3.1046 \(\int (2-5 x) \sqrt{x} \left (2+5 x+3 x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=210 \[ -\frac{10}{33} \sqrt{x} \left (3 x^2+5 x+2\right )^{5/2}+\frac{4}{231} \sqrt{x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac{4}{385} \sqrt{x} (39 x+55) \sqrt{3 x^2+5 x+2}-\frac{424 \sqrt{x} (3 x+2)}{1155 \sqrt{3 x^2+5 x+2}}-\frac{36 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{77 \sqrt{3 x^2+5 x+2}}+\frac{424 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{1155 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-424*Sqrt[x]*(2 + 3*x))/(1155*Sqrt[2 + 5*x + 3*x^2]) - (4*Sqrt[x]*(55 + 39*x)*S
qrt[2 + 5*x + 3*x^2])/385 + (4*Sqrt[x]*(65 + 84*x)*(2 + 5*x + 3*x^2)^(3/2))/231
- (10*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/33 + (424*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/
(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(1155*Sqrt[2 + 5*x + 3*x^2]) - (36*Sq
rt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(77*Sqrt
[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.350054, 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 \[ -\frac{10}{33} \sqrt{x} \left (3 x^2+5 x+2\right )^{5/2}+\frac{4}{231} \sqrt{x} (84 x+65) \left (3 x^2+5 x+2\right )^{3/2}-\frac{4}{385} \sqrt{x} (39 x+55) \sqrt{3 x^2+5 x+2}-\frac{424 \sqrt{x} (3 x+2)}{1155 \sqrt{3 x^2+5 x+2}}-\frac{36 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{77 \sqrt{3 x^2+5 x+2}}+\frac{424 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{1155 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-424*Sqrt[x]*(2 + 3*x))/(1155*Sqrt[2 + 5*x + 3*x^2]) - (4*Sqrt[x]*(55 + 39*x)*S
qrt[2 + 5*x + 3*x^2])/385 + (4*Sqrt[x]*(65 + 84*x)*(2 + 5*x + 3*x^2)^(3/2))/231
- (10*Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2))/33 + (424*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/
(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(1155*Sqrt[2 + 5*x + 3*x^2]) - (36*Sq
rt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(77*Sqrt
[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 37.9036, size = 199, normalized size = 0.95 \[ - \frac{212 \sqrt{x} \left (6 x + 4\right )}{1155 \sqrt{3 x^{2} + 5 x + 2}} + \frac{8 \sqrt{x} \left (1134 x + \frac{1755}{2}\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{6237} - \frac{16 \sqrt{x} \left (\frac{9477 x}{4} + \frac{13365}{4}\right ) \sqrt{3 x^{2} + 5 x + 2}}{93555} - \frac{10 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{33} + \frac{106 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{1155 \sqrt{3 x^{2} + 5 x + 2}} - \frac{9 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{77 \sqrt{3 x^{2} + 5 x + 2}} \]

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**(1/2),x)

[Out]

-212*sqrt(x)*(6*x + 4)/(1155*sqrt(3*x**2 + 5*x + 2)) + 8*sqrt(x)*(1134*x + 1755/
2)*(3*x**2 + 5*x + 2)**(3/2)/6237 - 16*sqrt(x)*(9477*x/4 + 13365/4)*sqrt(3*x**2
+ 5*x + 2)/93555 - 10*sqrt(x)*(3*x**2 + 5*x + 2)**(5/2)/33 + 106*sqrt((6*x + 4)/
(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(1155*sqrt(3*x**2 + 5*x + 2))
- 9*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(77*sqrt(
3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.274363, size = 173, normalized size = 0.82 \[ \frac{-424 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (58 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+4725 x^7+16065 x^6+17775 x^5+3497 x^4-6140 x^3-3106 x^2+520 x+424\right )}{1155 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((-424*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqr
t[2/3]/Sqrt[x]], 3/2] - 2*(424 + 520*x - 3106*x^2 - 6140*x^3 + 3497*x^4 + 17775*
x^5 + 16065*x^6 + 4725*x^7 + (58*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/
2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2]))/(1155*Sqrt[x]*Sqrt[2 + 5*x + 3
*x^2])

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Maple [A]  time = 0.016, size = 138, normalized size = 0.7 \[{\frac{2}{3465} \left ( -14175\,{x}^{7}-48195\,{x}^{6}-53325\,{x}^{5}+48\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -106\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -10491\,{x}^{4}+18420\,{x}^{3}+11226\,{x}^{2}+1620\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465/(3*x^2+5*x+2)^(1/2)/x^(1/2)*(-14175*x^7-48195*x^6-53325*x^5+48*(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))-106*(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))-10491*x^4+18420*x^3+11226*x^2+1620*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- 4 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}\right )\, dx - \int 19 x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac{7}{2}} \sqrt{3 x^{2} + 5 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-4*sqrt(x)*sqrt(3*x**2 + 5*x + 2), x) - Integral(19*x**(5/2)*sqrt(3*x*
*2 + 5*x + 2), x) - Integral(15*x**(7/2)*sqrt(3*x**2 + 5*x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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