3.1078 \(\int \frac{(2-5 x) \sqrt{x}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=179 \[ -\frac{198 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (297 x+250)}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{x} (37 x+30)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{245 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{198 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*Sqrt[x]*(30 + 37*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) - (198*Sqrt[x]*(2 + 3*x))/S
qrt[2 + 5*x + 3*x^2] + (2*Sqrt[x]*(250 + 297*x))/Sqrt[2 + 5*x + 3*x^2] + (198*Sq
rt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 +
5*x + 3*x^2] - (245*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sq
rt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi [A]  time = 0.291267, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{198 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (297 x+250)}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{x} (37 x+30)}{3 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{245 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{198 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[x]*(30 + 37*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) - (198*Sqrt[x]*(2 + 3*x))/S
qrt[2 + 5*x + 3*x^2] + (2*Sqrt[x]*(250 + 297*x))/Sqrt[2 + 5*x + 3*x^2] + (198*Sq
rt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 +
5*x + 3*x^2] - (245*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sq
rt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi in Sympy [A]  time = 32.4613, size = 170, normalized size = 0.95 \[ - \frac{99 \sqrt{x} \left (6 x + 4\right )}{\sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \sqrt{x} \left (37 x + 30\right )}{3 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{2 \sqrt{x} \left (891 x + 750\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{99 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{3 x^{2} + 5 x + 2}} - \frac{245 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-99*sqrt(x)*(6*x + 4)/sqrt(3*x**2 + 5*x + 2) - 2*sqrt(x)*(37*x + 30)/(3*(3*x**2
+ 5*x + 2)**(3/2)) + 2*sqrt(x)*(891*x + 750)/(3*sqrt(3*x**2 + 5*x + 2)) + 99*sqr
t((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(2*sqrt(3*x**2 +
5*x + 2)) - 245*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2
)/(4*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.526196, size = 165, normalized size = 0.92 \[ -\frac{47 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} x F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{198 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} x E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{2 \left (2205 x^3+5494 x^2+4470 x+1188\right )}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1188 + 4470*x + 5494*x^2 + 2205*x^3))/(3*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2)) -
((198*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]],
3/2])/Sqrt[2 + 5*x + 3*x^2] - ((47*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x*EllipticF[I*
ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/Sqrt[2 + 5*x + 3*x^2]

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Maple [A]  time = 0.03, size = 315, normalized size = 1.8 \[{\frac{1}{3\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 156\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-297\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+260\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-495\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+104\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -198\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +5346\,{x}^{4}+13410\,{x}^{3}+10990\,{x}^{2}+2940\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(156*(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))*x^2-297*(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))*x^2+260*(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))*x-495*(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))*x+104*(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))-198*(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))+5346*x^4+13410*x^3+10990*x^2+2940*x
)*(3*x^2+5*x+2)^(1/2)/x^(1/2)/(2+3*x)^2/(1+x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (5 \, x - 2\right )} \sqrt{x}}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x - 2)*sqrt(x)/((9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*sqrt(3*x^2 + 5
*x + 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{2 \sqrt{x}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{5 x^{\frac{3}{2}}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-2*sqrt(x)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x
+ 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x
**2 + 5*x + 2)), x) - Integral(5*x**(3/2)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x*
*3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 +
5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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