3.1064 \(\int \frac{(2-5 x) x^{7/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=197 \[ 20 \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{24 (3 x+2) \sqrt{x}}{\sqrt{3 x^2+5 x+2}}-\frac{20 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{24 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{5/2}}{3 \sqrt{3 x^2+5 x+2}}-\frac{64}{3} \sqrt{3 x^2+5 x+2} x^{3/2} \]

[Out]

(-24*Sqrt[x]*(2 + 3*x))/Sqrt[2 + 5*x + 3*x^2] + (2*x^(5/2)*(74 + 95*x))/(3*Sqrt[
2 + 5*x + 3*x^2]) + 20*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2] - (64*x^(3/2)*Sqrt[2 + 5*x
+ 3*x^2])/3 + (24*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[
x]], -1/2])/Sqrt[2 + 5*x + 3*x^2] - (20*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*
EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi [A]  time = 0.359635, antiderivative size = 197, 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 \[ 20 \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{24 (3 x+2) \sqrt{x}}{\sqrt{3 x^2+5 x+2}}-\frac{20 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{24 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{5/2}}{3 \sqrt{3 x^2+5 x+2}}-\frac{64}{3} \sqrt{3 x^2+5 x+2} x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-24*Sqrt[x]*(2 + 3*x))/Sqrt[2 + 5*x + 3*x^2] + (2*x^(5/2)*(74 + 95*x))/(3*Sqrt[
2 + 5*x + 3*x^2]) + 20*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2] - (64*x^(3/2)*Sqrt[2 + 5*x
+ 3*x^2])/3 + (24*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[
x]], -1/2])/Sqrt[2 + 5*x + 3*x^2] - (20*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*
EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi in Sympy [A]  time = 37.5524, size = 182, normalized size = 0.92 \[ \frac{2 x^{\frac{5}{2}} \left (95 x + 74\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{64 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}{3} - \frac{12 \sqrt{x} \left (6 x + 4\right )}{\sqrt{3 x^{2} + 5 x + 2}} + 20 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2} + \frac{6 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{\sqrt{3 x^{2} + 5 x + 2}} - \frac{5 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{\sqrt{3 x^{2} + 5 x + 2}} \]

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]

2*x**(5/2)*(95*x + 74)/(3*sqrt(3*x**2 + 5*x + 2)) - 64*x**(3/2)*sqrt(3*x**2 + 5*
x + 2)/3 - 12*sqrt(x)*(6*x + 4)/sqrt(3*x**2 + 5*x + 2) + 20*sqrt(x)*sqrt(3*x**2
+ 5*x + 2) + 6*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)
/sqrt(3*x**2 + 5*x + 2) - 5*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sq
rt(x)), -1/2)/sqrt(3*x**2 + 5*x + 2)

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Mathematica [C]  time = 0.251875, size = 156, normalized size = 0.79 \[ \frac{12 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-72 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (x^4-4 x^3+22 x^2+120 x+72\right )}{3 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(72 + 120*x + 22*x^2 - 4*x^3 + x^4) - (72*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3
+ 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (12*I)*Sqrt[2]*Sq
rt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2
])/(3*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.065, size = 123, normalized size = 0.6 \[{\frac{2}{3} \left ( 8\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -6\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -{x}^{4}+4\,{x}^{3}+86\,{x}^{2}+60\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(8*(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))-6*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*El
lipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-x^4+4*x^3+86*x^2+60*x)/x^(1/2)/(3*x^2+5*x+2
)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x^4 - 2*x^3)*sqrt(x)/(3*x^2 + 5*x + 2)^(3/2), 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**(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. \[ \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{7}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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