3.1074 \(\int \frac{(2-5 x) x^{9/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{7540}{81} \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{17512 (3 x+2) \sqrt{x}}{243 \sqrt{3 x^2+5 x+2}}-\frac{7540 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}+\frac{17512 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (645 x+536) x^{3/2}}{9 \sqrt{3 x^2+5 x+2}} \]

[Out]

(2*x^(7/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (17512*Sqrt[x]*(2 + 3*x))/
(243*Sqrt[2 + 5*x + 3*x^2]) - (4*x^(3/2)*(536 + 645*x))/(9*Sqrt[2 + 5*x + 3*x^2]
) + (7540*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/81 + (17512*Sqrt[2]*(1 + x)*Sqrt[(2 + 3
*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(243*Sqrt[2 + 5*x + 3*x^2]) - (75
40*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(81
*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.366505, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{7540}{81} \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{17512 (3 x+2) \sqrt{x}}{243 \sqrt{3 x^2+5 x+2}}-\frac{7540 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}+\frac{17512 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{243 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{7/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (645 x+536) x^{3/2}}{9 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((2 - 5*x)*x^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(2*x^(7/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (17512*Sqrt[x]*(2 + 3*x))/
(243*Sqrt[2 + 5*x + 3*x^2]) - (4*x^(3/2)*(536 + 645*x))/(9*Sqrt[2 + 5*x + 3*x^2]
) + (7540*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/81 + (17512*Sqrt[2]*(1 + x)*Sqrt[(2 + 3
*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(243*Sqrt[2 + 5*x + 3*x^2]) - (75
40*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(81
*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 39.298, size = 194, normalized size = 0.92 \[ \frac{2 x^{\frac{7}{2}} \left (95 x + 74\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{4 x^{\frac{3}{2}} \left (1935 x + 1608\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{8756 \sqrt{x} \left (6 x + 4\right )}{243 \sqrt{3 x^{2} + 5 x + 2}} + \frac{7540 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}{81} + \frac{4378 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{243 \sqrt{3 x^{2} + 5 x + 2}} - \frac{1885 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-5*x)*x**(9/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

2*x**(7/2)*(95*x + 74)/(9*(3*x**2 + 5*x + 2)**(3/2)) - 4*x**(3/2)*(1935*x + 1608
)/(27*sqrt(3*x**2 + 5*x + 2)) - 8756*sqrt(x)*(6*x + 4)/(243*sqrt(3*x**2 + 5*x +
2)) + 7540*sqrt(x)*sqrt(3*x**2 + 5*x + 2)/81 + 4378*sqrt((6*x + 4)/(x + 1))*(4*x
+ 4)*elliptic_e(atan(sqrt(x)), -1/2)/(243*sqrt(3*x**2 + 5*x + 2)) - 1885*sqrt((
6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(81*sqrt(3*x**2 + 5*
x + 2))

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Mathematica [C]  time = 0.402984, size = 177, normalized size = 0.84 \[ \frac{-5108 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-17512 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (135 x^5-1512 x^4+58590 x^3+155660 x^2+129880 x+35024\right )}{243 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 - 5*x)*x^(9/2))/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(35024 + 129880*x + 155660*x^2 + 58590*x^3 - 1512*x^4 + 135*x^5) - (17512*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] - (5108*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])/(243*Sqrt[x]*(2 + 5*x + 3*x^2
)^(3/2))

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Maple [A]  time = 0.034, size = 320, normalized size = 1.5 \[{\frac{2}{729\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 5472\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-13134\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+9120\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-21890\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+3648\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -8756\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -405\,{x}^{5}+240948\,{x}^{4}+612270\,{x}^{3}+504936\,{x}^{2}+135720\,x \right ){\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*x^(9/2)/(3*x^2+5*x+2)^(5/2),x)

[Out]

2/729*(3*x^2+5*x+2)^(1/2)*(5472*(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^2-13134*(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+9120*(
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-21890*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Elli
pticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*x+3648*(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))-8756*(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))-405*x
^5+240948*x^4+612270*x^3+504936*x^2+135720*x)/x^(1/2)/(2+3*x)^2/(1+x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (5 \, x - 2\right )} x^{\frac{9}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="maxima")

[Out]

-integrate((5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="fricas")

[Out]

integral(-(5*x^5 - 2*x^4)*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(-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**(9/2)/(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. \[ \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{9}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="giac")

[Out]

integrate(-(5*x - 2)*x^(9/2)/(3*x^2 + 5*x + 2)^(5/2), x)