3.1071 \(\int \frac{2-5 x}{x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=224 \[ \frac{2693 \sqrt{x} (3 x+2)}{30 \sqrt{3 x^2+5 x+2}}-\frac{2693 \sqrt{3 x^2+5 x+2}}{30 \sqrt{x}}+\frac{157 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{2693 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{157 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}}-\frac{191 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}+\frac{2 (45 x+38)}{x^{5/2} \sqrt{3 x^2+5 x+2}} \]

[Out]

(2693*Sqrt[x]*(2 + 3*x))/(30*Sqrt[2 + 5*x + 3*x^2]) + (2*(38 + 45*x))/(x^(5/2)*S
qrt[2 + 5*x + 3*x^2]) - (191*Sqrt[2 + 5*x + 3*x^2])/(5*x^(5/2)) + (157*Sqrt[2 +
5*x + 3*x^2])/(3*x^(3/2)) - (2693*Sqrt[2 + 5*x + 3*x^2])/(30*Sqrt[x]) - (2693*(1
+ x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(15*Sqrt[2]*Sqrt
[2 + 5*x + 3*x^2]) + (157*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[
x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.388695, antiderivative size = 224, 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 \[ \frac{2693 \sqrt{x} (3 x+2)}{30 \sqrt{3 x^2+5 x+2}}-\frac{2693 \sqrt{3 x^2+5 x+2}}{30 \sqrt{x}}+\frac{157 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{2693 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{15 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{157 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}}-\frac{191 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}}+\frac{2 (45 x+38)}{x^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2693*Sqrt[x]*(2 + 3*x))/(30*Sqrt[2 + 5*x + 3*x^2]) + (2*(38 + 45*x))/(x^(5/2)*S
qrt[2 + 5*x + 3*x^2]) - (191*Sqrt[2 + 5*x + 3*x^2])/(5*x^(5/2)) + (157*Sqrt[2 +
5*x + 3*x^2])/(3*x^(3/2)) - (2693*Sqrt[2 + 5*x + 3*x^2])/(30*Sqrt[x]) - (2693*(1
+ x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(15*Sqrt[2]*Sqrt
[2 + 5*x + 3*x^2]) + (157*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[
x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 42.2226, size = 207, normalized size = 0.92 \[ \frac{2693 \sqrt{x} \left (6 x + 4\right )}{60 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2693 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{120 \sqrt{3 x^{2} + 5 x + 2}} + \frac{157 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2693 \sqrt{3 x^{2} + 5 x + 2}}{30 \sqrt{x}} + \frac{157 \sqrt{3 x^{2} + 5 x + 2}}{3 x^{\frac{3}{2}}} + \frac{90 x + 76}{x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}} - \frac{191 \sqrt{3 x^{2} + 5 x + 2}}{5 x^{\frac{5}{2}}} \]

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)**(3/2),x)

[Out]

2693*sqrt(x)*(6*x + 4)/(60*sqrt(3*x**2 + 5*x + 2)) - 2693*sqrt((6*x + 4)/(x + 1)
)*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(120*sqrt(3*x**2 + 5*x + 2)) + 157*s
qrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(8*sqrt(3*x**2
+ 5*x + 2)) - 2693*sqrt(3*x**2 + 5*x + 2)/(30*sqrt(x)) + 157*sqrt(3*x**2 + 5*x +
2)/(3*x**(3/2)) + (90*x + 76)/(x**(5/2)*sqrt(3*x**2 + 5*x + 2)) - 191*sqrt(3*x*
*2 + 5*x + 2)/(5*x**(5/2))

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Mathematica [C]  time = 0.236756, size = 150, normalized size = 0.67 \[ \frac{-338 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+2693 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+4710 x^3+4412 x^2+110 x-12}{30 x^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-12 + 110*x + 4412*x^2 + 4710*x^3 + (2693*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 +
2/x]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (338*I)*Sqrt[2]*Sqrt
[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])
/(30*x^(5/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.031, size = 130, normalized size = 0.6 \[ -{\frac{1}{180} \left ( 3369\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-2693\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+48474\,{x}^{4}+52530\,{x}^{3}+5844\,{x}^{2}-660\,x+72 \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/180*(3369*(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^2-2693*(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+48474*x^4+52530*x^3+5844*x
^2-660*x+72)/x^(5/2)/(3*x^2+5*x+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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