3.2777 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=222 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{66055016 \sqrt{5 x+3} \sqrt{1-2 x}}{27783 \sqrt{3 x+2}}+\frac{950584 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 (3 x+2)^{3/2}}+\frac{20420 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}+\frac{512 \sqrt{5 x+3} \sqrt{1-2 x}}{81 (3 x+2)^{7/2}}-\frac{1986944 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783}-\frac{66055016 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783} \]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (512*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(81*(2 + 3*x)^(7/2)) + (20420*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(567*(2 +
3*x)^(5/2)) + (950584*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*(2 + 3*x)^(3/2)) + (66
055016*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*Sqrt[2 + 3*x]) - (66055016*Sqrt[11/3]
*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27783 - (1986944*Sqrt[11/3]*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27783

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Rubi [A]  time = 0.506174, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{14 \sqrt{5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac{66055016 \sqrt{5 x+3} \sqrt{1-2 x}}{27783 \sqrt{3 x+2}}+\frac{950584 \sqrt{5 x+3} \sqrt{1-2 x}}{3969 (3 x+2)^{3/2}}+\frac{20420 \sqrt{5 x+3} \sqrt{1-2 x}}{567 (3 x+2)^{5/2}}+\frac{512 \sqrt{5 x+3} \sqrt{1-2 x}}{81 (3 x+2)^{7/2}}-\frac{1986944 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783}-\frac{66055016 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{27783} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*Sqrt[3 + 5*x]),x]

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (512*Sqrt[1 - 2*x]*Sqr
t[3 + 5*x])/(81*(2 + 3*x)^(7/2)) + (20420*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(567*(2 +
3*x)^(5/2)) + (950584*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*(2 + 3*x)^(3/2)) + (66
055016*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*Sqrt[2 + 3*x]) - (66055016*Sqrt[11/3]
*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27783 - (1986944*Sqrt[11/3]*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27783

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Rubi in Sympy [A]  time = 48.5625, size = 201, normalized size = 0.91 \[ \frac{14 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{66055016 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{27783 \sqrt{3 x + 2}} + \frac{950584 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{3969 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{20420 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{567 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{512 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{81 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{66055016 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{83349} - \frac{1986944 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{83349} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(5/2)/(2+3*x)**(11/2)/(3+5*x)**(1/2),x)

[Out]

14*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(27*(3*x + 2)**(9/2)) + 66055016*sqrt(-2*x +
1)*sqrt(5*x + 3)/(27783*sqrt(3*x + 2)) + 950584*sqrt(-2*x + 1)*sqrt(5*x + 3)/(39
69*(3*x + 2)**(3/2)) + 20420*sqrt(-2*x + 1)*sqrt(5*x + 3)/(567*(3*x + 2)**(5/2))
+ 512*sqrt(-2*x + 1)*sqrt(5*x + 3)/(81*(3*x + 2)**(7/2)) - 66055016*sqrt(33)*el
liptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/83349 - 1986944*sqrt(33)*ellipt
ic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/83349

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Mathematica [A]  time = 0.384305, size = 111, normalized size = 0.5 \[ \frac{8 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2675228148 x^4+7223771916 x^3+7318104714 x^2+3296666850 x+557240459\right )}{4 (3 x+2)^{9/2}}+\sqrt{2} \left (8256877 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-4158805 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{83349} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*Sqrt[3 + 5*x]),x]

[Out]

(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(557240459 + 3296666850*x + 7318104714*x^2 +
7223771916*x^3 + 2675228148*x^4))/(4*(2 + 3*x)^(9/2)) + Sqrt[2]*(8256877*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 4158805*EllipticF[ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]], -33/2])))/83349

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Maple [C]  time = 0.033, size = 624, normalized size = 2.8 \[{\frac{2}{833490\,{x}^{2}+83349\,x-250047} \left ( 1347452820\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2675228148\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-7133941728\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-7133941728\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3170640768\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -528440128\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-39255728106\,{x}^{2}-27998280273\,x-5015164131 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(1/2),x)

[Out]

2/83349*(1347452820*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*
11^(1/2)*3^(1/2)*2^(1/2))*x^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-26752281
48*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*
2^(1/2))*x^4*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*Ellipt
icF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1-2
*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-7133941728*2^(1/2)*EllipticE(1/11*11^(1/2)
*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^3*(1-2*x)^(1/2)*(3+5*x)
^(1/2)*(2+3*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^
(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1
/2)-7133941728*2^(1/2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1
/2)*3^(1/2)*2^(1/2))*x^2*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^
(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/
2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-3170640768*2^(1/2)*EllipticE(1/1
1*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)
*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*x^6+266163520*2^(1/2)*(3+5*x)^(1/2)*(2+
3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^
(1/2)*3^(1/2)*2^(1/2))-528440128*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/
2)*EllipticE(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*11^(1/2)*3^(1/2)*2^(1/2))
[phone]*x^5+217137403836*x^4+55840372398*x^3-39255728106*x^2-27998280273*x
-5015164131)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(11/2)),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(11/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(11/2)),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(-2*x + 1)/((243*x^5 + 810*x^4 + 1080*x^3 + 720*x
^2 + 240*x + 32)*sqrt(5*x + 3)*sqrt(3*x + 2)), 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((1-2*x)**(5/2)/(2+3*x)**(11/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(11/2)),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(11/2)), x)