3.1079 \(\int \frac{2-5 x}{\sqrt{x} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=185 \[ \frac{715 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}-\frac{5 \sqrt{x} (429 x+361)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{295 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{715 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

[Out]

(2*Sqrt[x]*(38 + 45*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (715*Sqrt[x]*(2 + 3*x))/(3
*Sqrt[2 + 5*x + 3*x^2]) - (5*Sqrt[x]*(361 + 429*x))/(3*Sqrt[2 + 5*x + 3*x^2]) -
(715*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(
3*Sqrt[2 + 5*x + 3*x^2]) + (295*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elliptic
F[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi [A]  time = 0.287779, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{715 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}-\frac{5 \sqrt{x} (429 x+361)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{295 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{715 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(38 + 45*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (715*Sqrt[x]*(2 + 3*x))/(3
*Sqrt[2 + 5*x + 3*x^2]) - (5*Sqrt[x]*(361 + 429*x))/(3*Sqrt[2 + 5*x + 3*x^2]) -
(715*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(
3*Sqrt[2 + 5*x + 3*x^2]) + (295*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elliptic
F[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi in Sympy [A]  time = 32.0141, size = 168, normalized size = 0.91 \[ \frac{715 \sqrt{x} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\sqrt{x} \left (90 x + 76\right )}{3 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{\sqrt{x} \left (2145 x + 1805\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{715 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{12 \sqrt{3 x^{2} + 5 x + 2}} + \frac{295 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

715*sqrt(x)*(6*x + 4)/(6*sqrt(3*x**2 + 5*x + 2)) + sqrt(x)*(90*x + 76)/(3*(3*x**
2 + 5*x + 2)**(3/2)) - sqrt(x)*(2145*x + 1805)/(3*sqrt(3*x**2 + 5*x + 2)) - 715*
sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(12*sqrt(3*x**
2 + 5*x + 2)) + 295*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 = 0.403299, size = 167, normalized size = 0.9 \[ \frac{170 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+715 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+2 \left (2655 x^3+6615 x^2+5383 x+1430\right )}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(1430 + 5383*x + 6615*x^2 + 2655*x^3) + (715*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x
^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (170*I)*
Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSinh[Sqrt[2
/3]/Sqrt[x]], 3/2])/(3*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.033, size = 315, normalized size = 1.7 \[ -{\frac{1}{18\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 1125\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-2145\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+1875\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-3575\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+750\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -1430\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +38610\,{x}^{4}+96840\,{x}^{3}+79350\,{x}^{2}+21204\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*(1125*(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^2-2145*(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+1875*(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-3575*(
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+750*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Ellipt
icF(1/2*(6*x+4)^(1/2),I*2^(1/2))-1430*(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))+38610*x^4+96840*x^3+79350*x^
2+21204*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. \[ -\int \frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} \sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x - 2)/((9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*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)/(3*x**2+5*x+2)**(5/2)/x**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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