3.1082 \(\int \frac{2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=256 \[ -\frac{9521 \sqrt{x} (3 x+2)}{30 \sqrt{3 x^2+5 x+2}}+\frac{9521 \sqrt{3 x^2+5 x+2}}{30 \sqrt{x}}-\frac{1733 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{9521 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{1733 \sqrt{3 x^2+5 x+2}}{6 x^{3/2}}+\frac{1252 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}-\frac{1965 x+1541}{3 x^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(2*(38 + 45*x))/(3*x^(5/2)*(2 + 5*x + 3*x^2)^(3/2)) - (9521*Sqrt[x]*(2 + 3*x))/(
30*Sqrt[2 + 5*x + 3*x^2]) - (1541 + 1965*x)/(3*x^(5/2)*Sqrt[2 + 5*x + 3*x^2]) +
(1252*Sqrt[2 + 5*x + 3*x^2])/(5*x^(5/2)) - (1733*Sqrt[2 + 5*x + 3*x^2])/(6*x^(3/
2)) + (9521*Sqrt[2 + 5*x + 3*x^2])/(30*Sqrt[x]) + (9521*(1 + x)*Sqrt[(2 + 3*x)/(
1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(15*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2]) - (
1733*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2*Sqrt[2
]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.460869, 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 \[ -\frac{9521 \sqrt{x} (3 x+2)}{30 \sqrt{3 x^2+5 x+2}}+\frac{9521 \sqrt{3 x^2+5 x+2}}{30 \sqrt{x}}-\frac{1733 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{9521 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{1733 \sqrt{3 x^2+5 x+2}}{6 x^{3/2}}+\frac{1252 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}-\frac{1965 x+1541}{3 x^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - 5*x)/(x^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(2*(38 + 45*x))/(3*x^(5/2)*(2 + 5*x + 3*x^2)^(3/2)) - (9521*Sqrt[x]*(2 + 3*x))/(
30*Sqrt[2 + 5*x + 3*x^2]) - (1541 + 1965*x)/(3*x^(5/2)*Sqrt[2 + 5*x + 3*x^2]) +
(1252*Sqrt[2 + 5*x + 3*x^2])/(5*x^(5/2)) - (1733*Sqrt[2 + 5*x + 3*x^2])/(6*x^(3/
2)) + (9521*Sqrt[2 + 5*x + 3*x^2])/(30*Sqrt[x]) + (9521*(1 + x)*Sqrt[(2 + 3*x)/(
1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(15*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2]) - (
1733*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2*Sqrt[2
]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 48.9175, size = 235, normalized size = 0.92 \[ - \frac{9521 \sqrt{x} \left (6 x + 4\right )}{60 \sqrt{3 x^{2} + 5 x + 2}} + \frac{9521 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{120 \sqrt{3 x^{2} + 5 x + 2}} - \frac{1733 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{16 \sqrt{3 x^{2} + 5 x + 2}} + \frac{9521 \sqrt{3 x^{2} + 5 x + 2}}{30 \sqrt{x}} - \frac{1733 \sqrt{3 x^{2} + 5 x + 2}}{6 x^{\frac{3}{2}}} + \frac{90 x + 76}{3 x^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{1965 x + 1541}{3 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}} + \frac{1252 \sqrt{3 x^{2} + 5 x + 2}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-5*x)/x**(7/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

-9521*sqrt(x)*(6*x + 4)/(60*sqrt(3*x**2 + 5*x + 2)) + 9521*sqrt((6*x + 4)/(x + 1
))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(120*sqrt(3*x**2 + 5*x + 2)) - 1733
*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(16*sqrt(3*x*
*2 + 5*x + 2)) + 9521*sqrt(3*x**2 + 5*x + 2)/(30*sqrt(x)) - 1733*sqrt(3*x**2 + 5
*x + 2)/(6*x**(3/2)) + (90*x + 76)/(3*x**(5/2)*(3*x**2 + 5*x + 2)**(3/2)) - (196
5*x + 1541)/(3*x**(5/2)*sqrt(3*x**2 + 5*x + 2)) + 1252*sqrt(3*x**2 + 5*x + 2)/(5
*x**(5/2))

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Mathematica [C]  time = 0.428644, size = 177, normalized size = 0.69 \[ \frac{-6953 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{7/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-19042 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (77985 x^5+192342 x^4+154195 x^3+39836 x^2-130 x+12\right )}{60 x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - 5*x)/(x^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(-2*(12 - 130*x + 39836*x^2 + 154195*x^3 + 192342*x^4 + 77985*x^5) - (19042*I)*S
qrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(7/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/
3]/Sqrt[x]], 3/2] - (6953*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(7/2)*(2 + 5*x + 3*x^
2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(60*x^(5/2)*(2 + 5*x + 3*x^2)^(
3/2))

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Maple [A]  time = 0.04, size = 336, normalized size = 1.3 \[{\frac{1}{ \left ( 180+180\,x \right ) \left ( 2+3\,x \right ) } \left ( 7704\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{4}-28563\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{4}+12840\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}-47605\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}+5136\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-19042\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+514134\,{x}^{6}+1245870\,{x}^{5}+959610\,{x}^{4}+217350\,{x}^{3}-10512\,{x}^{2}+780\,x-72 \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(5/2),x)

[Out]

1/180/x^(5/2)*(7704*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Ellip
ticF(1/2*(6*x+4)^(1/2),I*2^(1/2))*x^4-28563*(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^4+12840*(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^3-47605*(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^3+5136*(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^2-19042*(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+514
134*x^6+1245870*x^5+959610*x^4+217350*x^3-10512*x^2+780*x-72)/(1+x)/(2+3*x)/(3*x
^2+5*x+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)),x, algorithm="maxima")

[Out]

-integrate((5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{{\left (9 \, x^{7} + 30 \, x^{6} + 37 \, x^{5} + 20 \, x^{4} + 4 \, x^{3}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)),x, algorithm="fricas")

[Out]

integral(-(5*x - 2)/((9*x^7 + 30*x^6 + 37*x^5 + 20*x^4 + 4*x^3)*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**(7/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)),x, algorithm="giac")

[Out]

integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(7/2)), x)