3.2689 \(\int \frac{(1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{21 (3 x+2)^{7/2}}+\frac{595324 \sqrt{5 x+3} \sqrt{1-2 x}}{46305 \sqrt{3 x+2}}+\frac{8516 \sqrt{5 x+3} \sqrt{1-2 x}}{6615 (3 x+2)^{3/2}}+\frac{82 \sqrt{5 x+3} \sqrt{1-2 x}}{315 (3 x+2)^{5/2}}-\frac{18016 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}-\frac{595324 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (82*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(315*(2 + 3*x)^(5/2)) + (8516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6615*(2 +
3*x)^(3/2)) + (595324*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(46305*Sqrt[2 + 3*x]) - (595
324*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/46305 - (18016
*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/46305

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Rubi [A]  time = 0.422274, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{3/2}}{21 (3 x+2)^{7/2}}+\frac{595324 \sqrt{5 x+3} \sqrt{1-2 x}}{46305 \sqrt{3 x+2}}+\frac{8516 \sqrt{5 x+3} \sqrt{1-2 x}}{6615 (3 x+2)^{3/2}}+\frac{82 \sqrt{5 x+3} \sqrt{1-2 x}}{315 (3 x+2)^{5/2}}-\frac{18016 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305}-\frac{595324 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{46305} \]

Antiderivative was successfully verified.

[In]  Int[((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(9/2),x]

[Out]

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (82*Sqrt[1 - 2*x]*Sqrt
[3 + 5*x])/(315*(2 + 3*x)^(5/2)) + (8516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(6615*(2 +
3*x)^(3/2)) + (595324*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(46305*Sqrt[2 + 3*x]) - (595
324*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/46305 - (18016
*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/46305

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Rubi in Sympy [A]  time = 38.238, size = 172, normalized size = 0.9 \[ - \frac{2 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{21 \left (3 x + 2\right )^{\frac{7}{2}}} + \frac{595324 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{46305 \sqrt{3 x + 2}} + \frac{8516 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{6615 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{82 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{315 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{595324 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{138915} - \frac{18016 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{138915} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1-2*x)**(3/2)*(3+5*x)**(1/2)/(2+3*x)**(9/2),x)

[Out]

-2*(-2*x + 1)**(3/2)*sqrt(5*x + 3)/(21*(3*x + 2)**(7/2)) + 595324*sqrt(-2*x + 1)
*sqrt(5*x + 3)/(46305*sqrt(3*x + 2)) + 8516*sqrt(-2*x + 1)*sqrt(5*x + 3)/(6615*(
3*x + 2)**(3/2)) + 82*sqrt(-2*x + 1)*sqrt(5*x + 3)/(315*(3*x + 2)**(5/2)) - 5953
24*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/138915 - 18016*sq
rt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 35/33)/138915

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Mathematica [A]  time = 0.329159, size = 106, normalized size = 0.55 \[ \frac{4 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (8036874 x^3+16342002 x^2+11095995 x+2510369\right )}{2 (3 x+2)^{7/2}}+\sqrt{2} \left (148831 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-74515 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{138915} \]

Antiderivative was successfully verified.

[In]  Integrate[((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(9/2),x]

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2510369 + 11095995*x + 16342002*x^2 + 803687
4*x^3))/(2*(2 + 3*x)^(7/2)) + Sqrt[2]*(148831*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]], -33/2] - 74515*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/1
38915

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Maple [C]  time = 0.029, size = 505, normalized size = 2.6 \[{\frac{2}{1389150\,{x}^{2}+138915\,x-416745} \left ( 4023810\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-8036874\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+8047620\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-16073748\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+5365080\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-10715832\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1192240\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -2381296\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}-38478963\,{x}^{2}-92332848\,x-22593321 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x)

[Out]

2/138915*(4023810*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*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)-8036874*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)+8047620*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)-16073748*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)+5365080*2^(1/2)*EllipticF(1/11*11^(1/2)*2^(1/2)*(3+5*x)^(1/2),1/2*I*1
1^(1/2)*3^(1/2)*2^(1/2))*x*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-10715832*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+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+1192240*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))-2381296*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))+
241106220*x^5+514370682*x^4+309573990*x^3-38478963*x^2-92332848*x-22593321)*(3+5
*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2),x, algorithm="fricas")

[Out]

integral(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/((81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16
)*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)**(3/2)*(3+5*x)**(1/2)/(2+3*x)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(-2*x + 1)^(3/2)/(3*x + 2)^(9/2), x)