3.2581 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{7/2}} \, dx\)

Optimal. Leaf size=170 \[ \frac{\sqrt{3 x^2+5 x+2} (43 x+32)}{25 (2 x+3)^{5/2}}+\frac{49 \sqrt{3 x^2+5 x+2}}{125 \sqrt{2 x+3}}+\frac{9 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{50 \sqrt{3 x^2+5 x+2}}-\frac{49 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{250 \sqrt{3 x^2+5 x+2}} \]

[Out]

(49*Sqrt[2 + 5*x + 3*x^2])/(125*Sqrt[3 + 2*x]) + ((32 + 43*x)*Sqrt[2 + 5*x + 3*x
^2])/(25*(3 + 2*x)^(5/2)) - (49*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[
Sqrt[3]*Sqrt[1 + x]], -2/3])/(250*Sqrt[2 + 5*x + 3*x^2]) + (9*Sqrt[3]*Sqrt[-2 -
5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(50*Sqrt[2 + 5*x + 3*
x^2])

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Rubi [A]  time = 0.336553, antiderivative size = 170, 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 \[ \frac{\sqrt{3 x^2+5 x+2} (43 x+32)}{25 (2 x+3)^{5/2}}+\frac{49 \sqrt{3 x^2+5 x+2}}{125 \sqrt{2 x+3}}+\frac{9 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{50 \sqrt{3 x^2+5 x+2}}-\frac{49 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{250 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(49*Sqrt[2 + 5*x + 3*x^2])/(125*Sqrt[3 + 2*x]) + ((32 + 43*x)*Sqrt[2 + 5*x + 3*x
^2])/(25*(3 + 2*x)^(5/2)) - (49*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[
Sqrt[3]*Sqrt[1 + x]], -2/3])/(250*Sqrt[2 + 5*x + 3*x^2]) + (9*Sqrt[3]*Sqrt[-2 -
5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(50*Sqrt[2 + 5*x + 3*
x^2])

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Rubi in Sympy [A]  time = 47.2244, size = 162, normalized size = 0.95 \[ - \frac{49 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{250 \sqrt{3 x^{2} + 5 x + 2}} + \frac{9 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{50 \sqrt{3 x^{2} + 5 x + 2}} + \frac{49 \sqrt{3 x^{2} + 5 x + 2}}{125 \sqrt{2 x + 3}} + \frac{\left (129 x + 96\right ) \sqrt{3 x^{2} + 5 x + 2}}{75 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-49*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(25
0*sqrt(3*x**2 + 5*x + 2)) + 9*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*s
qrt(6*x + 6)/2), -2/3)/(50*sqrt(3*x**2 + 5*x + 2)) + 49*sqrt(3*x**2 + 5*x + 2)/(
125*sqrt(2*x + 3)) + (129*x + 96)*sqrt(3*x**2 + 5*x + 2)/(75*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.44537, size = 182, normalized size = 1.07 \[ \frac{1290 x^3+3110 x^2+2460 x+22 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-49 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{7/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+640}{250 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(640 + 2460*x + 3110*x^2 + 1290*x^3 - 49*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*
x)^(7/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/
5] + 22*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])/(250*(3 + 2*x)^(5/2)*Sqrt[2 +
5*x + 3*x^2])

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Maple [B]  time = 0.046, size = 314, normalized size = 1.9 \[ -{\frac{1}{2500} \left ( 16\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}-196\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+48\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}-588\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+36\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -441\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -11760\,{x}^{4}-67780\,{x}^{3}-124200\,{x}^{2}-92220\,x-24040 \right ) \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2500*(16*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)-196*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)+48*1
5^(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)-588*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)+36*(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))-441*(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))-11760*x^4-67780*x^3-124200*x^2-92220*x-240
40)/(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. \[ -\int \frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{8 x^{3} \sqrt{2 x + 3} + 36 x^{2} \sqrt{2 x + 3} + 54 x \sqrt{2 x + 3} + 27 \sqrt{2 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3
) + 54*x*sqrt(2*x + 3) + 27*sqrt(2*x + 3)), x) - Integral(x*sqrt(3*x**2 + 5*x +
2)/(8*x**3*sqrt(2*x + 3) + 36*x**2*sqrt(2*x + 3) + 54*x*sqrt(2*x + 3) + 27*sqrt(
2*x + 3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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