3.1034 \(\int (2-5 x) x^{7/2} \sqrt{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=251 \[ \frac{157160 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{243243}-\frac{8 (502911 x+397265) \sqrt{3 x^2+5 x+2} \sqrt{x}}{2189187}+\frac{1543648 (3 x+2) \sqrt{x}}{6567561 \sqrt{3 x^2+5 x+2}}+\frac{349240 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2189187 \sqrt{3 x^2+5 x+2}}-\frac{1543648 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{6567561 \sqrt{3 x^2+5 x+2}}-\frac{10}{39} \left (3 x^2+5 x+2\right )^{3/2} x^{7/2}+\frac{656 \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}}{1287}-\frac{21620 \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}}{34749} \]

[Out]

(1543648*Sqrt[x]*(2 + 3*x))/(6567561*Sqrt[2 + 5*x + 3*x^2]) - (8*Sqrt[x]*(397265
+ 502911*x)*Sqrt[2 + 5*x + 3*x^2])/2189187 + (157160*Sqrt[x]*(2 + 5*x + 3*x^2)^
(3/2))/243243 - (21620*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/34749 + (656*x^(5/2)*(2
+ 5*x + 3*x^2)^(3/2))/1287 - (10*x^(7/2)*(2 + 5*x + 3*x^2)^(3/2))/39 - (1543648*
Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(65675
61*Sqrt[2 + 5*x + 3*x^2]) + (349240*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elli
pticF[ArcTan[Sqrt[x]], -1/2])/(2189187*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.480252, antiderivative size = 251, 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 \[ \frac{157160 \left (3 x^2+5 x+2\right )^{3/2} \sqrt{x}}{243243}-\frac{8 (502911 x+397265) \sqrt{3 x^2+5 x+2} \sqrt{x}}{2189187}+\frac{1543648 (3 x+2) \sqrt{x}}{6567561 \sqrt{3 x^2+5 x+2}}+\frac{349240 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{2189187 \sqrt{3 x^2+5 x+2}}-\frac{1543648 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{6567561 \sqrt{3 x^2+5 x+2}}-\frac{10}{39} \left (3 x^2+5 x+2\right )^{3/2} x^{7/2}+\frac{656 \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}}{1287}-\frac{21620 \left (3 x^2+5 x+2\right )^{3/2} x^{3/2}}{34749} \]

Antiderivative was successfully verified.

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

[Out]

(1543648*Sqrt[x]*(2 + 3*x))/(6567561*Sqrt[2 + 5*x + 3*x^2]) - (8*Sqrt[x]*(397265
+ 502911*x)*Sqrt[2 + 5*x + 3*x^2])/2189187 + (157160*Sqrt[x]*(2 + 5*x + 3*x^2)^
(3/2))/243243 - (21620*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/34749 + (656*x^(5/2)*(2
+ 5*x + 3*x^2)^(3/2))/1287 - (10*x^(7/2)*(2 + 5*x + 3*x^2)^(3/2))/39 - (1543648*
Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(65675
61*Sqrt[2 + 5*x + 3*x^2]) + (349240*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elli
pticF[ArcTan[Sqrt[x]], -1/2])/(2189187*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 49.3499, size = 236, normalized size = 0.94 \[ - \frac{10 x^{\frac{7}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{39} + \frac{656 x^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1287} - \frac{21620 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{34749} + \frac{771824 \sqrt{x} \left (6 x + 4\right )}{6567561 \sqrt{3 x^{2} + 5 x + 2}} - \frac{64 \sqrt{x} \left (\frac{7543665 x}{8} + \frac{5958975}{8}\right ) \sqrt{3 x^{2} + 5 x + 2}}{32837805} + \frac{157160 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{243243} - \frac{385912 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{6567561 \sqrt{3 x^{2} + 5 x + 2}} + \frac{87310 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2189187 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(1/2),x)

[Out]

-10*x**(7/2)*(3*x**2 + 5*x + 2)**(3/2)/39 + 656*x**(5/2)*(3*x**2 + 5*x + 2)**(3/
2)/1287 - 21620*x**(3/2)*(3*x**2 + 5*x + 2)**(3/2)/34749 + 771824*sqrt(x)*(6*x +
4)/(6567561*sqrt(3*x**2 + 5*x + 2)) - 64*sqrt(x)*(7543665*x/8 + 5958975/8)*sqrt
(3*x**2 + 5*x + 2)/32837805 + 157160*sqrt(x)*(3*x**2 + 5*x + 2)**(3/2)/243243 -
385912*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(656756
1*sqrt(3*x**2 + 5*x + 2)) + 87310*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(a
tan(sqrt(x)), -1/2)/(2189187*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.295191, size = 178, normalized size = 0.71 \[ \frac{-495928 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+1543648 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+2 \left (-7577955 x^8-10195794 x^7+671895 x^6+2892348 x^5+58374 x^4-141444 x^3+670548 x^2+2811400 x+1543648\right )}{6567561 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(1543648 + 2811400*x + 670548*x^2 - 141444*x^3 + 58374*x^4 + 2892348*x^5 + 67
1895*x^6 - 10195794*x^7 - 7577955*x^8) + (1543648*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sq
rt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (495928*I)*Sq
rt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[
x]], 3/2])/(6567561*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.139, size = 143, normalized size = 0.6 \[ -{\frac{2}{19702683} \left ( 22733865\,{x}^{8}[phone]\,{x}^{7}-2015685\,{x}^{6}-8677044\,{x}^{5}+633876\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -385912\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -175122\,{x}^{4}+424332\,{x}^{3}+4934772\,{x}^{2}+3143160\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*x^(7/2)*(3*x^2+5*x+2)^(1/2),x)

[Out]

-2/19702683/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(22733865*x^8+30587382*x^7-2015685*x^6-8
677044*x^5+633876*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Ellipti
cF(1/2*(6*x+4)^(1/2),I*2^(1/2))-385912*(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))-175122*x^4+424332*x^3+49347
72*x^2+3143160*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x^4 - 2*x^3)*sqrt(3*x^2 + 5*x + 2)*sqrt(x), 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)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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