3.1057 \(\int \frac{(2-5 x) x^{5/2}}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=200 \[ -\frac{412}{189} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{13688 (3 x+2) \sqrt{x}}{2835 \sqrt{3 x^2+5 x+2}}+\frac{412 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{189 \sqrt{3 x^2+5 x+2}}-\frac{13688 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2835 \sqrt{3 x^2+5 x+2}}-\frac{10}{21} \sqrt{3 x^2+5 x+2} x^{5/2}+\frac{128}{105} \sqrt{3 x^2+5 x+2} x^{3/2} \]

[Out]

(13688*Sqrt[x]*(2 + 3*x))/(2835*Sqrt[2 + 5*x + 3*x^2]) - (412*Sqrt[x]*Sqrt[2 + 5
*x + 3*x^2])/189 + (128*x^(3/2)*Sqrt[2 + 5*x + 3*x^2])/105 - (10*x^(5/2)*Sqrt[2
+ 5*x + 3*x^2])/21 - (13688*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[Ar
cTan[Sqrt[x]], -1/2])/(2835*Sqrt[2 + 5*x + 3*x^2]) + (412*Sqrt[2]*(1 + x)*Sqrt[(
2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(189*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.361984, antiderivative size = 200, 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 \[ -\frac{412}{189} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{13688 (3 x+2) \sqrt{x}}{2835 \sqrt{3 x^2+5 x+2}}+\frac{412 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{189 \sqrt{3 x^2+5 x+2}}-\frac{13688 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2835 \sqrt{3 x^2+5 x+2}}-\frac{10}{21} \sqrt{3 x^2+5 x+2} x^{5/2}+\frac{128}{105} \sqrt{3 x^2+5 x+2} x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(13688*Sqrt[x]*(2 + 3*x))/(2835*Sqrt[2 + 5*x + 3*x^2]) - (412*Sqrt[x]*Sqrt[2 + 5
*x + 3*x^2])/189 + (128*x^(3/2)*Sqrt[2 + 5*x + 3*x^2])/105 - (10*x^(5/2)*Sqrt[2
+ 5*x + 3*x^2])/21 - (13688*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[Ar
cTan[Sqrt[x]], -1/2])/(2835*Sqrt[2 + 5*x + 3*x^2]) + (412*Sqrt[2]*(1 + x)*Sqrt[(
2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(189*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 36.5883, size = 184, normalized size = 0.92 \[ - \frac{10 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}}{21} + \frac{128 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}{105} + \frac{6844 \sqrt{x} \left (6 x + 4\right )}{2835 \sqrt{3 x^{2} + 5 x + 2}} - \frac{412 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}{189} - \frac{3422 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2835 \sqrt{3 x^{2} + 5 x + 2}} + \frac{103 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{189 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*x**(5/2)*sqrt(3*x**2 + 5*x + 2)/21 + 128*x**(3/2)*sqrt(3*x**2 + 5*x + 2)/105
+ 6844*sqrt(x)*(6*x + 4)/(2835*sqrt(3*x**2 + 5*x + 2)) - 412*sqrt(x)*sqrt(3*x**
2 + 5*x + 2)/189 - 3422*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x
)), -1/2)/(2835*sqrt(3*x**2 + 5*x + 2)) + 103*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*
elliptic_f(atan(sqrt(x)), -1/2)/(189*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.231049, size = 160, normalized size = 0.8 \[ \frac{-7508 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+13688 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-4050 x^5+3618 x^4-3960 x^3+17076 x^2+56080 x+27376}{2835 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(27376 + 56080*x + 17076*x^2 - 3960*x^3 + 3618*x^4 - 4050*x^5 + (13688*I)*Sqrt[2
]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]],
3/2] - (7508*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcS
inh[Sqrt[2/3]/Sqrt[x]], 3/2])/(2835*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 128, normalized size = 0.6 \[ -{\frac{2}{8505} \left ( 6075\,{x}^{5}+7176\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -3422\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -5427\,{x}^{4}+5940\,{x}^{3}+35982\,{x}^{2}+18540\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/8505/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(6075*x^5+7176*(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))-3422*(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))-5427*x^4+5940*x^3+35982*x^2+18540*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x^3 - 2*x^2)*sqrt(x)/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**(5/2)/(3*x**2+5*x+2)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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