3.2589 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=175 \[ \frac{(337 x+408) \left (3 x^2+5 x+2\right )^{3/2}}{75 (2 x+3)^{5/2}}-\frac{(181 x+614) \sqrt{3 x^2+5 x+2}}{50 \sqrt{2 x+3}}-\frac{243 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{20 \sqrt{3 x^2+5 x+2}}+\frac{2779 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{100 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

-((614 + 181*x)*Sqrt[2 + 5*x + 3*x^2])/(50*Sqrt[3 + 2*x]) + ((408 + 337*x)*(2 +
5*x + 3*x^2)^(3/2))/(75*(3 + 2*x)^(5/2)) + (2779*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
E[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(100*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (243
*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
0*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.322431, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{(337 x+408) \left (3 x^2+5 x+2\right )^{3/2}}{75 (2 x+3)^{5/2}}-\frac{(181 x+614) \sqrt{3 x^2+5 x+2}}{50 \sqrt{2 x+3}}-\frac{243 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{20 \sqrt{3 x^2+5 x+2}}+\frac{2779 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{100 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-((614 + 181*x)*Sqrt[2 + 5*x + 3*x^2])/(50*Sqrt[3 + 2*x]) + ((408 + 337*x)*(2 +
5*x + 3*x^2)^(3/2))/(75*(3 + 2*x)^(5/2)) + (2779*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
E[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(100*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (243
*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
0*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 47.3154, size = 165, normalized size = 0.94 \[ \frac{2779 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{300 \sqrt{3 x^{2} + 5 x + 2}} - \frac{243 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{20 \sqrt{3 x^{2} + 5 x + 2}} - \frac{\left (543 x + 1842\right ) \sqrt{3 x^{2} + 5 x + 2}}{150 \sqrt{2 x + 3}} + \frac{\left (337 x + 408\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{75 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(7/2),x)

[Out]

2779*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3
00*sqrt(3*x**2 + 5*x + 2)) - 243*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2
)*sqrt(6*x + 6)/2), -2/3)/(20*sqrt(3*x**2 + 5*x + 2)) - (543*x + 1842)*sqrt(3*x*
*2 + 5*x + 2)/(150*sqrt(2*x + 3)) + (337*x + 408)*(3*x**2 + 5*x + 2)**(3/2)/(75*
(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.521767, size = 192, normalized size = 1.1 \[ \frac{-900 x^5+16800 x^4+100610 x^3+190440 x^2+147790 x-592 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{7/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+2779 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{7/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+40260}{300 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(40260 + 147790*x + 190440*x^2 + 100610*x^3 + 16800*x^4 - 900*x^5 + 2779*Sqrt[5]
*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[Arc
Sin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 592*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*
x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/
5])/(300*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.024, size = 319, normalized size = 1.8 \[ -{\frac{1}{3000} \left ( 3464\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+11116\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+10392\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+33348\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+7794\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +25011\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +9000\,{x}^{5}+498960\,{x}^{4}+2106380\,{x}^{3}+3375700\,{x}^{2}+2357120\,x+597840 \right ) \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3000*(3464*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^2*(-
30*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+11116*15^(1/2)*EllipticE(1/5*15^(1/2
)*(3+2*x)^(1/2),1/3*15^(1/2))*x^2*(-30*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+
10392*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x*(-2-2*x)^(1/
2)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)+33348*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^
(1/2),1/3*15^(1/2))*x*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)+7794*(3+2*x)
^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(
1/2),1/3*15^(1/2))+25011*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*
EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+9000*x^5+498960*x^4+2106380*x
^3+3375700*x^2+2357120*x+597840)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x^{3} - 10 \, x^{2} - 23 \, x - 10\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x^3 - 10*x^2 - 23*x - 10)*sqrt(3*x^2 + 5*x + 2)/((8*x^3 + 36*x^2 +
54*x + 27)*sqrt(2*x + 3)), 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((5-x)*(3*x**2+5*x+2)**(3/2)/(3+2*x)**(7/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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