3.2600 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{9/2}} \, dx\)

Optimal. Leaf size=207 \[ -\frac{(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}-\frac{(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{5/2}}+\frac{(1823 x+6292) \sqrt{3 x^2+5 x+2}}{140 \sqrt{2 x+3}}+\frac{2505 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{56 \sqrt{3 x^2+5 x+2}}-\frac{4091 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{40 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((6292 + 1823*x)*Sqrt[2 + 5*x + 3*x^2])/(140*Sqrt[3 + 2*x]) - ((3354 + 2531*x)*(
2 + 5*x + 3*x^2)^(3/2))/(210*(3 + 2*x)^(5/2)) - ((43 + 7*x)*(2 + 5*x + 3*x^2)^(5
/2))/(35*(3 + 2*x)^(7/2)) - (4091*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3
]*Sqrt[1 + x]], -2/3])/(40*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (2505*Sqrt[3]*Sqrt[-
2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(56*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi [A]  time = 0.413291, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{(7 x+43) \left (3 x^2+5 x+2\right )^{5/2}}{35 (2 x+3)^{7/2}}-\frac{(2531 x+3354) \left (3 x^2+5 x+2\right )^{3/2}}{210 (2 x+3)^{5/2}}+\frac{(1823 x+6292) \sqrt{3 x^2+5 x+2}}{140 \sqrt{2 x+3}}+\frac{2505 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{56 \sqrt{3 x^2+5 x+2}}-\frac{4091 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{40 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((6292 + 1823*x)*Sqrt[2 + 5*x + 3*x^2])/(140*Sqrt[3 + 2*x]) - ((3354 + 2531*x)*(
2 + 5*x + 3*x^2)^(3/2))/(210*(3 + 2*x)^(5/2)) - ((43 + 7*x)*(2 + 5*x + 3*x^2)^(5
/2))/(35*(3 + 2*x)^(7/2)) - (4091*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3
]*Sqrt[1 + x]], -2/3])/(40*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (2505*Sqrt[3]*Sqrt[-
2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(56*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi in Sympy [A]  time = 57.5934, size = 194, normalized size = 0.94 \[ - \frac{4091 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{120 \sqrt{3 x^{2} + 5 x + 2}} + \frac{2505 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{56 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\left (27345 x + 94380\right ) \sqrt{3 x^{2} + 5 x + 2}}{2100 \sqrt{2 x + 3}} - \frac{\left (12655 x + 16770\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{1050 \left (2 x + 3\right )^{\frac{5}{2}}} - \frac{\left (7 x + 43\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{35 \left (2 x + 3\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4091*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(
120*sqrt(3*x**2 + 5*x + 2)) + 2505*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt
(2)*sqrt(6*x + 6)/2), -2/3)/(56*sqrt(3*x**2 + 5*x + 2)) + (27345*x + 94380)*sqrt
(3*x**2 + 5*x + 2)/(2100*sqrt(2*x + 3)) - (12655*x + 16770)*(3*x**2 + 5*x + 2)**
(3/2)/(1050*(2*x + 3)**(5/2)) - (7*x + 43)*(3*x**2 + 5*x + 2)**(5/2)/(35*(2*x +
3)**(7/2))

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Mathematica [A]  time = 0.561081, size = 202, normalized size = 0.98 \[ -\frac{4536 x^7-29736 x^6+158172 x^5+2140148 x^4+6437058 x^3+8516152 x^2+5250234 x-6092 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+28637 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{9/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+1223436}{840 (2 x+3)^{7/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1223436 + 5250234*x + 8516152*x^2 + 6437058*x^3 + 2140148*x^4 + 158172*x^5 - 2
9736*x^6 + 4536*x^7 + 28637*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt
[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 6092*Sqr
t[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(9/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF
[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(840*(3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3*x^
2])

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Maple [B]  time = 0.027, size = 423, normalized size = 2. \[{\frac{1}{8400} \left ( 71504\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+229096\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+321768\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+1030932\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}-45360\,{x}^{7}+482652\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+1546398\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+297360\,{x}^{6}+241326\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +773199\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+18693600 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8400*(71504*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^3*(3
+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+229096*15^(1/2)*EllipticE(1/5*15^(1/
2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^3*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)
+321768*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^2*(-30*x-2
0)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+1030932*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)-4536
0*x^7+482652*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)+1546398*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)+297
360*x^6+241326*(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))+773199*(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))+121640
40*x^5+63364040*x^4+140670340*x^3+157107500*x^2+86673480*x+18693600)/(3*x^2+5*x+
2)^(1/2)/(3+2*x)^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^5 - 15*x^4 - 113*x^3 - 165*x^2 - 96*x - 20)*sqrt(3*x^2 + 5*x + 2)
/((16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*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)**(5/2)/(3+2*x)**(9/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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