3.1044 \(\int (2-5 x) x^{5/2} \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=256 \[ -\frac{4660 \sqrt{x} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac{8 \sqrt{x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac{8 \sqrt{x} (205407 x+190465) \sqrt{3 x^2+5 x+2}}{10945935}-\frac{497824 \sqrt{x} (3 x+2)}{32837805 \sqrt{3 x^2+5 x+2}}-\frac{61736 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2189187 \sqrt{3 x^2+5 x+2}}+\frac{497824 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{32837805 \sqrt{3 x^2+5 x+2}}-\frac{2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{136}{351} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2} \]

[Out]

(-497824*Sqrt[x]*(2 + 3*x))/(32837805*Sqrt[2 + 5*x + 3*x^2]) - (8*Sqrt[x]*(19046
5 + 205407*x)*Sqrt[2 + 5*x + 3*x^2])/10945935 + (8*Sqrt[x]*(27010 + 32921*x)*(2
+ 5*x + 3*x^2)^(3/2))/243243 - (4660*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/11583 + (1
36*x^(3/2)*(2 + 5*x + 3*x^2)^(5/2))/351 - (2*x^(5/2)*(2 + 5*x + 3*x^2)^(5/2))/9
+ (497824*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/
2])/(32837805*Sqrt[2 + 5*x + 3*x^2]) - (61736*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1
+ x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2189187*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.46721, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{4660 \sqrt{x} \left (3 x^2+5 x+2\right )^{5/2}}{11583}+\frac{8 \sqrt{x} (32921 x+27010) \left (3 x^2+5 x+2\right )^{3/2}}{243243}-\frac{8 \sqrt{x} (205407 x+190465) \sqrt{3 x^2+5 x+2}}{10945935}-\frac{497824 \sqrt{x} (3 x+2)}{32837805 \sqrt{3 x^2+5 x+2}}-\frac{61736 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2189187 \sqrt{3 x^2+5 x+2}}+\frac{497824 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{32837805 \sqrt{3 x^2+5 x+2}}-\frac{2}{9} x^{5/2} \left (3 x^2+5 x+2\right )^{5/2}+\frac{136}{351} x^{3/2} \left (3 x^2+5 x+2\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(-497824*Sqrt[x]*(2 + 3*x))/(32837805*Sqrt[2 + 5*x + 3*x^2]) - (8*Sqrt[x]*(19046
5 + 205407*x)*Sqrt[2 + 5*x + 3*x^2])/10945935 + (8*Sqrt[x]*(27010 + 32921*x)*(2
+ 5*x + 3*x^2)^(3/2))/243243 - (4660*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/11583 + (1
36*x^(3/2)*(2 + 5*x + 3*x^2)^(5/2))/351 - (2*x^(5/2)*(2 + 5*x + 3*x^2)^(5/2))/9
+ (497824*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/
2])/(32837805*Sqrt[2 + 5*x + 3*x^2]) - (61736*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1
+ x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2189187*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 49.7056, size = 245, normalized size = 0.96 \[ - \frac{2 x^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{9} + \frac{136 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{351} - \frac{248912 \sqrt{x} \left (6 x + 4\right )}{32837805 \sqrt{3 x^{2} + 5 x + 2}} + \frac{32 \sqrt{x} \left (\frac{1481445 x}{4} + \frac{607725}{2}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{10945935} - \frac{64 \sqrt{x} \left (\frac{3081105 x}{8} + \frac{2856975}{8}\right ) \sqrt{3 x^{2} + 5 x + 2}}{164189025} - \frac{4660 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{11583} + \frac{124456 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{32837805 \sqrt{3 x^{2} + 5 x + 2}} - \frac{15434 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2189187 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(3/2),x)

[Out]

-2*x**(5/2)*(3*x**2 + 5*x + 2)**(5/2)/9 + 136*x**(3/2)*(3*x**2 + 5*x + 2)**(5/2)
/351 - 248912*sqrt(x)*(6*x + 4)/(32837805*sqrt(3*x**2 + 5*x + 2)) + 32*sqrt(x)*(
1481445*x/4 + 607725/2)*(3*x**2 + 5*x + 2)**(3/2)/10945935 - 64*sqrt(x)*(3081105
*x/8 + 2856975/8)*sqrt(3*x**2 + 5*x + 2)/164189025 - 4660*sqrt(x)*(3*x**2 + 5*x
+ 2)**(5/2)/11583 + 124456*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqr
t(x)), -1/2)/(32837805*sqrt(3*x**2 + 5*x + 2)) - 15434*sqrt((6*x + 4)/(x + 1))*(
4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(2189187*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.304509, size = 183, normalized size = 0.71 \[ \frac{-497824 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (214108 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )[phone] x^9+320800095 x^8+337486905 x^7+69664455 x^6-83323080 x^5-37601118 x^4+91620 x^3-273876 x^2+318520 x+497824\right )}{32837805 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((-497824*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[
Sqrt[2/3]/Sqrt[x]], 3/2] - 2*(497824 + 318520*x - 273876*x^2 + 91620*x^3 - 37601
118*x^4 - 83323080*x^5 + 69664455*x^6 + 337486905*x^7 + 320800095*x^8 + 98513415
*x^9 + (214108*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*Arc
Sinh[Sqrt[2/3]/Sqrt[x]], 3/2]))/(32837805*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.029, size = 148, normalized size = 0.6 \[ -{\frac{2}{98513415} \left ( 295540245\,{x}^{9}[phone]\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}-249969240\,{x}^{5}+89652\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +124456\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -112803354\,{x}^{4}+274860\,{x}^{3}-3061836\,{x}^{2}-2778120\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/98513415/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(295540245*x^9+962400285*x^8+1012460715*
x^7+208993365*x^6-249969240*x^5+89652*(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))+124456*(6*x+4)^(1/2)*(3+3*x)
^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-1128033
54*x^4+274860*x^3-3061836*x^2-2778120*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (15 \, x^{5} + 19 \, x^{4} - 4 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(15*x^5 + 19*x^4 - 4*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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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