3.1072 \(\int \frac{(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=256 \[ \frac{211144 \sqrt{3 x^2+5 x+2} \sqrt{x}}{5103}-\frac{1521056 (3 x+2) \sqrt{x}}{76545 \sqrt{3 x^2+5 x+2}}-\frac{211144 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}+\frac{1521056 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{76545 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (1685 x+1484) x^{7/2}}{27 \sqrt{3 x^2+5 x+2}}+\frac{45820}{567} \sqrt{3 x^2+5 x+2} x^{5/2}-\frac{167336 \sqrt{3 x^2+5 x+2} x^{3/2}}{2835} \]

[Out]

(2*x^(11/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (1521056*Sqrt[x]*(2 + 3*x
))/(76545*Sqrt[2 + 5*x + 3*x^2]) - (4*x^(7/2)*(1484 + 1685*x))/(27*Sqrt[2 + 5*x
+ 3*x^2]) + (211144*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/5103 - (167336*x^(3/2)*Sqrt[2
+ 5*x + 3*x^2])/2835 + (45820*x^(5/2)*Sqrt[2 + 5*x + 3*x^2])/567 + (1521056*Sqr
t[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(76545*Sq
rt[2 + 5*x + 3*x^2]) - (211144*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF
[ArcTan[Sqrt[x]], -1/2])/(5103*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.487256, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{211144 \sqrt{3 x^2+5 x+2} \sqrt{x}}{5103}-\frac{1521056 (3 x+2) \sqrt{x}}{76545 \sqrt{3 x^2+5 x+2}}-\frac{211144 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{5103 \sqrt{3 x^2+5 x+2}}+\frac{1521056 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{76545 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{11/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}-\frac{4 (1685 x+1484) x^{7/2}}{27 \sqrt{3 x^2+5 x+2}}+\frac{45820}{567} \sqrt{3 x^2+5 x+2} x^{5/2}-\frac{167336 \sqrt{3 x^2+5 x+2} x^{3/2}}{2835} \]

Antiderivative was successfully verified.

[In]  Int[((2 - 5*x)*x^(13/2))/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(2*x^(11/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (1521056*Sqrt[x]*(2 + 3*x
))/(76545*Sqrt[2 + 5*x + 3*x^2]) - (4*x^(7/2)*(1484 + 1685*x))/(27*Sqrt[2 + 5*x
+ 3*x^2]) + (211144*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/5103 - (167336*x^(3/2)*Sqrt[2
+ 5*x + 3*x^2])/2835 + (45820*x^(5/2)*Sqrt[2 + 5*x + 3*x^2])/567 + (1521056*Sqr
t[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(76545*Sq
rt[2 + 5*x + 3*x^2]) - (211144*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF
[ArcTan[Sqrt[x]], -1/2])/(5103*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 50.1616, size = 238, normalized size = 0.93 \[ \frac{2 x^{\frac{11}{2}} \left (95 x + 74\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{4 x^{\frac{7}{2}} \left (1685 x + 1484\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} + \frac{45820 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}}{567} - \frac{167336 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}{2835} - \frac{760528 \sqrt{x} \left (6 x + 4\right )}{76545 \sqrt{3 x^{2} + 5 x + 2}} + \frac{211144 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}{5103} + \frac{380264 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{76545 \sqrt{3 x^{2} + 5 x + 2}} - \frac{52786 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{5103 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-5*x)*x**(13/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

2*x**(11/2)*(95*x + 74)/(9*(3*x**2 + 5*x + 2)**(3/2)) - 4*x**(7/2)*(1685*x + 148
4)/(27*sqrt(3*x**2 + 5*x + 2)) + 45820*x**(5/2)*sqrt(3*x**2 + 5*x + 2)/567 - 167
336*x**(3/2)*sqrt(3*x**2 + 5*x + 2)/2835 - 760528*sqrt(x)*(6*x + 4)/(76545*sqrt(
3*x**2 + 5*x + 2)) + 211144*sqrt(x)*sqrt(3*x**2 + 5*x + 2)/5103 + 380264*sqrt((6
*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(76545*sqrt(3*x**2 +
5*x + 2)) - 52786*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1
/2)/(5103*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.414158, size = 187, normalized size = 0.73 \[ \frac{-1646104 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-1521056 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (18225 x^7-70956 x^6+262710 x^5-2106756 x^4-2967300 x^3+5504080 x^2+8876240 x+3042112\right )}{76545 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 - 5*x)*x^(13/2))/(2 + 5*x + 3*x^2)^(5/2),x]

[Out]

(-2*(3042112 + 8876240*x + 5504080*x^2 - 2967300*x^3 - 2106756*x^4 + 262710*x^5
- 70956*x^6 + 18225*x^7) - (1521056*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] - (1646104*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])/(76545*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.069, size = 330, normalized size = 1.3 \[ -{\frac{2}{229635\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 1328364\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+1140792\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+2213940\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+1901320\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+54675\,{x}^{7}+885576\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +760528\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -212868\,{x}^{6}+788130\,{x}^{5}-26854524\,{x}^{4}-77349420\,{x}^{3}-67906368\,{x}^{2}-19002960\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*x^(13/2)/(3*x^2+5*x+2)^(5/2),x)

[Out]

-2/229635*(1328364*(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))*x^2+1140792*(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+2213940*(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+1901320*(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+54675*x^7+885576*(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))+760528*(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
))-212868*x^6+788130*x^5-26854524*x^4-77349420*x^3-67906368*x^2-19002960*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. \[ -\int \frac{{\left (5 \, x - 2\right )} x^{\frac{13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)*x^(13/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="maxima")

[Out]

-integrate((5*x - 2)*x^(13/2)/(3*x^2 + 5*x + 2)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (5 \, x^{7} - 2 \, x^{6}\right )} \sqrt{x}}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)*x^(13/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="fricas")

[Out]

integral(-(5*x^7 - 2*x^6)*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**(13/2)/(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. \[ \int -\frac{{\left (5 \, x - 2\right )} x^{\frac{13}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)*x^(13/2)/(3*x^2 + 5*x + 2)^(5/2),x, algorithm="giac")

[Out]

integrate(-(5*x - 2)*x^(13/2)/(3*x^2 + 5*x + 2)^(5/2), x)