3.1063 \(\int \frac{2-5 x}{x^{7/2} \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=196 \[ \frac{66 \sqrt{x} (3 x+2)}{5 \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}+\frac{9 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5 \sqrt{3 x^2+5 x+2}}+\frac{3 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}} \]

[Out]

(66*Sqrt[x]*(2 + 3*x))/(5*Sqrt[2 + 5*x + 3*x^2]) - (2*Sqrt[2 + 5*x + 3*x^2])/(5*
x^(5/2)) + (3*Sqrt[2 + 5*x + 3*x^2])/x^(3/2) - (66*Sqrt[2 + 5*x + 3*x^2])/(5*Sqr
t[x]) - (66*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -
1/2])/(5*Sqrt[2 + 5*x + 3*x^2]) + (9*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[A
rcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.34382, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{66 \sqrt{x} (3 x+2)}{5 \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{3 x^2+5 x+2}}{5 \sqrt{x}}+\frac{9 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{66 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5 \sqrt{3 x^2+5 x+2}}+\frac{3 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{5 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(66*Sqrt[x]*(2 + 3*x))/(5*Sqrt[2 + 5*x + 3*x^2]) - (2*Sqrt[2 + 5*x + 3*x^2])/(5*
x^(5/2)) + (3*Sqrt[2 + 5*x + 3*x^2])/x^(3/2) - (66*Sqrt[2 + 5*x + 3*x^2])/(5*Sqr
t[x]) - (66*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -
1/2])/(5*Sqrt[2 + 5*x + 3*x^2]) + (9*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[A
rcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 36.4737, size = 182, normalized size = 0.93 \[ \frac{33 \sqrt{x} \left (6 x + 4\right )}{5 \sqrt{3 x^{2} + 5 x + 2}} - \frac{33 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{10 \sqrt{3 x^{2} + 5 x + 2}} + \frac{9 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} - \frac{66 \sqrt{3 x^{2} + 5 x + 2}}{5 \sqrt{x}} + \frac{3 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}} - \frac{2 \sqrt{3 x^{2} + 5 x + 2}}{5 x^{\frac{5}{2}}} \]

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)**(1/2),x)

[Out]

33*sqrt(x)*(6*x + 4)/(5*sqrt(3*x**2 + 5*x + 2)) - 33*sqrt((6*x + 4)/(x + 1))*(4*
x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(10*sqrt(3*x**2 + 5*x + 2)) + 9*sqrt((6*x
+ 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(8*sqrt(3*x**2 + 5*x +
2)) - 66*sqrt(3*x**2 + 5*x + 2)/(5*sqrt(x)) + 3*sqrt(3*x**2 + 5*x + 2)/x**(3/2)
- 2*sqrt(3*x**2 + 5*x + 2)/(5*x**(5/2))

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Mathematica [C]  time = 0.227298, size = 150, normalized size = 0.77 \[ \frac{-87 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+132 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{7/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+90 x^3+138 x^2+40 x-8}{10 x^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-8 + 40*x + 138*x^2 + 90*x^3 + (132*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x
^(7/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (87*I)*Sqrt[2]*Sqrt[1 + x^
(-1)]*Sqrt[3 + 2/x]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(10*x^
(5/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.028, size = 130, normalized size = 0.7 \[ -{\frac{1}{10} \left ( 51\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-22\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+396\,{x}^{4}+570\,{x}^{3}+126\,{x}^{2}-40\,x+8 \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)/x^(7/2)/(3*x^2+5*x+2)^(1/2),x)

[Out]

-1/10*(51*(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^2-22*(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+396*x^4+570*x^3+126*x^2-40*x+8)
/(3*x^2+5*x+2)^(1/2)/x^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2-5*x)/x**(7/2)/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-2/(x**(7/2)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(5/(x**(5/2)*sqrt(3
*x**2 + 5*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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