2)*(2 + 5*x + 3*x^2
)^(3/2))/33 + (261784*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[S
qrt[x]], -1/2])/(841995*Sqrt[2 + 5*x + 3*x^2]) - (13016*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(56133*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 42.6801, size = 212, normalized size = 0.93 \[ - \frac{10 x^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{33} + \frac{532 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{891} - \frac{130892 \sqrt{x} \left (6 x + 4\right )}{841995 \sqrt{3 x^{2} + 5 x + 2}} + \frac{32 \sqrt{x} \left (\frac{222939 x}{4} + 43395\right ) \sqrt{3 x^{2} + 5 x + 2}}{841995} - \frac{4420 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{6237} + \frac{65446 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{841995 \sqrt{3 x^{2} + 5 x + 2}} - \frac{3254 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{56133 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-5*x)*x**(5/2)*(3*x**2+5*x+2)**(1/2),x)

[Out]

-10*x**(5/2)*(3*x**2 + 5*x + 2)**(3/2)/33 + 532*x**(3/2)*(3*x**2 + 5*x + 2)**(3/
2)/891 - 130892*sqrt(x)*(6*x + 4)/(841995*sqrt(3*x**2 + 5*x + 2)) + 32*sqrt(x)*(
222939*x/4 + 43395)*sqrt(3*x**2 + 5*x + 2)/841995 - 4420*sqrt(x)*(3*x**2 + 5*x +
2)**(3/2)/6237 + 65446*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x
)), -1/2)/(841995*sqrt(3*x**2 + 5*x + 2)) - 3254*sqrt((6*x + 4)/(x + 1))*(4*x +
4)*elliptic_f(atan(sqrt(x)), -1/2)/(56133*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.242404, size = 170, normalized size = 0.75 \[ \frac{66544 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-261784 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2296350 x^7-3129840 x^6+271350 x^5+947916 x^4+39780 x^3-198168 x^2-918440 x-523568}{841995 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - 5*x)*x^(5/2)*Sqrt[2 + 5*x + 3*x^2],x]

[Out]

(-523568 - 918440*x - 198168*x^2 + 39780*x^3 + 947916*x^4 + 271350*x^5 - 3129840
*x^6 - 2296350*x^7 - (261784*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*E
llipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (66544*I)*Sqrt[2]*Sqrt[1 + x^(-1)]
*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(841995*Sqr
t[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.028, size = 138, normalized size = 0.6 \[{\frac{2}{2525985} \left ( -3444525\,{x}^{7}-4694760\,{x}^{6}+407025\,{x}^{5}+98718\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -65446\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +1421874\,{x}^{4}+59670\,{x}^{3}+880776\,{x}^{2}+585720\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*x^(5/2)*(3*x^2+5*x+2)^(1/2),x)

[Out]

2/2525985/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(-3444525*x^7-4694760*x^6+407025*x^5+98718
*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticF(1/2*(6*x+4)^(1
/2),I*2^(1/2))-65446*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Elli
pticE(1/2*(6*x+4)^(1/2),I*2^(1/2))+1421874*x^4+59670*x^3+880776*x^2+585720*x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} x^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2),x, algorithm="maxima")

[Out]

-integrate(sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (5 \, x^{3} - 2 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2),x, algorithm="fricas")

[Out]

integral(-(5*x^3 - 2*x^2)*sqrt(3*x^2 + 5*x + 2)*sqrt(x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2-5*x)*x**(5/2)*(3*x**2+5*x+2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )} x^{\frac{5}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2),x, algorithm="giac")

[Out]

integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)*x^(5/2), x)