3.2810 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{272 \sqrt{1-2 x} \sqrt{5 x+3}}{441 \sqrt{3 x+2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{63 (3 x+2)^{3/2}}-\frac{202}{441} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{272}{441} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(63*(2 + 3*x)^(3/2)) - (272*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(441*Sqrt[2 + 3*x]) + (272*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/441 - (202*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/441

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Rubi [A]  time = 0.262089, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{272 \sqrt{1-2 x} \sqrt{5 x+3}}{441 \sqrt{3 x+2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{63 (3 x+2)^{3/2}}-\frac{202}{441} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{272}{441} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(63*(2 + 3*x)^(3/2)) - (272*Sqrt[1 - 2*x]*Sqrt[3
+ 5*x])/(441*Sqrt[2 + 3*x]) + (272*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1
- 2*x]], 35/33])/441 - (202*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/441

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Rubi in Sympy [A]  time = 25.4723, size = 114, normalized size = 0.88 \[ - \frac{272 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{441 \sqrt{3 x + 2}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{63 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{272 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1323} - \frac{202 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1323} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-272*sqrt(-2*x + 1)*sqrt(5*x + 3)/(441*sqrt(3*x + 2)) + 2*sqrt(-2*x + 1)*sqrt(5*
x + 3)/(63*(3*x + 2)**(3/2)) + 272*sqrt(33)*elliptic_e(asin(sqrt(21)*sqrt(-2*x +
1)/7), 35/33)/1323 - 202*sqrt(33)*elliptic_f(asin(sqrt(21)*sqrt(-2*x + 1)/7), 3
5/33)/1323

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Mathematica [A]  time = 0.265843, size = 97, normalized size = 0.75 \[ \frac{-\frac{6 \sqrt{1-2 x} \sqrt{5 x+3} (408 x+265)}{(3 x+2)^{3/2}}+3605 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-272 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1323} \]

Antiderivative was successfully verified.

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

[Out]

((-6*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(265 + 408*x))/(2 + 3*x)^(3/2) - 272*Sqrt[2]*El
lipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 3605*Sqrt[2]*EllipticF[ArcSin
[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/1323

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Maple [C]  time = 0.029, size = 267, normalized size = 2.1 \[ -{\frac{1}{13230\,{x}^{2}+1323\,x-3969} \left ( 10815\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-816\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+7210\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -544\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +24480\,{x}^{3}+18348\,{x}^{2}-5754\,x-4770 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1323*(10815*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)-816*2^(1/2)*Ell
ipticE(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)+7210*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))-544*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))+24480*x^3+18348*
x^2-5754*x-4770)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x + 3)^(3/2)/((9*x^2 + 12*x + 4)*sqrt(3*x + 2)*sqrt(-2*x + 1)), 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((3+5*x)**(3/2)/(2+3*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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