3.1052 \(\int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{11/2}} \, dx\)

Optimal. Leaf size=210 \[ \frac{5438 \sqrt{3 x^2+5 x+2}}{315 \sqrt{x}}-\frac{5438 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}-\frac{899 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{5438 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}}-\frac{4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}+\frac{(4055 x+1446) \sqrt{3 x^2+5 x+2}}{315 x^{5/2}} \]

[Out]

(-5438*Sqrt[x]*(2 + 3*x))/(315*Sqrt[2 + 5*x + 3*x^2]) + (5438*Sqrt[2 + 5*x + 3*x
^2])/(315*Sqrt[x]) + ((1446 + 4055*x)*Sqrt[2 + 5*x + 3*x^2])/(315*x^(5/2)) - (4*
(7 - 15*x)*(2 + 5*x + 3*x^2)^(3/2))/(63*x^(9/2)) + (5438*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(315*Sqrt[2 + 5*x + 3*x^2]) -
(899*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(21*Sqrt
[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.338617, 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 \[ \frac{5438 \sqrt{3 x^2+5 x+2}}{315 \sqrt{x}}-\frac{5438 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}-\frac{899 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{5438 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}}-\frac{4 (7-15 x) \left (3 x^2+5 x+2\right )^{3/2}}{63 x^{9/2}}+\frac{(4055 x+1446) \sqrt{3 x^2+5 x+2}}{315 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(11/2),x]

[Out]

(-5438*Sqrt[x]*(2 + 3*x))/(315*Sqrt[2 + 5*x + 3*x^2]) + (5438*Sqrt[2 + 5*x + 3*x
^2])/(315*Sqrt[x]) + ((1446 + 4055*x)*Sqrt[2 + 5*x + 3*x^2])/(315*x^(5/2)) - (4*
(7 - 15*x)*(2 + 5*x + 3*x^2)^(3/2))/(63*x^(9/2)) + (5438*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(315*Sqrt[2 + 5*x + 3*x^2]) -
(899*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(21*Sqrt
[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 37.4154, size = 196, normalized size = 0.93 \[ - \frac{2719 \sqrt{x} \left (6 x + 4\right )}{315 \sqrt{3 x^{2} + 5 x + 2}} + \frac{2719 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{630 \sqrt{3 x^{2} + 5 x + 2}} - \frac{899 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{168 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5438 \sqrt{3 x^{2} + 5 x + 2}}{315 \sqrt{x}} + \frac{2 \left (\frac{4055 x}{2} + 723\right ) \sqrt{3 x^{2} + 5 x + 2}}{315 x^{\frac{5}{2}}} - \frac{2 \left (- 30 x + 14\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{63 x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-5*x)*(3*x**2+5*x+2)**(3/2)/x**(11/2),x)

[Out]

-2719*sqrt(x)*(6*x + 4)/(315*sqrt(3*x**2 + 5*x + 2)) + 2719*sqrt((6*x + 4)/(x +
1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(630*sqrt(3*x**2 + 5*x + 2)) - 899
*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(168*sqrt(3*x
**2 + 5*x + 2)) + 5438*sqrt(3*x**2 + 5*x + 2)/(315*sqrt(x)) + 2*(4055*x/2 + 723)
*sqrt(3*x**2 + 5*x + 2)/(315*x**(5/2)) - 2*(-30*x + 14)*(3*x**2 + 5*x + 2)**(3/2
)/(63*x**(9/2))

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Mathematica [C]  time = 0.234928, size = 160, normalized size = 0.76 \[ \frac{-2609 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{11/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-10876 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{11/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+29730 x^5+64706 x^4+44480 x^3+7424 x^2-3200 x-1120}{630 x^{9/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 - 5*x)*(2 + 5*x + 3*x^2)^(3/2))/x^(11/2),x]

[Out]

(-1120 - 3200*x + 7424*x^2 + 44480*x^3 + 64706*x^4 + 29730*x^5 - (10876*I)*Sqrt[
2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(11/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]
], 3/2] - (2609*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(11/2)*EllipticF[I*A
rcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(630*x^(9/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 140, normalized size = 0.7 \[{\frac{1}{1890} \left ( 2829\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{4}-5438\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{4}+97884\,{x}^{6}+252330\,{x}^{5}+259374\,{x}^{4}+133440\,{x}^{3}+22272\,{x}^{2}-9600\,x-3360 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}{x}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*(3*x^2+5*x+2)^(3/2)/x^(11/2),x)

[Out]

1/1890*(2829*(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^4-5438*(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^4+97884*x^6+252330*x^5+25937
4*x^4+133440*x^3+22272*x^2-9600*x-3360)/(3*x^2+5*x+2)^(1/2)/x^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(11/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{11}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(11/2),x, algorithm="fricas")

[Out]

integral(-(15*x^3 + 19*x^2 - 4)*sqrt(3*x^2 + 5*x + 2)/x^(11/2), 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)*(3*x**2+5*x+2)**(3/2)/x**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}{\left (5 \, x - 2\right )}}{x^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(11/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(3/2)*(5*x - 2)/x^(11/2), x)