3.1051 \(\int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{633 \sqrt{x} (3 x+2)}{7 \sqrt{3 x^2+5 x+2}}-\frac{783 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}}+\frac{633 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}}-\frac{4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}+\frac{3 (133 x+22) \sqrt{3 x^2+5 x+2}}{7 x^{3/2}} \]

[Out]

(-633*Sqrt[x]*(2 + 3*x))/(7*Sqrt[2 + 5*x + 3*x^2]) + (3*(22 + 133*x)*Sqrt[2 + 5*
x + 3*x^2])/(7*x^(3/2)) - (4*(1 - 2*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*x^(7/2)) + (6
33*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(7*
Sqrt[2 + 5*x + 3*x^2]) - (783*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[
ArcTan[Sqrt[x]], -1/2])/(7*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.29809, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{633 \sqrt{x} (3 x+2)}{7 \sqrt{3 x^2+5 x+2}}-\frac{783 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}}+\frac{633 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{3 x^2+5 x+2}}-\frac{4 (1-2 x) \left (3 x^2+5 x+2\right )^{3/2}}{7 x^{7/2}}+\frac{3 (133 x+22) \sqrt{3 x^2+5 x+2}}{7 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-633*Sqrt[x]*(2 + 3*x))/(7*Sqrt[2 + 5*x + 3*x^2]) + (3*(22 + 133*x)*Sqrt[2 + 5*
x + 3*x^2])/(7*x^(3/2)) - (4*(1 - 2*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*x^(7/2)) + (6
33*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(7*
Sqrt[2 + 5*x + 3*x^2]) - (783*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[
ArcTan[Sqrt[x]], -1/2])/(7*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 32.3865, size = 173, normalized size = 0.93 \[ - \frac{633 \sqrt{x} \left (6 x + 4\right )}{14 \sqrt{3 x^{2} + 5 x + 2}} + \frac{633 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{28 \sqrt{3 x^{2} + 5 x + 2}} - \frac{783 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{28 \sqrt{3 x^{2} + 5 x + 2}} + \frac{2 \left (\frac{1995 x}{2} + 165\right ) \sqrt{3 x^{2} + 5 x + 2}}{35 x^{\frac{3}{2}}} - \frac{2 \left (- 20 x + 10\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{35 x^{\frac{7}{2}}} \]

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**(9/2),x)

[Out]

-633*sqrt(x)*(6*x + 4)/(14*sqrt(3*x**2 + 5*x + 2)) + 633*sqrt((6*x + 4)/(x + 1))
*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(28*sqrt(3*x**2 + 5*x + 2)) - 783*sqr
t((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(28*sqrt(3*x**2 +
5*x + 2)) + 2*(1995*x/2 + 165)*sqrt(3*x**2 + 5*x + 2)/(35*x**(3/2)) - 2*(-20*x
+ 10)*(3*x**2 + 5*x + 2)**(3/2)/(35*x**(7/2))

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Mathematica [C]  time = 0.248089, size = 163, normalized size = 0.87 \[ \frac{-150 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-633 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (315 x^5+384 x^4-19 x^3-72 x^2+24 x+8\right )}{7 x^{7/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(8 + 24*x - 72*x^2 - 19*x^3 + 384*x^4 + 315*x^5) - (633*I)*Sqrt[2]*Sqrt[1 +
x^(-1)]*Sqrt[3 + 2/x]*x^(9/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (15
0*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[2/3
]/Sqrt[x]], 3/2])/(7*x^(7/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 135, normalized size = 0.7 \[{\frac{1}{14} \left ( 111\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}-211\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}+2538\,{x}^{5}+4794\,{x}^{4}+2608\,{x}^{3}+288\,{x}^{2}-96\,x-32 \right ){x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*(111*(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^3-211*(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^3+2538*x^5+4794*x^4+2608*x^3+288
*x^2-96*x-32)/(3*x^2+5*x+2)^(1/2)/x^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{9}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(15*x^3 + 19*x^2 - 4)*sqrt(3*x^2 + 5*x + 2)/x^(9/2), 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**(9/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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